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The Samuel realcompactification
Accepted Manuscript The Samuel realcompactification
M. Isabel Garrido, Ana S. Meroño
PII: DOI: Reference:
S0166-8641(18)30212-8 https://doi.org/10.1016/j.topol.2018.03.033 TOPOL 6452
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Topology and its Applications
Received date: Revised date: Accepted date:
31 December 2016 4 December 2017 5 February 2018
Please cite this article in press as: M. Isabel Garrido, A.S. Meroño, The Samuel realcompactification, Topol. Appl. (2018), https://doi.org/10.1016/j.topol.2018.03.033
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THE SAMUEL REALCOMPACTIFICATION ˜ M. ISABEL GARRIDO AND ANA S. MERONO* Abstract. For a uniform space (X, μ), we introduce a realcompactification of X by means of the family Uμ (X) of all the real-valued uniformly continuous functions on X, in the same way that the known Samuel compactification of the space is given by Uμ∗ (X) the set of all the bounded functions in Uμ (X). We will call it “the Samuel realcompactification” by several resemblances to the Samuel compactification. In this paper, we present different ways to construct such realcompactification as well as we study the corresponding problem of knowing when a uniform space is Samuel realcompact, that is, when it (topologically) coincides with its Samuel realcompactification. At this respect, we obtain as main result a theorem of Katˇ etov-Shirota type, given in terms of a property of completeness, recently introduced by the authors, called Bourbaki-completeness.
1. Introduction A realcompactification of a Tychonoff space X is a realcompact space Y in which X is densely embedded. For instance, the well-known Hewitt-Nachbin realcompactification υX. Recall that υX is characterized as the smallest realcompactification of X (in the usual order on the family of all realcompactifications) such that every real-valued continuous function f ∈ C(X) can be continuously extended to it [9]. In the frame of uniform spaces, since we can also consider Uμ (X), the set of all the real-valued uniformly continuous functions on the uniform space (X, μ), it is natural to ask what is the smallest realcompactification of X such that every function f ∈ Uμ (X) can be continuously extended to it. Here, following the ideas of [3], we will introduce this realcompactification that we denote by H(Uμ (X)) because in fact it can be represented as the set of all the real homomorphisms on the unital vector lattice Uμ (X). Moreover, we will see that it also coincides, as a topological space, with the completion of the uniform space (X, wUμ (X) ), where wUμ (X) denotes the weak uniformity on X generated by Uμ (X) (Theorem 1). We will call H(Uμ (X)) the Samuel realcompactification of (X, μ) in likeness to the Samuel compactification sμ X. Recall that the Samuel compactification of a uniform space (X, μ), also known as the Smirnov compactification, is the smallest (real)compactification of X such that every bounded real-valued uniformly continuous function f ∈ Uμ∗ (X) can be continuously extended to it ([20]). The resemblance between the Samuel realcompactification and the Samuel compactification is not only due to their characterization as the smallest realcompactification, respectively compactification, such that every real-valued, respectively bounded, uniformly continuous function can be continuously extended to it. In fact, as well as every compactification of a Tychonoff space X can be considered as a Samuel compactification for some totally bounded (precompact) uniformity on X [2] (see also [10]), every realcompactification of X can be considered as a Samuel realcompactification for some uniformity on it, as we will explain below (Theorem 2). A Tychonoff space X is realcompact whenever X = υX, and we can similarly define, for a uniform space (X, μ), that (X, μ) is Samuel realcompact if X = H(Uμ (X)). In this paper, we give a uniform analogue to the well-known Katˇetov-Shirota Theorem ([21], see also [9]) which deals with realcompactness of Tychonoff spaces. This classical theorem states that a Tychonoff space X is realcompact if and only if X is completely uniformizable and every closed discrete subspace has non-measurable cardinal. In this line, we will obtain an analogous result that asserts that a uniform space is Samuel realcompact if and only if it is Bourbaki-complete and every (closed) uniformly discrete subspace has non-measurable cardinal (Theorem 12). The property of Bourbaki-completeness was introduced and studied by the authors in [5] (see also [6] and [12]), and it is a uniform property stronger than usual completeness. We say that a uniform space is Bourbaki-complete if every Bourbaki-Cauchy filter, defined later, clusters. 2010 Mathematics Subject Classification. Primary 54D60, 54E15; Secondary 54D35, 54A20, 54A25. Key words and phrases. Uniform space, realcompactification, real-valued uniformly continuous function, Samuel realcompactification, Cauchy filter, Bourbaki-Cauchy filter, Bourbaki-completeness. *Corresponding author. Partially supported by MINECO Project-MTM2015-65825-P (Spain). 1
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
It turns out that Bourbaki-Cauchy filters are a useful tool to study Samuel realcompactness. In fact, the Samuel realcompactification, considered as the completion of (X, wUμ (X) ), is always a Bourbaki-complete uniform space (Theorem 7). Furthermore, assuming that the uniform partitions of (X, μ) have non-measurable cardinality, the Cauchy filters of (X, wUμ (X) ) are precisely the Bourbaki-Cauchy filters of (X, μ) (Proposition 10), and then it follows that (X, wUμ (X) ) is complete if and only of (X, μ) is Bourbaki-complete (Theorem 11). This will be the key not only for proving our main result but also for finding another description of the Samuel realcompactification H(Uμ (X)). Namely, H(Uμ (X)) is the subset of sμ X formed by the cluster points of the Bourbaki-Cauchy filters in (X, μ) (Theorem 18). Very recently, in [7], we proved the same result for Samuel realcompactness but in the frame of metric spaces. The techniques used in [7] are not the same as those used here, since we worked in [7] with a family of metrics defined on the metric space (X, d) all of them uniformly equivalent to the initial metric d. We must notice that in [19], Rice and Reynolds obtained in some sense a similar Katˇetov-Shirota type theorem. More precisely, these authors characterized the completeness of (X, wUμ (X) ) by means of the uniformity generated by all the star-finite uniform covers of (X, μ). We are going to explain in Section 3 the similarities between both results. Moreover, and also in this line, Huˇsek and Pulgar´ın gave in [13] a kind of a uniform Katetov-Shirota result for the so-called uniformly 0-dimensional uniform spaces. We shall show how that result can be easily derived from our results. In the last section, we show that the Samuel realcompactification H(Uμ (X)) also admits a characterization in terms of a certain compatible uniformity on X, which has a base consisting of countable covers. This characterization is reminiscent of some classical results of Shirota [21]. 2. Preliminary results about realcompactifications We start this section with some basic facts about realcompactifications, that can be found mainly in [3]. Thus, the classical way of generating realcompactifications of a Tychonoff space X is the following. First, we take a family L of real-valued continuous functions, that we suppose having the algebraic structure of unital vector lattice and separating points from closed sets of X. Then, we embed (homeomorphically) X into the product space of real lines RL , through the evaluation map e : X → RL x e(x) = (f (x))f ∈L . Next, we take the closure of X in R . We will denote this closure by H(L) because it is exactly the set of all the real unital homomorphisms on L. Since H(L) is closed in RL then it is in fact a realcompactification of X. If we just take the bounded functions L∗ = L ∩ C ∗ (X) in L (where C ∗ (X) is the family of bounded real-valued continuous functions) we get that H(L∗ ) is now a compactification of X. Similarly as for compactifications, we can consider a partial order ≤ on the set R(X) of all the realcompactifications of X. Namely, for two realcompactifications α1 X and α2 X, we write α1 X ≤ α2 X whenever there is a continuous mapping h : α2 X → α1 X leaving X pointwise fixed. We say that α1 X and α2 X are equivalent (and we write α1 X = α2 X) whenever α1 X ≤ α2 X and α2 X ≤ α1 X, and this implies the existence of a homeomorphism between α1 X and α2 X leaving X pointwise fixed. With the partial order ≤ defined above, the pair (R(X), ≤) is a complete upper semi-lattice where the largest element is exactly the Hewitt-Nachbin realcompactification H(C(X)) = υX. Recall that, for the family K(X) of all compactifications of X, the pair (K(X), ≤) is also a complete upper semi-lattice where, the ˇ largest element is now the Stone-Cech compactification H(C ∗ (X)) = βX. ∗ In particular, H(L) (resp. H(L )) is characterized (up to equivalence) as the smallest realcompactification (resp. compactification) of X such that every function f ∈ L can be continuously extended to it. Besides, it is easy to see that H(L) can be considered as a topological subspace of H(L∗ ). Thus, we can write L
(♣)
X ⊂ H(L) ⊂ H(L∗ ).
In the frame of uniform spaces, when we deal with the the Samuel realcompactification and the Samuel compactification defined in the introduction, we have that they are respectively H(Uμ (X)) and sμ X = H(Uμ∗ (X)), and also that H(Uμ (X)) ≤ υX and sμ X ≤ βX. Moreover, by (♣) we know that X ⊂ υX ⊂ βX and X ⊂ H(Uμ (X)) ⊂ sμ X. In both cases, for realcompactifications and compactifications, it is known that (R(X), ≤), (respectively (K(X), ≤)) is a complete lattice if and only if X is locally compact. In that case, the smallest element in both
THE SAMUEL REALCOMPACTIFICATION
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lattices is the Alexandroff or the one-point compactification of X which is generated by the family of all the real-valued continuous functions which are constant at infinity [17]. We will say that a Tychonoff space X is L-realcompact, for the unital vector lattice of real-valued continuous functions L, if X = H(L). Thus, a space X is realcompact if and only if X = υX and X is compact if and only if X = βX. Clearly every L-realcompact space is realcompact. We will see later that, in general, when we are considering different lattices L and L , if X is L-realcompact then it is not necessarily true that X is L -realcompact. For instance, there are realcompact spaces which are not realcompact for other lattices L different from C(X). However, when a space is compact then it is L∗ -compact for any lattice L. On the other hand, for a unital vector lattice L ⊂ C(X) we can consider wL the weak uniformity in X which is the weakest uniformity making each function in L uniformly continuous [22]. When L separates points and closed sets in X, then this Hausdorff uniformity is compatible with the topology of X. If we endow X with the weak uniformity wL and RL with the product uniformity, then the evaluation map e : X → RL is uniformly continuous and the inverse map e−1 : e(X) → X is also uniformly continuous. Thus, X is uniformly embedded in RL . Since RL endowed with the product uniformity is a complete uniform space, the closure H(L) of X in RL is the completion of (X, wL ) (by uniqueness of the completion). We can summarize all of this as follows. Theorem 1. Let L ⊂ C(X) be a unital vector lattice separating points and closed sets in the Tychonoff space X. The realcompactification H(L) of X is (topologically) homeomorphic to the completion of the uniform space (X, wL ) where wL is the weak uniformity generated by L. If we apply this result to the above defined realcompactifications we get that υX is homeomorphic to the completion of (X, wC(X) ), βX to the completion of (X, wC ∗ (X) ), H(Uμ (X)) to the completion of (X, wUμ (X) ) and sμ X to the completion of (X, wUμ∗ (X) ). In general, when we have a realcompactification Y of X which is not generated by an explicit lattice L, if C(Y ) is the family of all the real-valued continuous functions on Y , then (Y, wC(Y ) ) is complete [9]. Precisely it is the completion of (X, wC(Y )|X ) where C(Y )|X consists of the restrictions to X of the functions in C(Y ) [11]. Moreover we can describe C(Y )|X as the set of those continuous real-valued functions on X which preserve Cauchy filters of the weak uniformity wC(Y )|X [1]. Recall that a function between uniform spaces preserves Cauchy filters whenever it maps Cauchy filters into Cauchy filters. These functions are usually called Cauchy continuous (see [16]). Note that we have UwC(Y )|X (X) = C(Y )|X , because every uniformly continuous functions preserves Cauchy filters (or nets) [1]. Therefore the following result obtains. Theorem 2. Let Y be a realcompactification of the Tychonoff space X. Then Y is (up to equivalence) the Samuel realcompactification of the uniform space (X, wC(Y )|X ). For instance, the Hewitt-Nachbin realcompactification υX can be considered as the Samuel realcompactification of the uniform space (X, wC(X) ) [9]. Equivalently, υX is the Samuel realcompactification of the uniform space (X, u) where u is the universal uniformity on X, because the family of real-valued uniformly continuous ˇ functions on (X, u) is exactly C(X). In the same way, the Stone-Cech compactification βX is the Samuel compactification of the uniform space (X, wC ∗ (X) ). Now, we are going to finish this section with some easy results in this topic, which we will use later. Theorem 3. Let (X, μ) be a uniform space. For every compatible uniformity ν on X, if wUμ (X) ν μ then H(Uμ (X)) = H(Uν (X)). Besides, sμ X = sν X. Proof. This follows easy from the fact that Uμ (X) = UwUμ (X) (X) ⊂ Uν (X) ⊂ Uμ (X). The same is true for bounded functions. and μ Theorem 4. Let (X, ) be the completion of a uniform space (X, μ). Then H(Uμ (X)) = H(Uμ (X)) sμ X = sμ X. Proof. It is known that the functions in Uμ (X) are exactly the restrictions of the real-valued uniformly con μ tinuous functions of the completion (X, ) (see [22]). Thus, by density, the result follows. As in the previous theorem, the same is true for bounded functions.
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3. Main results We start this section by recalling the notion of Bourbaki-completeness for uniform spaces which was introduced and studied by the authors in [5], in the frame of metric spaces, and in [6] for some special uniformities. So let U ∈ μ, a uniform cover in (X, μ). For U ∈ U let us write St1 (U, U ) = St(U, U) := {V ∈ U : V ∩ U = ∅} and put Stm (U, U ) = St(Stm−1 (U, U), U ), m ≥ 2 St∞ (U, U) =
∞
Stm (U, U).
m=1
Definition 5. A filter F in the uniform space (X, μ) is said to be Bourbaki-Cauchy if for every U ∈ μ there exist m ∈ N and U ∈ U such that F ⊂ Stm (U, U), for some F ∈ F (i.e. Stm (U, U) ∈ F). Moreover, we say that (X, μ) is Bourbaki-complete whenever every Bourbaki-Cauchy filter in X clusters. It is easy to see that every Cauchy filter is Bourbaki-Cauchy, and therefore every Bourbaki-complete uniform space is complete. In general the reverse implication is not true. For example, every infinite-dimensional Banach space is non-Bourbaki-complete with the uniformity given by its norm (see [5]). In fact, we know that a normed space is Bourbaki-complete if and only if it has finite dimension [5]. More examples can be found in [12]. On the other hand, it is easy to check that a subspace of a Bourbaki-complete uniform space is Bourbakicomplete if and only if it is closed. Furthermore this uniform property is also productive as next result proves. Proposition 6. Any nonempty product of uniform spaces is Bourbaki-complete if and only if each factor is Bourbaki-complete. Proof. Suppose ΠXi is Bourbaki-complete. Since each factor Xi is (uniformly) homeomorphic to a closed subspace of this product, then it must be Bourbaki-complete, as we have said above. On the other hand, suppose Xi is Bourbaki-complete for every i ∈ I, and let F a Bourbaki-Cauchy filter in the product. Take H an ultrafilter containing F. Clearly, H is also Bourbaki-Cauchy and then its projection into Xi will be a Bourbaki-Cauchy ultrafilter, for every i ∈ I. Now, from the Bourbaki-completeness of every factor, this projection must converges to a point in Xi . Therefore, H also converges to a point in the product, and this means, in particular, that the initial filter F clusters, as we wanted. Theorem 7. Let (X, μ) be a uniform space. Then its Samuel realcompactification H(Uμ (X)), considered as the completion of (X, wUμ (X) ), is Bourbaki-complete. Proof. First note that the completion of (X, wUμ (X) ) is precisely the closed subspace H(Uμ (X)) ⊂ RUμ (X) with the uniformity inherited by the product space RUμ (X) . And then the proof follows at once from the above Proposition 6, since R is Bourbaki-complete. The next result, an easy corollary of the above, is in line with those contained in [6] comparing both completeness properties for some special uniformities. Theorem 8. Let (X, μ) be a uniform space. Then (X, wUμ (X) ) is complete if and only if (X, wUμ (X) ) is Bourbaki-complete. Now we are going to see an interesting relationship between the Bourbaki-Cauchy filters in the space (X, μ) and its Samuel realcompactification. Proposition 9. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). Then X ⊂ Z ⊂ H(Uμ (X)). Proof. Clearly X ⊂ Z. So, let F be a Bourbaki-Cauchy filter in (X, μ) and ξ ∈ sμ X a cluster point of F. Consider the filter H in X generated by the family {F ∩ V : F ∈ F, V is a neighborhood of ξ in sμ X}. Then H is also Bourbaki-Cauchy in (X, μ), since it contains F. Taking into account that the identity mapping id : (X, μ) → (X, wUμ (X) ) is uniformly continuous, H is Bourbaki-Cauchy in (X, wUμ (X) ). Now, from Theorem 7, this filter must cluster in H(Uμ (X)). And we finish, noting that ξ is the only cluster point of H.
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Now, we are going to study the problem of the Samuel realcompactness of a uniform space, and we will see that a Katˇetov-Shirota type theorem can be obtained where Bourbaki-completeness will play the role of completeness in the classical one. Recall that every uniform countable cover {Un : n ∈ Z} ∈ μ such that Un ∩ Um = ∅ whenever |n − m| > 1 belongs also to the weak uniformity wUμ (X) (see [14]). This kind of covers were called linear covers by Isbell in [14], and 2-finite covers by Garrido and Montalvo in [4]. Proposition 10. Let (X, μ) be a uniform space such that every uniform partition of (X, μ) has non-measurable cardinal. Then every Cauchy filter of (X, wUμ (X) ) is a Bourbaki-Cauchy filter of (X, μ). Proof. Let F be a Cauchy filter in (X, wUμ (X) ) and let U ∈ μ. Note that the family {St∞ (U, U) : U ∈ U} is a uniform partition of X. We are going to prove that there exists U ∈ U such that St∞ (U, U) ∈ F. Observe that we can fix representative elements Ui ∈ U, i ∈ I, such that X = {St∞ (Ui , U ) : i ∈ I} and St∞ (Ui , U) ∩ St∞ (Uj , U) = ∅ whenever i = j, i, j ∈ I. Next, define J := J ⊂ I : {St∞ (Ui , U ) : i ∈ J} ∈ F . We want to prove that J is an ultrafilter in I with the countable intersection property. Clearly, we have ∅∈ / J . Let J ⊂ I and suppose that J ∈ / J . In order to see that I − J ∈ J , consider ∞ W= {St (Ui , U ) : i ∈ J}, {St∞ (Ui , U ) : i ∈ I − J} . Since W is a uniform finite partition of X then W ∈ wUμ (X) . Now as F is a Cauchy filter of (X, wUμ (X) ) and {St∞ (Ui , U) : i ∈ J} ∈ / F, we have {St∞ (Ui , U ) : i ∈ I − J} ∈ F. Now we prove that J satisfies the countable intersection property. Take {Jn : n ∈ N} ⊂ J and suppose without loss of generality it is a strictly decreasing family. If {Jn : n ∈ N} = ∅, then {I − Jn : n ∈ N} forms a cover on I. Define the family of sets W = {Wn : n ∈ N} by Wn = {St∞ (Ui , U ) : i ∈ Jn−1 − Jn }, where J0 = I. Then, W is a countable uniform partition of X and therefore, W ∈ wUμ (X) . Since F is a ∞ ) then W ∈ F for some n ∈ N. Since W ⊂ {St (Ui , U ) : i ∈ I − Jn } then Cauchy filter of (X, w n n U (X) μ {St∞ (Ui , U) : i ∈ I− Jn } ∈ F, and therefore I − Jn ∈ J . But this is a contradiction because J is a filter and Jn ∈ J . Hence, {Jn : n ∈ N} = ∅ and J has the countable intersection property. By hypothesis, the uniform partition {St∞ (Ui , U ) : i ∈ I} of X, has non-measurable cardinal, and therefore the discrete space I is realcompact. Hence the ultrafilter J must be fixed, that is, there exists i0 ∈ I such that /F i0 ∈ J . But now we have {i0 } ∈ J and hence St∞ (Ui0 , U ) ∈ F. Note that {St∞ (Ui , U ) : i ∈ I − {i0 }} ∈ To complete the proof that F is Bourbaki-Cauchy in (X, μ), it suffices to find some m ∈ N such that Stm (Ui0 , U) ∈ F. We define a cover A = {An : n ∈ {0} ∪ N} of X as follows: A0 = {St∞ (Ui , U ) : i ∈ I − {i0 }}, A1 = St(Ui0 , U ), A2 = St2 (Ui0 , U ), n An = {V ∈ U : V ∩ St (Ui0 , U ) = ∅, V ∩ Stn−2 (Ui0 , U ) = ∅} if n ≥ 3. Then A is a countable uniform cover satisfying that Ai ∩ Aj = ∅ whenever |i − j| > 1 and hence, A ∈ wUμ (X) . /F Again, since F is Cauchy in (X, wUμ (X) ) there exists some n ∈ {0} ∪ N such that An ∈ F. We have A0 ∈ and hence n = 0. Since An ⊂ Stn+1 (Ui0 , U), then Stn+1 (Ui0 , U ) ∈ F. Theorem 11. Let (X, μ) be a uniform space such that every uniform partition of X has non-measurable cardinal. Then, (X, μ) is Bourbaki-complete if and only if (X, wUμ (X) ) is complete. Proof. First, suppose that (X, wUμ (X) ) is complete then, from Theorem 8, it is Bourbaki-complete. Now, as the uniformity μ is finer than wUμ (X) , then (X, μ) will be also Bourbaki-complete. Reciprocally, suppose (X, μ) is Bourbaki-complete and let F be a Cauchy filter in (X, wUμ (X) ). From Proposition 10, we have that F is a Bourbaki-Cauchy filter in (X, μ), and hence it clusters. Since any Cauchy filter with a cluster point converges, (X, wUμ (X) ) is complete. We have already all the ingredients in order to establish our main result in this section.
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ˇtov-Shirota type theorem) Let (X, μ) be a uniform space. Then (X, μ) is Samuel Theorem 12. (Kate realcompact if and only if (X, μ) is Bourbaki-complete and every uniformly discrete subspace of X has nonmeasurable cardinal. Proof. If (X, μ) is Samuel realcompact then it is realcompact and hence every discrete closed subspace has nonmeasurable cardinal [9]. In particular, every uniformly discrete subspace must have non-measurable cardinal since it is in addition closed. On the other hand, from the identity X = H(Uμ (X)), it follows that (X, wUμ (X) ) coincides with its completion and then (X, wUμ (X) ) is complete and, by Theorem 11, (X, μ) is Bourbakicomplete. Note that we can apply that theorem because every uniform partition produces a uniformly discrete subspace with the same cardinal. Conversely, again from Theorem 11, the Bourbaki-completeness of (X, μ) together with the assumption of non-measurability imply the completeness of (X, wUμ (X) ). Therefore X = H(Uμ (X)), and this means that (X, μ) is Samuel realcompact. Remark 13. It is clear that in Theorem 12 only the non-measurability of the uniform partitions is needed and not the (stronger) condition of non-measurability of the uniformly discrete subspaces. On the other hand, the condition of non-measurable cardinality can not be deleted in the three previous results. Indeed, if X is a set with a measurable cardinal endowed with the uniformity μ given by the 0-1 metric, then the corresponding uniform (metric) discrete space (X, μ) is Bourbaki-complete ([5]), but not Samuel realcompact since it is not even realcompact. In this case X = υX = H(U μ(X)), (X, wUμ (X) ) is not complete, and there exist Cauchy filters in (X, wUμ (X) ) that are not Bourbaki-Cauchy filters in (X, μ). We must notice now that there is a similar precedent of the above Katˇetov-Shirota type theorem in a paper by Rice and Reynolds [19]. In fact, in this work the authors characterized the completeness of (X, wUμ (X) ) by means of the uniformity generated by all the star-finite uniform covers of (X, μ). Next, we are going to explain the similarities between both results. Recall that a cover A of a set X is said to be star-finite if every element A ∈ A meets at most finitely many A ∈ A. For every uniform space (X, μ) the family of all star-finite uniform covers generates a uniformity on X compatible with the topology induced by μ [15]. We will denote this uniformity by sf μ. Note that, since R with the usual uniformity has a base by star-finite (in fact, linear) covers, then wUμ (X) sf μ μ. Lemma 14. Let (X, μ) be a uniform space. Then, (a) Every Cauchy filter of (X, sf μ) is a Bourbaki-Cauchy filter of (X, μ). (b) Every Bourbaki-Cauchy ultrafilter of (X, μ) is a Cauchy ultrafilter of (X, sf μ). Proof. (a) Let F be a Cauchy filter of (X, sf μ) and let U ∈ μ. Next, we produce a star-finite uniform cover as follows. As in the proof of Proposition 10, let {St∞ (Ui , U ), i ∈ I} the uniform partition of X generated by U. Fix i ∈ I, write A0 (Ui ) = Ui = St0 (Ui , U ), and for every n ∈ N let An (Ui ) = {V ∈ U : V ⊂ Stn (Ui , U ), V − Stn−1 (Ui , U ) = ∅}. Then, it is easy to see that the family A(U) = {An (Ui ), i ∈ I, n ≥ 0} ∈ sf μ. By hypothesis there exist some i ∈ I and n ≥ 0 such that An (Ui ) ∈ F. Since An (Ui ) ⊂ Stn (Ui , U ), then Stn (Ui , U ) ∈ F and we can deduce that F is in fact a Bourbaki-Cauchy filter of (X, μ). (b) Let F be a Bourbaki-Cauchy ultrafilter of (X, μ). Then it is clear that F is also a Bourbaki-Cauchy ultrafilter of (X, sf μ). Now, let U ∈ sf μ be a star-finite uniform cover of (X, μ). Then, for some m ∈ N and U ∈ U , Stm (U, U) ∈ F. Since U is star-finite, we can choose finitely many Ui ∈ U, i = 1, ..., k, such that k Stm (U, U) ⊂ i=1 Ui . Since F is an ultrafilter then Ui ∈ F, for some i = 1, ..., k, and therefore F is a Cauchy ultrafilter in (X, sf μ). Proposition 15. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). Then Z is (topologically) homeomorphic to the completion of (X, sf μ). s Proof. Let (X, f μ) denote the completion of (X, sf μ). Since wUμ (X) sf μ μ then, by Theorems 3 and 4, = Z. it is clear that X is a (topological) subspace of sμ X. We are going to see that X Firstly, by Lemma 14, it is clear that X ⊂ Z. Conversely, let F be a Bourbaki-Cauchy filter in (X, μ) and ξ ∈ Z a cluster point of F. Consider an ultrafilter H in X containing the family {F ∩ V : F ∈ F, V is a neighborhood of ξ in sμ X}. Then, H is also Bourbaki-Cauchy in (X, μ) because it contains F. Again, by Lemma 14, H is Cauchy in (X, sf μ) and then ξ ∈ X.
THE SAMUEL REALCOMPACTIFICATION
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Theorem 16. A uniform space (X, μ) is Bourbaki-complete if and only if (X, sf μ) is complete. In particular, every complete uniform space with a base of star-finite uniform covers is Bourbaki-complete. Examples of uniformities having a star-finite base are, for instance, the weak uniformities wL generated by a family of functions L. More precisely, these weak uniformities have star-finite and countable bases composed by finite intersections of linear covers. In fact, the intersection of a finite number of covers of the form {f −1 ((n − 1) · ε, (n + 1) · ε), n ∈ Z}, for f ∈ L and ε > 0. Observe that the finite intersection of countable star-finite covers is still countable and star-finite. Thus, the results of Theorem 7 and Theorem 8 can be considered as particular cases of last result. Having in mind that in a uniform space every uniform cover has non-measurable cardinality if and only if every uniformly discrete subspace has non measurable cardinality [8], the above Theorem 16 is the link between our Katˇetov-Shirota result (Theorem 12) and the following result by Rice and Reynolds. Theorem 17. (Reynolds-Rice [19]) Let (X, μ) be a uniform space such that every uniform cover has nonmeasurable cardinal. Then (X, wUμ (X) ) is complete if and only if (X, sf μ) is complete. We point out now some slight difference between our result and the result of Rice and Reynolds. It is clear that also, as in Theorem 12, only the non-measurability of the uniform partitions is needed in last Theorem 17, and not the (stronger) condition of the non-measurability of the cardinality of the uniform covers. The next result characterizes precisely the property of a uniform space that every uniform partition has nonmeasurable cardinality which is not in general equivalent to satisfy that every uniformly discrete subspace has non-measurable cardinality. However, this equivalence is true whenever we consider the uniformity sf μ. Theorem 18. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). The following statements are equivalent: (1) Every uniform partition of (X, μ) has non-measurable cardinal. (2) Z = H(Uμ (X)). (3) Z is realcompact. (4) Every uniformly discrete subspace of (X, sf μ) has non-measurable cardinal. (5) Every uniform partition of (X, sf μ) has non-measurable cardinal. Proof. (1) ⇒ (2). By Proposition 9, we have Z ⊂ H(Uμ (X)). The converse follows from Proposition 10. (2) ⇒ (3). This implication is trivial. (3) ⇒ (4). By Proposition 15, Z is homeomorphic to the completion of (X, sf μ). Since every uniformly s discrete subspace of (X, sf μ) is a closed discrete subspace of the completion (X, f μ), then it follows that every uniformly discrete subspace of (X, sf μ) has non-measurable cardinal. (4) ⇒ (5). This follows taking into account that every uniform partition determines a uniformly discrete subspace with the same cardinal. (5) ⇒ (1). It is clear since (X, μ) and (X, sf μ) share the same uniform partitions. Note that the last result provides another interesting description of the Samuel realcompactification for those uniform spaces fulfilling condition (1). Namely, the Samuel realcompactification of these spaces are formed by the cluster points in the Samuel compactification of their Bourbaki-Cauchy filters, or equivalently, it is exactly the completion of (X, sf μ). Note that there are many spaces with this property, for instance, connected spaces, or more generally uniformly connected, separable, Lindel¨ of, and many other spaces. Examples of uniform spaces such that every uniform partition has non-measurable cardinal but having a uniform cover of measurable cardinality are easy to find, just think of connected spaces containing a uniformly discrete subspace of measurable cardinal as, for instance, the metric hedgehog H(κ) or any p (κ) where κ is a measurable cardinal. Observe also that the equivalence between (3) and (4) in last theorem reminds of the result in [9, Theorem 15.21] that asserts that the completion of a uniform space is realcompact if and only if every uniform discrete subspace has non-measurable cardinal. Note that for the uniformity sf μ we can even consider the weaker condition appearing in (5). But, in general, we can not change uniform partitions for uniformly discrete subspaces. We finish this section recalling a result of Katˇetov-Shirota type given by Huˇsek and Pulgar´ın in the setting of uniform spaces with weak uniformities ([13]). They define a uniform space (X, μ) to be uniformly realcomplete when (X, wUμ X ) is complete. Thus, in that paper, the following result is proved for the particular case of uniformly 0-dimensional spaces, where a uniform space is uniformly 0-dimensional if it has a base for the uniformity composed by partitions (for instance, every uniformly discrete space).
8
˜ M. ISABEL GARRIDO AND ANA S. MERONO*
Theorem 19. (Huˇ sek-Pulgar´ın [13]) A uniformly 0-dimensional uniform space is uniformly realcomplete if and only if the space is complete and it does not have uniformly discrete subspaces of measurable cardinality. Proof. The proof follows easily from Theorems 12 and 16.
4. Uniformities with countable uniform covers In his famous paper [21], Shirota proved that the Hewitt-Nachbin realcompactification υX of a space X is homeomorphic to the completion of (X, eu) where eu is the uniformity having as a base all the normal covers of X. Thus he was proving that the completion of (X, wC(X) ) and the completion of (X, eu) are (topologically) equivalent even if the uniformities wC(X) and eu are not. Later on, Isbell proved (see [15]) a similar result in the frame of the so called locally fine uniform spaces ([8]). Recall that to be locally fine is equivalent to the more intuitive notion of being subfine, i.e., to be a uniform subspace of a fine space [18]. The result of Isbell states that for a locally fine uniform space (X, μ) every Cauchy filter in (X, wUμ(X) ) is Cauchy in (X, eμ), where eμ is the compatible uniformity on X having as a base all the countable covers of μ ([8]). Then we can say now that, for a locally fine uniform space (X, μ), the completion of (X, eμ) is homeomorphic to its Samuel realcompactification H(Uμ (X)). In particular, since every fine space is locally fine, we can apply the last result to the fine uniformity u on a Tychonoff space X. Thus, we can deduce easily the above result of Shirota, because for fine uniform spaces, every continuous function is uniformly continuous. Observe that, in general, locally fine uniform spaces do not satisfy that every continuous function is uniformly continuous. In fact, locally fine uniform spaces are characterized as those uniform spaces satisfying that every uniformly locally uniformly continuous function into a metric space is uniformly continuous [18]. Therefore, for locally fine uniform spaces, the Hewitt-Nachbin realcompactification and the Samuel realcompactification are not necessarily equivalent. For instance, this is the case of every totally bounded realcompact and non-compact uniform space, as the real interval (0, 1) with the usual metric. That the completion of (X, eμ) is a realcompactification of X follows from the fact that it is also complete with the uniformity having as a base all the countable uniform covers of its fine uniformity, and therefore, by the Shirota result, it is realcompact. In [19], Rice and Reynolds characterize those uniform spaces (X, μ) satisfying that (X, eμ) is complete. Precisely, they prove that whenever every uniform cover of the uniform space (X, μ) has non-measurable cardinal, then (X, eμ) is complete if and only if (X, pf μ) is complete, where pf μ denotes the compatible uniformity on X having as a base all the point-finite covers of μ. Recall that a cover U is point-finite if every x ∈ X just belongs to finitely many elements of U . In the next example we show that, in general, the completion of (X, eμ) as well as the completion of (X, pf μ) are not homeomorphic to the Samuel realcompactification H(Uμ (X)). To that purpose we need to use some results and concepts from [5]. Recall that a uniform space (X, μ) is Bourbaki-bounded if for every U ∈ μ there exists m ∈ N and finitely k many Ui ∈ U , i = 1, ..., k, such that X = i=1 Stm (Ui , U ). Observe that it was precisely the notion of Bourbaki-boundedness that originated the study of Bourbaki-Cauchy filters, nets and sequences, as well as Bourbaki-completeness (see [5] and [6]). In particular in [6] it is proved that a uniform space is compact if and only if it is Bourbaki-bounded and Bourbaki-complete. Example 20. Let (X, d) be a separable non-compact complete metric space which is Bourbaki-bounded for the metric uniformity. For instance the metric hedgehog H(ℵ0 ) of countably many spines, or any non-compact closed and bounded subset of the classical Hilbert space 2 . Then the metric uniformity given by d coincides with the uniformity eμd by the Lindel¨of property. Therefore (X, eμd ) is complete. However, by Theorem 11, (X, wUd (X) (X)) is not complete because (X, d) fails to be Bourbaki-complete. Otherwise, since (X, d) is Bourbaki-bounded, X would be compact, which is false. The above example shows that there are uniform spaces such that (X, pf μ) is complete but which are not Bourbaki-complete. On the other hand, the completion of (X, eμ) is not necessarily homeomorphic to the Samuel realcompactification H(Uμ (X)) because completeness is not enough. In fact, we need Bourbakicompleteness as the following two results show. Note that sf (eμ) and e(sf μ) will denote respectively the uniformity having as a base all the star-finite covers of eμ and all the countable covers of sf μ. Theorem 21. Let (X, μ) be a uniform space. The following spaces are (topologically) homeomorphic:
THE SAMUEL REALCOMPACTIFICATION
(1) (2) (3) (4)
9
H(Uμ (X)). The subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, eμ). The completion of (X, sf (eμ)). The subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, e(sf μ)).
Proof. Applying Theorem 3 several times, we have that H(Uμ (X)) = H(Ueμ (X)) = H(Usf μ (X)) = H(Usf (eμ) (X)) = H(Ue(sf μ) (X)). The same is true for the Samuel compactification. Since it is not difficult to see that every uniform partition of (X, eμ) is countable the equivalence of (1) and (2) follows from Theorem 18. Analogously the equivalence of (1) and (4) follows also at once. Finally, the equivalence of (2) and (3) is just Theorem 15. Theorem 22. Let (X, μ) be a uniform space. The following statements are equivalent: (1) X is Samuel realcompact. (2) (X, eμ) is Bourbaki-complete. (3) (X, sf (eμ)) is complete. (4) (X, e(sf μ)) is complete. Proof. We only need to prove (4) ⇒ (1). If (X, e(sf μ)) is complete then (X, sf μ) is complete since the identity map id : (X, e(sf μ)) → (X, sf μ) is uniformly continuous. In addition, X is a realcompact space since, as we have said before, the completion of every uniformity of type eμ is a realcompactification of X. Thus, every uniform cover of (X, sf μ) has non-measurable cardinal and then, by Theorem 17, X is Samuel realcompact. Remark 23. In condition (4) of Theorem 21, it would be interesting to be able to consider the completion of (X, e(sf μ)) instead of the set of cluster points of its Bourbaki-Cauchy filters, as it happens in condition (4) of Theorem 22. However, since we cannot assure that the uniformity e(sf μ) has a star-finite basis we can not refine the result (observe that the same question is noted in [19]). Thus, according to Theorem 22, we wonder if the difference between above results lies in the non-measurability of the cardinality of the uniform covers. Nevertheless, for uniformly 0-dimensional we have that e(sf μ) = eμ = sf (eμ), because whenever (X, μ) is uniformly 0-dimensional then (X, eμ) is also uniformly 0-dimensional. Acknowledgements We would like to thank the referee for his/her careful revision and interesting suggestions. References [1] J. Bors´ık, Mappings preserving Cauchy nets, Tatra Mt. Math. Publ. 19 (2000) 63-73. [2] I.S. G´ al, Proximity relations and precompact structures. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 62 (1959) 304-326. [3] M.I. Garrido and J.A. Jaramillo, Homomorphism on function lattices, Monatsh. Math. 141 (2004) 127-146. [4] M.I. Garrido and F. Montalvo, Countable covers and uniform closure, Rend. Istit. Mat. Univ. Trieste 30 (1999) 91-102. [5] M.I. Garrido and A.S. Mero˜ no, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733-758. [6] M.I. Garrido and A.S. Mero˜ no, On paracompactness, completeness and boundedness in uniform spaces, Topology Appl. 203 (2016) 98-107. [7] M.I. Garrido and A.S. Mero˜ no, The Samuel realcompactification of a metric space, J. Math. Anal. Appl. 456 (2017) 1013-1039. [8] S. Ginsburg and J.R. Isbell, Some operators on uniform spaces, Trans. Amer. Math. Soc. 93 (1959) 145-168. [9] L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics 43, Springer-Verlag, New York, 1976. [10] L. Google and M. Megrelishvili, Semigroup actions: proximities, compactifications and normality, Topology Proc. 35 (2010) 37-71. [11] A.W. Hager, D.G. Johnson, A note on certain subalgebras of C(X), Canadian J. Math. 20 (1968) 389-393. ˇ [12] A. Hohti, H. Junnila and A.S. Mero˜ no, On strongly Cech-complete spaces, manuscript. [13] M. Huˇsek and A. Pulgar´ın, Banach-Stone-like theorems for lattices of uniformly continuous functions, Quaest. Math. 35 (2012) 417-430. [14] J.R Isbell, Euclidean and weak uniformities, Pacific J. Math 8 (1958) 67-86. [15] J.R Isbell, Uniform spaces, Math. Surveys 12, Amer. Math. Soc. 1964. [16] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps, Marcel Dekker, New York, 1989. [17] J. Mack, M. Rayburn and G. Woods, Lattices of topological extensions, Trans. Amer. Math. Soc. 189 (1974) 163-174. [18] J. Pelant, Locally fine uniformities and normal covers, Czechoslovak Math. J. 37 (1987) 181-187. [19] G.D. Reynolds and M.D. Rice, Completenes and covering properties of uniform spaces, Quart. J. Math. Oxford 29 (1978) 367-374. [20] P. Samuel, Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948) 100-132. [21] T. Shirota, A class of topological spaces, Osaka Math. J. 4 (1952) 23-40. [22] S. Willard, General Topology, Dover Publications, Mineola, NY, 2004.
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
´ tica Interdisciplinar (IMI), Departamento de Geometr´ıa y Topolog´ıa, Universidad (Garrido) Instituto de Matema Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected] ´ lisis Matema ´ tico, Universidad Complutense de Madrid, 28040 Madrid, Spain (Mero˜ no*) Departamento de Ana E-mail address: [email protected]
M. Isabel Garrido, Ana S. Meroño
PII: DOI: Reference:
S0166-8641(18)30212-8 https://doi.org/10.1016/j.topol.2018.03.033 TOPOL 6452
To appear in:
Topology and its Applications
Received date: Revised date: Accepted date:
31 December 2016 4 December 2017 5 February 2018
Please cite this article in press as: M. Isabel Garrido, A.S. Meroño, The Samuel realcompactification, Topol. Appl. (2018), https://doi.org/10.1016/j.topol.2018.03.033
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THE SAMUEL REALCOMPACTIFICATION ˜ M. ISABEL GARRIDO AND ANA S. MERONO* Abstract. For a uniform space (X, μ), we introduce a realcompactification of X by means of the family Uμ (X) of all the real-valued uniformly continuous functions on X, in the same way that the known Samuel compactification of the space is given by Uμ∗ (X) the set of all the bounded functions in Uμ (X). We will call it “the Samuel realcompactification” by several resemblances to the Samuel compactification. In this paper, we present different ways to construct such realcompactification as well as we study the corresponding problem of knowing when a uniform space is Samuel realcompact, that is, when it (topologically) coincides with its Samuel realcompactification. At this respect, we obtain as main result a theorem of Katˇ etov-Shirota type, given in terms of a property of completeness, recently introduced by the authors, called Bourbaki-completeness.
1. Introduction A realcompactification of a Tychonoff space X is a realcompact space Y in which X is densely embedded. For instance, the well-known Hewitt-Nachbin realcompactification υX. Recall that υX is characterized as the smallest realcompactification of X (in the usual order on the family of all realcompactifications) such that every real-valued continuous function f ∈ C(X) can be continuously extended to it [9]. In the frame of uniform spaces, since we can also consider Uμ (X), the set of all the real-valued uniformly continuous functions on the uniform space (X, μ), it is natural to ask what is the smallest realcompactification of X such that every function f ∈ Uμ (X) can be continuously extended to it. Here, following the ideas of [3], we will introduce this realcompactification that we denote by H(Uμ (X)) because in fact it can be represented as the set of all the real homomorphisms on the unital vector lattice Uμ (X). Moreover, we will see that it also coincides, as a topological space, with the completion of the uniform space (X, wUμ (X) ), where wUμ (X) denotes the weak uniformity on X generated by Uμ (X) (Theorem 1). We will call H(Uμ (X)) the Samuel realcompactification of (X, μ) in likeness to the Samuel compactification sμ X. Recall that the Samuel compactification of a uniform space (X, μ), also known as the Smirnov compactification, is the smallest (real)compactification of X such that every bounded real-valued uniformly continuous function f ∈ Uμ∗ (X) can be continuously extended to it ([20]). The resemblance between the Samuel realcompactification and the Samuel compactification is not only due to their characterization as the smallest realcompactification, respectively compactification, such that every real-valued, respectively bounded, uniformly continuous function can be continuously extended to it. In fact, as well as every compactification of a Tychonoff space X can be considered as a Samuel compactification for some totally bounded (precompact) uniformity on X [2] (see also [10]), every realcompactification of X can be considered as a Samuel realcompactification for some uniformity on it, as we will explain below (Theorem 2). A Tychonoff space X is realcompact whenever X = υX, and we can similarly define, for a uniform space (X, μ), that (X, μ) is Samuel realcompact if X = H(Uμ (X)). In this paper, we give a uniform analogue to the well-known Katˇetov-Shirota Theorem ([21], see also [9]) which deals with realcompactness of Tychonoff spaces. This classical theorem states that a Tychonoff space X is realcompact if and only if X is completely uniformizable and every closed discrete subspace has non-measurable cardinal. In this line, we will obtain an analogous result that asserts that a uniform space is Samuel realcompact if and only if it is Bourbaki-complete and every (closed) uniformly discrete subspace has non-measurable cardinal (Theorem 12). The property of Bourbaki-completeness was introduced and studied by the authors in [5] (see also [6] and [12]), and it is a uniform property stronger than usual completeness. We say that a uniform space is Bourbaki-complete if every Bourbaki-Cauchy filter, defined later, clusters. 2010 Mathematics Subject Classification. Primary 54D60, 54E15; Secondary 54D35, 54A20, 54A25. Key words and phrases. Uniform space, realcompactification, real-valued uniformly continuous function, Samuel realcompactification, Cauchy filter, Bourbaki-Cauchy filter, Bourbaki-completeness. *Corresponding author. Partially supported by MINECO Project-MTM2015-65825-P (Spain). 1
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
It turns out that Bourbaki-Cauchy filters are a useful tool to study Samuel realcompactness. In fact, the Samuel realcompactification, considered as the completion of (X, wUμ (X) ), is always a Bourbaki-complete uniform space (Theorem 7). Furthermore, assuming that the uniform partitions of (X, μ) have non-measurable cardinality, the Cauchy filters of (X, wUμ (X) ) are precisely the Bourbaki-Cauchy filters of (X, μ) (Proposition 10), and then it follows that (X, wUμ (X) ) is complete if and only of (X, μ) is Bourbaki-complete (Theorem 11). This will be the key not only for proving our main result but also for finding another description of the Samuel realcompactification H(Uμ (X)). Namely, H(Uμ (X)) is the subset of sμ X formed by the cluster points of the Bourbaki-Cauchy filters in (X, μ) (Theorem 18). Very recently, in [7], we proved the same result for Samuel realcompactness but in the frame of metric spaces. The techniques used in [7] are not the same as those used here, since we worked in [7] with a family of metrics defined on the metric space (X, d) all of them uniformly equivalent to the initial metric d. We must notice that in [19], Rice and Reynolds obtained in some sense a similar Katˇetov-Shirota type theorem. More precisely, these authors characterized the completeness of (X, wUμ (X) ) by means of the uniformity generated by all the star-finite uniform covers of (X, μ). We are going to explain in Section 3 the similarities between both results. Moreover, and also in this line, Huˇsek and Pulgar´ın gave in [13] a kind of a uniform Katetov-Shirota result for the so-called uniformly 0-dimensional uniform spaces. We shall show how that result can be easily derived from our results. In the last section, we show that the Samuel realcompactification H(Uμ (X)) also admits a characterization in terms of a certain compatible uniformity on X, which has a base consisting of countable covers. This characterization is reminiscent of some classical results of Shirota [21]. 2. Preliminary results about realcompactifications We start this section with some basic facts about realcompactifications, that can be found mainly in [3]. Thus, the classical way of generating realcompactifications of a Tychonoff space X is the following. First, we take a family L of real-valued continuous functions, that we suppose having the algebraic structure of unital vector lattice and separating points from closed sets of X. Then, we embed (homeomorphically) X into the product space of real lines RL , through the evaluation map e : X → RL x e(x) = (f (x))f ∈L . Next, we take the closure of X in R . We will denote this closure by H(L) because it is exactly the set of all the real unital homomorphisms on L. Since H(L) is closed in RL then it is in fact a realcompactification of X. If we just take the bounded functions L∗ = L ∩ C ∗ (X) in L (where C ∗ (X) is the family of bounded real-valued continuous functions) we get that H(L∗ ) is now a compactification of X. Similarly as for compactifications, we can consider a partial order ≤ on the set R(X) of all the realcompactifications of X. Namely, for two realcompactifications α1 X and α2 X, we write α1 X ≤ α2 X whenever there is a continuous mapping h : α2 X → α1 X leaving X pointwise fixed. We say that α1 X and α2 X are equivalent (and we write α1 X = α2 X) whenever α1 X ≤ α2 X and α2 X ≤ α1 X, and this implies the existence of a homeomorphism between α1 X and α2 X leaving X pointwise fixed. With the partial order ≤ defined above, the pair (R(X), ≤) is a complete upper semi-lattice where the largest element is exactly the Hewitt-Nachbin realcompactification H(C(X)) = υX. Recall that, for the family K(X) of all compactifications of X, the pair (K(X), ≤) is also a complete upper semi-lattice where, the ˇ largest element is now the Stone-Cech compactification H(C ∗ (X)) = βX. ∗ In particular, H(L) (resp. H(L )) is characterized (up to equivalence) as the smallest realcompactification (resp. compactification) of X such that every function f ∈ L can be continuously extended to it. Besides, it is easy to see that H(L) can be considered as a topological subspace of H(L∗ ). Thus, we can write L
(♣)
X ⊂ H(L) ⊂ H(L∗ ).
In the frame of uniform spaces, when we deal with the the Samuel realcompactification and the Samuel compactification defined in the introduction, we have that they are respectively H(Uμ (X)) and sμ X = H(Uμ∗ (X)), and also that H(Uμ (X)) ≤ υX and sμ X ≤ βX. Moreover, by (♣) we know that X ⊂ υX ⊂ βX and X ⊂ H(Uμ (X)) ⊂ sμ X. In both cases, for realcompactifications and compactifications, it is known that (R(X), ≤), (respectively (K(X), ≤)) is a complete lattice if and only if X is locally compact. In that case, the smallest element in both
THE SAMUEL REALCOMPACTIFICATION
3
lattices is the Alexandroff or the one-point compactification of X which is generated by the family of all the real-valued continuous functions which are constant at infinity [17]. We will say that a Tychonoff space X is L-realcompact, for the unital vector lattice of real-valued continuous functions L, if X = H(L). Thus, a space X is realcompact if and only if X = υX and X is compact if and only if X = βX. Clearly every L-realcompact space is realcompact. We will see later that, in general, when we are considering different lattices L and L , if X is L-realcompact then it is not necessarily true that X is L -realcompact. For instance, there are realcompact spaces which are not realcompact for other lattices L different from C(X). However, when a space is compact then it is L∗ -compact for any lattice L. On the other hand, for a unital vector lattice L ⊂ C(X) we can consider wL the weak uniformity in X which is the weakest uniformity making each function in L uniformly continuous [22]. When L separates points and closed sets in X, then this Hausdorff uniformity is compatible with the topology of X. If we endow X with the weak uniformity wL and RL with the product uniformity, then the evaluation map e : X → RL is uniformly continuous and the inverse map e−1 : e(X) → X is also uniformly continuous. Thus, X is uniformly embedded in RL . Since RL endowed with the product uniformity is a complete uniform space, the closure H(L) of X in RL is the completion of (X, wL ) (by uniqueness of the completion). We can summarize all of this as follows. Theorem 1. Let L ⊂ C(X) be a unital vector lattice separating points and closed sets in the Tychonoff space X. The realcompactification H(L) of X is (topologically) homeomorphic to the completion of the uniform space (X, wL ) where wL is the weak uniformity generated by L. If we apply this result to the above defined realcompactifications we get that υX is homeomorphic to the completion of (X, wC(X) ), βX to the completion of (X, wC ∗ (X) ), H(Uμ (X)) to the completion of (X, wUμ (X) ) and sμ X to the completion of (X, wUμ∗ (X) ). In general, when we have a realcompactification Y of X which is not generated by an explicit lattice L, if C(Y ) is the family of all the real-valued continuous functions on Y , then (Y, wC(Y ) ) is complete [9]. Precisely it is the completion of (X, wC(Y )|X ) where C(Y )|X consists of the restrictions to X of the functions in C(Y ) [11]. Moreover we can describe C(Y )|X as the set of those continuous real-valued functions on X which preserve Cauchy filters of the weak uniformity wC(Y )|X [1]. Recall that a function between uniform spaces preserves Cauchy filters whenever it maps Cauchy filters into Cauchy filters. These functions are usually called Cauchy continuous (see [16]). Note that we have UwC(Y )|X (X) = C(Y )|X , because every uniformly continuous functions preserves Cauchy filters (or nets) [1]. Therefore the following result obtains. Theorem 2. Let Y be a realcompactification of the Tychonoff space X. Then Y is (up to equivalence) the Samuel realcompactification of the uniform space (X, wC(Y )|X ). For instance, the Hewitt-Nachbin realcompactification υX can be considered as the Samuel realcompactification of the uniform space (X, wC(X) ) [9]. Equivalently, υX is the Samuel realcompactification of the uniform space (X, u) where u is the universal uniformity on X, because the family of real-valued uniformly continuous ˇ functions on (X, u) is exactly C(X). In the same way, the Stone-Cech compactification βX is the Samuel compactification of the uniform space (X, wC ∗ (X) ). Now, we are going to finish this section with some easy results in this topic, which we will use later. Theorem 3. Let (X, μ) be a uniform space. For every compatible uniformity ν on X, if wUμ (X) ν μ then H(Uμ (X)) = H(Uν (X)). Besides, sμ X = sν X. Proof. This follows easy from the fact that Uμ (X) = UwUμ (X) (X) ⊂ Uν (X) ⊂ Uμ (X). The same is true for bounded functions. and μ Theorem 4. Let (X, ) be the completion of a uniform space (X, μ). Then H(Uμ (X)) = H(Uμ (X)) sμ X = sμ X. Proof. It is known that the functions in Uμ (X) are exactly the restrictions of the real-valued uniformly con μ tinuous functions of the completion (X, ) (see [22]). Thus, by density, the result follows. As in the previous theorem, the same is true for bounded functions.
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
3. Main results We start this section by recalling the notion of Bourbaki-completeness for uniform spaces which was introduced and studied by the authors in [5], in the frame of metric spaces, and in [6] for some special uniformities. So let U ∈ μ, a uniform cover in (X, μ). For U ∈ U let us write St1 (U, U ) = St(U, U) := {V ∈ U : V ∩ U = ∅} and put Stm (U, U ) = St(Stm−1 (U, U), U ), m ≥ 2 St∞ (U, U) =
∞
Stm (U, U).
m=1
Definition 5. A filter F in the uniform space (X, μ) is said to be Bourbaki-Cauchy if for every U ∈ μ there exist m ∈ N and U ∈ U such that F ⊂ Stm (U, U), for some F ∈ F (i.e. Stm (U, U) ∈ F). Moreover, we say that (X, μ) is Bourbaki-complete whenever every Bourbaki-Cauchy filter in X clusters. It is easy to see that every Cauchy filter is Bourbaki-Cauchy, and therefore every Bourbaki-complete uniform space is complete. In general the reverse implication is not true. For example, every infinite-dimensional Banach space is non-Bourbaki-complete with the uniformity given by its norm (see [5]). In fact, we know that a normed space is Bourbaki-complete if and only if it has finite dimension [5]. More examples can be found in [12]. On the other hand, it is easy to check that a subspace of a Bourbaki-complete uniform space is Bourbakicomplete if and only if it is closed. Furthermore this uniform property is also productive as next result proves. Proposition 6. Any nonempty product of uniform spaces is Bourbaki-complete if and only if each factor is Bourbaki-complete. Proof. Suppose ΠXi is Bourbaki-complete. Since each factor Xi is (uniformly) homeomorphic to a closed subspace of this product, then it must be Bourbaki-complete, as we have said above. On the other hand, suppose Xi is Bourbaki-complete for every i ∈ I, and let F a Bourbaki-Cauchy filter in the product. Take H an ultrafilter containing F. Clearly, H is also Bourbaki-Cauchy and then its projection into Xi will be a Bourbaki-Cauchy ultrafilter, for every i ∈ I. Now, from the Bourbaki-completeness of every factor, this projection must converges to a point in Xi . Therefore, H also converges to a point in the product, and this means, in particular, that the initial filter F clusters, as we wanted. Theorem 7. Let (X, μ) be a uniform space. Then its Samuel realcompactification H(Uμ (X)), considered as the completion of (X, wUμ (X) ), is Bourbaki-complete. Proof. First note that the completion of (X, wUμ (X) ) is precisely the closed subspace H(Uμ (X)) ⊂ RUμ (X) with the uniformity inherited by the product space RUμ (X) . And then the proof follows at once from the above Proposition 6, since R is Bourbaki-complete. The next result, an easy corollary of the above, is in line with those contained in [6] comparing both completeness properties for some special uniformities. Theorem 8. Let (X, μ) be a uniform space. Then (X, wUμ (X) ) is complete if and only if (X, wUμ (X) ) is Bourbaki-complete. Now we are going to see an interesting relationship between the Bourbaki-Cauchy filters in the space (X, μ) and its Samuel realcompactification. Proposition 9. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). Then X ⊂ Z ⊂ H(Uμ (X)). Proof. Clearly X ⊂ Z. So, let F be a Bourbaki-Cauchy filter in (X, μ) and ξ ∈ sμ X a cluster point of F. Consider the filter H in X generated by the family {F ∩ V : F ∈ F, V is a neighborhood of ξ in sμ X}. Then H is also Bourbaki-Cauchy in (X, μ), since it contains F. Taking into account that the identity mapping id : (X, μ) → (X, wUμ (X) ) is uniformly continuous, H is Bourbaki-Cauchy in (X, wUμ (X) ). Now, from Theorem 7, this filter must cluster in H(Uμ (X)). And we finish, noting that ξ is the only cluster point of H.
THE SAMUEL REALCOMPACTIFICATION
5
Now, we are going to study the problem of the Samuel realcompactness of a uniform space, and we will see that a Katˇetov-Shirota type theorem can be obtained where Bourbaki-completeness will play the role of completeness in the classical one. Recall that every uniform countable cover {Un : n ∈ Z} ∈ μ such that Un ∩ Um = ∅ whenever |n − m| > 1 belongs also to the weak uniformity wUμ (X) (see [14]). This kind of covers were called linear covers by Isbell in [14], and 2-finite covers by Garrido and Montalvo in [4]. Proposition 10. Let (X, μ) be a uniform space such that every uniform partition of (X, μ) has non-measurable cardinal. Then every Cauchy filter of (X, wUμ (X) ) is a Bourbaki-Cauchy filter of (X, μ). Proof. Let F be a Cauchy filter in (X, wUμ (X) ) and let U ∈ μ. Note that the family {St∞ (U, U) : U ∈ U} is a uniform partition of X. We are going to prove that there exists U ∈ U such that St∞ (U, U) ∈ F. Observe that we can fix representative elements Ui ∈ U, i ∈ I, such that X = {St∞ (Ui , U ) : i ∈ I} and St∞ (Ui , U) ∩ St∞ (Uj , U) = ∅ whenever i = j, i, j ∈ I. Next, define J := J ⊂ I : {St∞ (Ui , U ) : i ∈ J} ∈ F . We want to prove that J is an ultrafilter in I with the countable intersection property. Clearly, we have ∅∈ / J . Let J ⊂ I and suppose that J ∈ / J . In order to see that I − J ∈ J , consider ∞ W= {St (Ui , U ) : i ∈ J}, {St∞ (Ui , U ) : i ∈ I − J} . Since W is a uniform finite partition of X then W ∈ wUμ (X) . Now as F is a Cauchy filter of (X, wUμ (X) ) and {St∞ (Ui , U) : i ∈ J} ∈ / F, we have {St∞ (Ui , U ) : i ∈ I − J} ∈ F. Now we prove that J satisfies the countable intersection property. Take {Jn : n ∈ N} ⊂ J and suppose without loss of generality it is a strictly decreasing family. If {Jn : n ∈ N} = ∅, then {I − Jn : n ∈ N} forms a cover on I. Define the family of sets W = {Wn : n ∈ N} by Wn = {St∞ (Ui , U ) : i ∈ Jn−1 − Jn }, where J0 = I. Then, W is a countable uniform partition of X and therefore, W ∈ wUμ (X) . Since F is a ∞ ) then W ∈ F for some n ∈ N. Since W ⊂ {St (Ui , U ) : i ∈ I − Jn } then Cauchy filter of (X, w n n U (X) μ {St∞ (Ui , U) : i ∈ I− Jn } ∈ F, and therefore I − Jn ∈ J . But this is a contradiction because J is a filter and Jn ∈ J . Hence, {Jn : n ∈ N} = ∅ and J has the countable intersection property. By hypothesis, the uniform partition {St∞ (Ui , U ) : i ∈ I} of X, has non-measurable cardinal, and therefore the discrete space I is realcompact. Hence the ultrafilter J must be fixed, that is, there exists i0 ∈ I such that /F i0 ∈ J . But now we have {i0 } ∈ J and hence St∞ (Ui0 , U ) ∈ F. Note that {St∞ (Ui , U ) : i ∈ I − {i0 }} ∈ To complete the proof that F is Bourbaki-Cauchy in (X, μ), it suffices to find some m ∈ N such that Stm (Ui0 , U) ∈ F. We define a cover A = {An : n ∈ {0} ∪ N} of X as follows: A0 = {St∞ (Ui , U ) : i ∈ I − {i0 }}, A1 = St(Ui0 , U ), A2 = St2 (Ui0 , U ), n An = {V ∈ U : V ∩ St (Ui0 , U ) = ∅, V ∩ Stn−2 (Ui0 , U ) = ∅} if n ≥ 3. Then A is a countable uniform cover satisfying that Ai ∩ Aj = ∅ whenever |i − j| > 1 and hence, A ∈ wUμ (X) . /F Again, since F is Cauchy in (X, wUμ (X) ) there exists some n ∈ {0} ∪ N such that An ∈ F. We have A0 ∈ and hence n = 0. Since An ⊂ Stn+1 (Ui0 , U), then Stn+1 (Ui0 , U ) ∈ F. Theorem 11. Let (X, μ) be a uniform space such that every uniform partition of X has non-measurable cardinal. Then, (X, μ) is Bourbaki-complete if and only if (X, wUμ (X) ) is complete. Proof. First, suppose that (X, wUμ (X) ) is complete then, from Theorem 8, it is Bourbaki-complete. Now, as the uniformity μ is finer than wUμ (X) , then (X, μ) will be also Bourbaki-complete. Reciprocally, suppose (X, μ) is Bourbaki-complete and let F be a Cauchy filter in (X, wUμ (X) ). From Proposition 10, we have that F is a Bourbaki-Cauchy filter in (X, μ), and hence it clusters. Since any Cauchy filter with a cluster point converges, (X, wUμ (X) ) is complete. We have already all the ingredients in order to establish our main result in this section.
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
ˇtov-Shirota type theorem) Let (X, μ) be a uniform space. Then (X, μ) is Samuel Theorem 12. (Kate realcompact if and only if (X, μ) is Bourbaki-complete and every uniformly discrete subspace of X has nonmeasurable cardinal. Proof. If (X, μ) is Samuel realcompact then it is realcompact and hence every discrete closed subspace has nonmeasurable cardinal [9]. In particular, every uniformly discrete subspace must have non-measurable cardinal since it is in addition closed. On the other hand, from the identity X = H(Uμ (X)), it follows that (X, wUμ (X) ) coincides with its completion and then (X, wUμ (X) ) is complete and, by Theorem 11, (X, μ) is Bourbakicomplete. Note that we can apply that theorem because every uniform partition produces a uniformly discrete subspace with the same cardinal. Conversely, again from Theorem 11, the Bourbaki-completeness of (X, μ) together with the assumption of non-measurability imply the completeness of (X, wUμ (X) ). Therefore X = H(Uμ (X)), and this means that (X, μ) is Samuel realcompact. Remark 13. It is clear that in Theorem 12 only the non-measurability of the uniform partitions is needed and not the (stronger) condition of non-measurability of the uniformly discrete subspaces. On the other hand, the condition of non-measurable cardinality can not be deleted in the three previous results. Indeed, if X is a set with a measurable cardinal endowed with the uniformity μ given by the 0-1 metric, then the corresponding uniform (metric) discrete space (X, μ) is Bourbaki-complete ([5]), but not Samuel realcompact since it is not even realcompact. In this case X = υX = H(U μ(X)), (X, wUμ (X) ) is not complete, and there exist Cauchy filters in (X, wUμ (X) ) that are not Bourbaki-Cauchy filters in (X, μ). We must notice now that there is a similar precedent of the above Katˇetov-Shirota type theorem in a paper by Rice and Reynolds [19]. In fact, in this work the authors characterized the completeness of (X, wUμ (X) ) by means of the uniformity generated by all the star-finite uniform covers of (X, μ). Next, we are going to explain the similarities between both results. Recall that a cover A of a set X is said to be star-finite if every element A ∈ A meets at most finitely many A ∈ A. For every uniform space (X, μ) the family of all star-finite uniform covers generates a uniformity on X compatible with the topology induced by μ [15]. We will denote this uniformity by sf μ. Note that, since R with the usual uniformity has a base by star-finite (in fact, linear) covers, then wUμ (X) sf μ μ. Lemma 14. Let (X, μ) be a uniform space. Then, (a) Every Cauchy filter of (X, sf μ) is a Bourbaki-Cauchy filter of (X, μ). (b) Every Bourbaki-Cauchy ultrafilter of (X, μ) is a Cauchy ultrafilter of (X, sf μ). Proof. (a) Let F be a Cauchy filter of (X, sf μ) and let U ∈ μ. Next, we produce a star-finite uniform cover as follows. As in the proof of Proposition 10, let {St∞ (Ui , U ), i ∈ I} the uniform partition of X generated by U. Fix i ∈ I, write A0 (Ui ) = Ui = St0 (Ui , U ), and for every n ∈ N let An (Ui ) = {V ∈ U : V ⊂ Stn (Ui , U ), V − Stn−1 (Ui , U ) = ∅}. Then, it is easy to see that the family A(U) = {An (Ui ), i ∈ I, n ≥ 0} ∈ sf μ. By hypothesis there exist some i ∈ I and n ≥ 0 such that An (Ui ) ∈ F. Since An (Ui ) ⊂ Stn (Ui , U ), then Stn (Ui , U ) ∈ F and we can deduce that F is in fact a Bourbaki-Cauchy filter of (X, μ). (b) Let F be a Bourbaki-Cauchy ultrafilter of (X, μ). Then it is clear that F is also a Bourbaki-Cauchy ultrafilter of (X, sf μ). Now, let U ∈ sf μ be a star-finite uniform cover of (X, μ). Then, for some m ∈ N and U ∈ U , Stm (U, U) ∈ F. Since U is star-finite, we can choose finitely many Ui ∈ U, i = 1, ..., k, such that k Stm (U, U) ⊂ i=1 Ui . Since F is an ultrafilter then Ui ∈ F, for some i = 1, ..., k, and therefore F is a Cauchy ultrafilter in (X, sf μ). Proposition 15. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). Then Z is (topologically) homeomorphic to the completion of (X, sf μ). s Proof. Let (X, f μ) denote the completion of (X, sf μ). Since wUμ (X) sf μ μ then, by Theorems 3 and 4, = Z. it is clear that X is a (topological) subspace of sμ X. We are going to see that X Firstly, by Lemma 14, it is clear that X ⊂ Z. Conversely, let F be a Bourbaki-Cauchy filter in (X, μ) and ξ ∈ Z a cluster point of F. Consider an ultrafilter H in X containing the family {F ∩ V : F ∈ F, V is a neighborhood of ξ in sμ X}. Then, H is also Bourbaki-Cauchy in (X, μ) because it contains F. Again, by Lemma 14, H is Cauchy in (X, sf μ) and then ξ ∈ X.
THE SAMUEL REALCOMPACTIFICATION
7
Theorem 16. A uniform space (X, μ) is Bourbaki-complete if and only if (X, sf μ) is complete. In particular, every complete uniform space with a base of star-finite uniform covers is Bourbaki-complete. Examples of uniformities having a star-finite base are, for instance, the weak uniformities wL generated by a family of functions L. More precisely, these weak uniformities have star-finite and countable bases composed by finite intersections of linear covers. In fact, the intersection of a finite number of covers of the form {f −1 ((n − 1) · ε, (n + 1) · ε), n ∈ Z}, for f ∈ L and ε > 0. Observe that the finite intersection of countable star-finite covers is still countable and star-finite. Thus, the results of Theorem 7 and Theorem 8 can be considered as particular cases of last result. Having in mind that in a uniform space every uniform cover has non-measurable cardinality if and only if every uniformly discrete subspace has non measurable cardinality [8], the above Theorem 16 is the link between our Katˇetov-Shirota result (Theorem 12) and the following result by Rice and Reynolds. Theorem 17. (Reynolds-Rice [19]) Let (X, μ) be a uniform space such that every uniform cover has nonmeasurable cardinal. Then (X, wUμ (X) ) is complete if and only if (X, sf μ) is complete. We point out now some slight difference between our result and the result of Rice and Reynolds. It is clear that also, as in Theorem 12, only the non-measurability of the uniform partitions is needed in last Theorem 17, and not the (stronger) condition of the non-measurability of the cardinality of the uniform covers. The next result characterizes precisely the property of a uniform space that every uniform partition has nonmeasurable cardinality which is not in general equivalent to satisfy that every uniformly discrete subspace has non-measurable cardinality. However, this equivalence is true whenever we consider the uniformity sf μ. Theorem 18. Let (X, μ) be a uniform space and Z the subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, μ). The following statements are equivalent: (1) Every uniform partition of (X, μ) has non-measurable cardinal. (2) Z = H(Uμ (X)). (3) Z is realcompact. (4) Every uniformly discrete subspace of (X, sf μ) has non-measurable cardinal. (5) Every uniform partition of (X, sf μ) has non-measurable cardinal. Proof. (1) ⇒ (2). By Proposition 9, we have Z ⊂ H(Uμ (X)). The converse follows from Proposition 10. (2) ⇒ (3). This implication is trivial. (3) ⇒ (4). By Proposition 15, Z is homeomorphic to the completion of (X, sf μ). Since every uniformly s discrete subspace of (X, sf μ) is a closed discrete subspace of the completion (X, f μ), then it follows that every uniformly discrete subspace of (X, sf μ) has non-measurable cardinal. (4) ⇒ (5). This follows taking into account that every uniform partition determines a uniformly discrete subspace with the same cardinal. (5) ⇒ (1). It is clear since (X, μ) and (X, sf μ) share the same uniform partitions. Note that the last result provides another interesting description of the Samuel realcompactification for those uniform spaces fulfilling condition (1). Namely, the Samuel realcompactification of these spaces are formed by the cluster points in the Samuel compactification of their Bourbaki-Cauchy filters, or equivalently, it is exactly the completion of (X, sf μ). Note that there are many spaces with this property, for instance, connected spaces, or more generally uniformly connected, separable, Lindel¨ of, and many other spaces. Examples of uniform spaces such that every uniform partition has non-measurable cardinal but having a uniform cover of measurable cardinality are easy to find, just think of connected spaces containing a uniformly discrete subspace of measurable cardinal as, for instance, the metric hedgehog H(κ) or any p (κ) where κ is a measurable cardinal. Observe also that the equivalence between (3) and (4) in last theorem reminds of the result in [9, Theorem 15.21] that asserts that the completion of a uniform space is realcompact if and only if every uniform discrete subspace has non-measurable cardinal. Note that for the uniformity sf μ we can even consider the weaker condition appearing in (5). But, in general, we can not change uniform partitions for uniformly discrete subspaces. We finish this section recalling a result of Katˇetov-Shirota type given by Huˇsek and Pulgar´ın in the setting of uniform spaces with weak uniformities ([13]). They define a uniform space (X, μ) to be uniformly realcomplete when (X, wUμ X ) is complete. Thus, in that paper, the following result is proved for the particular case of uniformly 0-dimensional spaces, where a uniform space is uniformly 0-dimensional if it has a base for the uniformity composed by partitions (for instance, every uniformly discrete space).
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
Theorem 19. (Huˇ sek-Pulgar´ın [13]) A uniformly 0-dimensional uniform space is uniformly realcomplete if and only if the space is complete and it does not have uniformly discrete subspaces of measurable cardinality. Proof. The proof follows easily from Theorems 12 and 16.
4. Uniformities with countable uniform covers In his famous paper [21], Shirota proved that the Hewitt-Nachbin realcompactification υX of a space X is homeomorphic to the completion of (X, eu) where eu is the uniformity having as a base all the normal covers of X. Thus he was proving that the completion of (X, wC(X) ) and the completion of (X, eu) are (topologically) equivalent even if the uniformities wC(X) and eu are not. Later on, Isbell proved (see [15]) a similar result in the frame of the so called locally fine uniform spaces ([8]). Recall that to be locally fine is equivalent to the more intuitive notion of being subfine, i.e., to be a uniform subspace of a fine space [18]. The result of Isbell states that for a locally fine uniform space (X, μ) every Cauchy filter in (X, wUμ(X) ) is Cauchy in (X, eμ), where eμ is the compatible uniformity on X having as a base all the countable covers of μ ([8]). Then we can say now that, for a locally fine uniform space (X, μ), the completion of (X, eμ) is homeomorphic to its Samuel realcompactification H(Uμ (X)). In particular, since every fine space is locally fine, we can apply the last result to the fine uniformity u on a Tychonoff space X. Thus, we can deduce easily the above result of Shirota, because for fine uniform spaces, every continuous function is uniformly continuous. Observe that, in general, locally fine uniform spaces do not satisfy that every continuous function is uniformly continuous. In fact, locally fine uniform spaces are characterized as those uniform spaces satisfying that every uniformly locally uniformly continuous function into a metric space is uniformly continuous [18]. Therefore, for locally fine uniform spaces, the Hewitt-Nachbin realcompactification and the Samuel realcompactification are not necessarily equivalent. For instance, this is the case of every totally bounded realcompact and non-compact uniform space, as the real interval (0, 1) with the usual metric. That the completion of (X, eμ) is a realcompactification of X follows from the fact that it is also complete with the uniformity having as a base all the countable uniform covers of its fine uniformity, and therefore, by the Shirota result, it is realcompact. In [19], Rice and Reynolds characterize those uniform spaces (X, μ) satisfying that (X, eμ) is complete. Precisely, they prove that whenever every uniform cover of the uniform space (X, μ) has non-measurable cardinal, then (X, eμ) is complete if and only if (X, pf μ) is complete, where pf μ denotes the compatible uniformity on X having as a base all the point-finite covers of μ. Recall that a cover U is point-finite if every x ∈ X just belongs to finitely many elements of U . In the next example we show that, in general, the completion of (X, eμ) as well as the completion of (X, pf μ) are not homeomorphic to the Samuel realcompactification H(Uμ (X)). To that purpose we need to use some results and concepts from [5]. Recall that a uniform space (X, μ) is Bourbaki-bounded if for every U ∈ μ there exists m ∈ N and finitely k many Ui ∈ U , i = 1, ..., k, such that X = i=1 Stm (Ui , U ). Observe that it was precisely the notion of Bourbaki-boundedness that originated the study of Bourbaki-Cauchy filters, nets and sequences, as well as Bourbaki-completeness (see [5] and [6]). In particular in [6] it is proved that a uniform space is compact if and only if it is Bourbaki-bounded and Bourbaki-complete. Example 20. Let (X, d) be a separable non-compact complete metric space which is Bourbaki-bounded for the metric uniformity. For instance the metric hedgehog H(ℵ0 ) of countably many spines, or any non-compact closed and bounded subset of the classical Hilbert space 2 . Then the metric uniformity given by d coincides with the uniformity eμd by the Lindel¨of property. Therefore (X, eμd ) is complete. However, by Theorem 11, (X, wUd (X) (X)) is not complete because (X, d) fails to be Bourbaki-complete. Otherwise, since (X, d) is Bourbaki-bounded, X would be compact, which is false. The above example shows that there are uniform spaces such that (X, pf μ) is complete but which are not Bourbaki-complete. On the other hand, the completion of (X, eμ) is not necessarily homeomorphic to the Samuel realcompactification H(Uμ (X)) because completeness is not enough. In fact, we need Bourbakicompleteness as the following two results show. Note that sf (eμ) and e(sf μ) will denote respectively the uniformity having as a base all the star-finite covers of eμ and all the countable covers of sf μ. Theorem 21. Let (X, μ) be a uniform space. The following spaces are (topologically) homeomorphic:
THE SAMUEL REALCOMPACTIFICATION
(1) (2) (3) (4)
9
H(Uμ (X)). The subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, eμ). The completion of (X, sf (eμ)). The subspace of sμ X of all the cluster points of the Bourbaki-Cauchy filters of (X, e(sf μ)).
Proof. Applying Theorem 3 several times, we have that H(Uμ (X)) = H(Ueμ (X)) = H(Usf μ (X)) = H(Usf (eμ) (X)) = H(Ue(sf μ) (X)). The same is true for the Samuel compactification. Since it is not difficult to see that every uniform partition of (X, eμ) is countable the equivalence of (1) and (2) follows from Theorem 18. Analogously the equivalence of (1) and (4) follows also at once. Finally, the equivalence of (2) and (3) is just Theorem 15. Theorem 22. Let (X, μ) be a uniform space. The following statements are equivalent: (1) X is Samuel realcompact. (2) (X, eμ) is Bourbaki-complete. (3) (X, sf (eμ)) is complete. (4) (X, e(sf μ)) is complete. Proof. We only need to prove (4) ⇒ (1). If (X, e(sf μ)) is complete then (X, sf μ) is complete since the identity map id : (X, e(sf μ)) → (X, sf μ) is uniformly continuous. In addition, X is a realcompact space since, as we have said before, the completion of every uniformity of type eμ is a realcompactification of X. Thus, every uniform cover of (X, sf μ) has non-measurable cardinal and then, by Theorem 17, X is Samuel realcompact. Remark 23. In condition (4) of Theorem 21, it would be interesting to be able to consider the completion of (X, e(sf μ)) instead of the set of cluster points of its Bourbaki-Cauchy filters, as it happens in condition (4) of Theorem 22. However, since we cannot assure that the uniformity e(sf μ) has a star-finite basis we can not refine the result (observe that the same question is noted in [19]). Thus, according to Theorem 22, we wonder if the difference between above results lies in the non-measurability of the cardinality of the uniform covers. Nevertheless, for uniformly 0-dimensional we have that e(sf μ) = eμ = sf (eμ), because whenever (X, μ) is uniformly 0-dimensional then (X, eμ) is also uniformly 0-dimensional. Acknowledgements We would like to thank the referee for his/her careful revision and interesting suggestions. References [1] J. Bors´ık, Mappings preserving Cauchy nets, Tatra Mt. Math. Publ. 19 (2000) 63-73. [2] I.S. G´ al, Proximity relations and precompact structures. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 62 (1959) 304-326. [3] M.I. Garrido and J.A. Jaramillo, Homomorphism on function lattices, Monatsh. Math. 141 (2004) 127-146. [4] M.I. Garrido and F. Montalvo, Countable covers and uniform closure, Rend. Istit. Mat. Univ. Trieste 30 (1999) 91-102. [5] M.I. Garrido and A.S. Mero˜ no, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733-758. [6] M.I. Garrido and A.S. Mero˜ no, On paracompactness, completeness and boundedness in uniform spaces, Topology Appl. 203 (2016) 98-107. [7] M.I. Garrido and A.S. Mero˜ no, The Samuel realcompactification of a metric space, J. Math. Anal. Appl. 456 (2017) 1013-1039. [8] S. Ginsburg and J.R. Isbell, Some operators on uniform spaces, Trans. Amer. Math. Soc. 93 (1959) 145-168. [9] L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics 43, Springer-Verlag, New York, 1976. [10] L. Google and M. Megrelishvili, Semigroup actions: proximities, compactifications and normality, Topology Proc. 35 (2010) 37-71. [11] A.W. Hager, D.G. Johnson, A note on certain subalgebras of C(X), Canadian J. Math. 20 (1968) 389-393. ˇ [12] A. Hohti, H. Junnila and A.S. Mero˜ no, On strongly Cech-complete spaces, manuscript. [13] M. Huˇsek and A. Pulgar´ın, Banach-Stone-like theorems for lattices of uniformly continuous functions, Quaest. Math. 35 (2012) 417-430. [14] J.R Isbell, Euclidean and weak uniformities, Pacific J. Math 8 (1958) 67-86. [15] J.R Isbell, Uniform spaces, Math. Surveys 12, Amer. Math. Soc. 1964. [16] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps, Marcel Dekker, New York, 1989. [17] J. Mack, M. Rayburn and G. Woods, Lattices of topological extensions, Trans. Amer. Math. Soc. 189 (1974) 163-174. [18] J. Pelant, Locally fine uniformities and normal covers, Czechoslovak Math. J. 37 (1987) 181-187. [19] G.D. Reynolds and M.D. Rice, Completenes and covering properties of uniform spaces, Quart. J. Math. Oxford 29 (1978) 367-374. [20] P. Samuel, Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948) 100-132. [21] T. Shirota, A class of topological spaces, Osaka Math. J. 4 (1952) 23-40. [22] S. Willard, General Topology, Dover Publications, Mineola, NY, 2004.
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˜ M. ISABEL GARRIDO AND ANA S. MERONO*
´ tica Interdisciplinar (IMI), Departamento de Geometr´ıa y Topolog´ıa, Universidad (Garrido) Instituto de Matema Complutense de Madrid, 28040 Madrid, Spain E-mail address: [email protected] ´ lisis Matema ´ tico, Universidad Complutense de Madrid, 28040 Madrid, Spain (Mero˜ no*) Departamento de Ana E-mail address: [email protected]