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Sequential properties of lexicographic products
Topology and its Applications 168 (2014) 94–102
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Topology and its Applications www.elsevier.com/locate/topol
Sequential properties of lexicographic products F. Azarpanah, M. Etebar Department of Mathematics, Chamran University, Ahvaz, Iran
a r t i c l e
i n f o
MSC: 54F05 Keywords: Lexicographic product Almost P -point P + (P − )-point Sequential space Sequentially connected space Quasi F -space
a b s t r a c t In this article, using the characterization of almost P -points of a linearly ordered topological space (LOTS) in terms of sequences, we observe that in the category of linearly ordered topological spaces, quasi F -spaces and almost P -spaces coincide. This coincidence gives examples of quasi F -spaces with no F -points. We also use the characterization of sequentially connected LOTS in terms of almost P -points to show that whenever each LOTS Xn has first and last elements, the lexicographic product ∞ each n=1 Xn is sequentially connected if and only if each Xn is. Whenever X Xn is a LOTS without first and last elements, thenit is shown that ∞ n is n=1 always a sequential space. The lexicographic product α<ω1 Xα , where ω1 is the first uncountable ordinal, isalso investigated and it is shown that if each Xα contains at least two points, then α<ω1 Xα is always an almost P -space (a quasi F -space) but it is neither sequential nor sequentially connected. Using this lexicographic product, we give an example of a quasi F -space in which the set of F -points and the set of ω1 , does not have first and last non-F -points are dense. Whenever each Xα , α < elements, we show that the lexicographic product α<ω1 Xα is a P -space without isolated points. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For each f in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff (Tychonoff) space X, Z(f ) = {x ∈ X: f (x) = 0} is called a zeroset and X \ Z(f ) is called a cozeroset. A point x in a completely regular Hausdorff space X is said to be a P -point (an almost P -point) if every Gδ -set or every zeroset containing x is a neighborhood of x (has a nonempty interior) and X is called a P -space (an almost P -space) if every point of X is a P -point (an almost P -point), see [1,6,8] for more details and properties of these spaces. A point x in a completely regular Hausdorff space X is said to be an F -point if the ideal Ox = {f ∈ C(X): x ∈ intβX clβX Z(f )} is prime, where βX is the Stone–Čech compactification of X. A space X is called an F -space if every point of βX is an F -point. It is well-known that a space X is an F -space if and only if every cozeroset is C ∗ -embedded in X, see Theorem 14.25 in [6]. A quasi F -space E-mail addresses: [email protected] (F. Azarpanah), [email protected] (M. Etebar). http://dx.doi.org/10.1016/j.topol.2014.02.011 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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is a space in which every dense cozeroset is C ∗ -embedded, see [3] for more properties and characterizations of quasi F -spaces. Throughout this paper, topological spaces under consideration are linearly ordered spaces and the reader is referred to [4] and [6] for undefined terms and notations. A linearly ordered topological space (LOTS) is a triple (X, τ, <), where < is a linear ordering of the set X and τ is the usual open interval topology defined by <. For every x in a LOTS X, x+ (x− ) denotes the immediate successor (predecessor) of x, if it exists and the set of all points y ∈ X satisfying y > x (y < x) is denoted by (x, →) ((←, x)). A subset S of a LOTS X is said to be cofinal if, for every x ∈ X, there exists s ∈ S such that s x. It is well-known that every LOTS is a normal Hausdorff space, see [4] and a LOTS is connected if and only if it is Dedekind-complete (i.e., every nonempty subset with an upper bound has a supremum or equivalently, every nonempty subset with a lower bound has an infimum) and it does not have consecutive elements, see [6, 3O]. Whenever (W, ) is a well-ordered set and for every α ∈ W , (Xα , α ) is a linearly ordered set, then the lexicographic product of the family {(Xα , α ): α ∈ W } is the set of all points x = (xα )α∈W with the order < defined as follows: if x = (xα )α∈W and y = (yα )α∈W have x = y, let σ be the first element of the set {α ∈ W : xα = yα } in the ordering ≺ and then define x < y provided xσ <σ yσ . In what follows, we shall omit putting subscripts on the various orderings because context will make clear which ordering is meant. The lexicographic product will be denoted by α∈W Xα . In this paper we consider two familiar well-ordered sets as index sets: ω1 , the set of all countable ordinals and N, the set of all natural numbers. We also assume that each factor space Xα in the lexicographic product α∈W Xα contains at least two elements. In a LOTS X, we call a point x a P + -point if every Gδ -set containing x contains an interval [x, y) for some x < y ∈ X. P − -points will be defined similarly. If we define O+ (x) = {f ∈ C(X): [x, y) ⊆ Z(f ), for some y ∈ X, x < y} and similarly O− (x) = {f ∈ C(X): (y, x] ⊆ Z(f ), for some y ∈ X, x > y}, then clearly x is a P + -point (P − -point) if and only if O+ (x) = Mx (O− (x) = Mx ), where Mx = {f ∈ C(X): f (x) = 0}. A LOTS X is said to be P + -space (P − -space) if every point of X is a P + -point (P − -point) and it is clear that a LOTS X is a P -space if and only if it is both a P + -space and a P − -space. The space of countable ordinals is an example of a P + -space which is not P − -space, whence it is not a P -space, see [4] and [6] for more details of the space of ordinals. A subset A of a topological space X is said to be sequentially open if whenever {xn } is a sequence in X that converges to a point in A, then {xn } is eventually in A. Similarly a subset B of X is called sequentially closed if whenever there is a sequence in B that converges to some point x ∈ X, then x ∈ B. We recall that a space X is a sequential space if every sequentially closed (open) subset of X is closed (open). It is well-known that a LOTS is sequential if and only if it is first countable, see [4] and [9]. A space X is said to be sequentially connected if X cannot be expressed as the union of two nonempty disjoint sequentially closed (open) sets. Our aim in this article is to reveal the importance of almost P -points and their role in characterizing of some sequential properties of lexicographic products. In Section 2, we give some preliminary results and cite some facts from [2]. By the characterization of almost P -points of a LOTS in terms of sequences, we observe that quasi F -spaces and almost P -spaces coincide in the category of linearly ordered spaces. Using the characterization of a sequential LOTS and a sequentially connected LOTS in terms of almost P -points, we observe that a LOTS is sequentially connected if and only if it is connected without any almost P -points. In Sections 3 and 4 we study the lexicographic product α∈W Xα , where the well-ordered set W is either the set of natural numbers N or the set of countable ordinals ω1 . We observe that the existence of almost P -points of the lexicographic product n∈N Xn depends on first and last elements of the factor spaces. Using this lexicographic product, we give examples of P + -spaces and P − -spaces in which the set of almost P -points is dense. Whenever the set of countable ordinals is considered as an index set, we will show that the lexicographic product is always an almost P -space and whenever each factor space does not have first and last elements, then this lexicographic product will be a P -space without isolated points. Sequential
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connectedness of the lexicographic countable products is also investigated and it is shown that if each Xn ∞ has first and last elements, then n=1 Xn is sequentially connected if and only if each Xn is. Whenever ∞ each Xn is a LOTS without first and last elements, we show that the lexicographic product n=1 Xn is always a sequential space. Finally, using lexicographic products and the coincidence of quasi F -spaces and almost P -spaces in the category of linearly ordered spaces, we give an example of a quasi F -space (an almost P -space) without F -points (P -points) and an example of a quasi F -space (an almost P -space) in which the set of F -points (P -points) and the set of non-F -points (non-P -points) are dense. 2. Preliminaries In this section, using the characterization of almost P -points in terms of sequences, we observe that a LOTS is an almost P -space if and only if it is a quasi F -space. By the characterization of almost P -points in terms of sequences and the characterization of a sequentially connected LOTS in terms of almost P -points in this section together with the results in Section 3, enable us to prove the main results of the article in Section 4. In [6, 5O], it is shown that a point x of a LOTS is a P -point if and only if it is neither the limit of an increasing sequence nor the limit of a decreasing sequence. By the following lemma, we also characterize almost P -points of a LOTS in terms of sequences. We note that whenever x is not the first (last) point of a LOTS X and x is not the limit of an increasing (a decreasing) sequence in X and moreover x− (x+ ) does not exist, then every Gδ -set containing x, contains a nonempty interval (α, x] ([x, α)) for some α < x (α > x). On the other hand, for a point x ∈ X, whenever x− or x+ exists and x is both the limit of an increasing sequence and the limit of a decreasing sequence in X, then either x is an isolated point or the singleton {x} is a Gδ -set with empty interior. Using this argument the proof of the following result is evident. Lemma 2.1. A non-isolated point x in a LOTS X is an almost P -point if and only if one of the following statements holds. (a) x− does not exist and x is not the limit of an increasing sequence. (b) x+ does not exist and x is not the limit of a decreasing sequence. We note that whenever x is the first (last) element of X, then the condition (b) ( (a)) holds. We also cite the following lemma from [3] which is needed in the proof of the next theorem. Lemma 2.2. Let X be a completely regular Hausdorff quasi F -space and x ∈ X be a Gδ -point. Then x is not the limit of a distinct sequence in X. In particular, if X is a quasi F -space satisfying the first axiom of countability, then it is discrete. Now we prove the coincidence of quasi F -spaces and almost P -spaces in the category of linearly ordered spaces; for a different proof, see [2]. Theorem 2.3. A LOTS is an almost P -space if and only if it is a quasi F -space. Proof. Since the only dense cozeroset in an almost P -space X is X itself, clearly every almost P -space is a quasi F -space. Now let X be a quasi F -space and x ∈ X. If x+ and x− exist, then x is an isolated point and hence it is an almost P -point. Therefore suppose that one of x+ and x− does not exist, say x+ . If x is the limit of a decreasing sequence, then x is a Gδ -point and by Lemma 2.2, x is again an isolated point. Whenever x is not the limit of a decreasing sequence, then x is an almost P -point, by Lemma 2.1. Now
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suppose that both x+ and x− do not exist. If x is the limit of a decreasing sequence and it is also the limit of an increasing sequence, then x is again a Gδ -point and hence x is an isolated point, by Lemma 2.2. Whenever one of these cases does not happen, x should be an almost P -point, by Lemma 2.1. Therefore X is an almost P -space and we are through. 2 Now using Theorem 2.3, examples of linearly ordered almost P -spaces with no P -points (see Example 2.9 in [2] and the example given in the first paragraph of Section 4 in [8]) are also examples of quasi F -spaces with no F -points. It is well-known that every first countable space is a Fréchet space (a space X is called a Fréchet space if for every A ⊆ X and every x ∈ clX A, there exists a sequence of points of A converging to x) and every Fréchet space is a sequential space, see [4, Theorem 1.6.14]. The converse is also true for any LOTS, i.e., every sequential LOTS is first countable, see [9]. It is also easy to see that a LOTS X is sequential if and only if every point of X is a Gδ -point. Now using these well-known facts and Lemma 2.1, the following proposition is evident. Proposition 2.4. A LOTS X is a sequential space if and only if every almost P -point of X is an isolated point. Clearly every disconnected space is sequentially disconnected, but not conversely. In fact a connected space may be sequentially disconnected. The following lemma characterizes connected sequentially disconnected linearly ordered spaces in terms of almost P -points, see [2] for the proof. Corollary 2.5. A LOTS X is sequentially connected if and only if X is connected without any almost P -point. Using Proposition 2.4 and Corollary 2.5, the following corollary is evident. Corollary 2.6. If a LOTS is sequentially connected, then it is sequential. 3. P -points and almost P -points in lexicographic products In this section we investigate almost P -points and P -points of the lexicographic products. We note that n the existence of almost P -points and P -points of the lexicographic finite product i=1 Xi only depends on n the last factor space Xn . In fact if x = (x1 , x2 , . . . , xn ) ∈ i=1 Xi , then x− (x+ ) exists if and only if x− n (x+ P -point (P -point) if and only if xn is an almost P -point (P -point). n ) exists in Xn and x is an almost ∞ But for a lexicographic product n=1 Xn , we will observe that the existence of almost P -points depends on first and last elements of the factor spaces. For lexicographic products α<ω1 Xα , we show that its every point is an almost P -point. First we need the following lemma. Lemma 3.1. Suppose that for every α < ω1 , Xα is a LOTS with at least two points. Let X be the lexicographic product α<ω1 Xα and for each x ∈ X, define F (x) = {α < ω1 : xα is not the first element of Xα } and L(x) = {α < ω1 : xα is not the last element of Xα }. (a) If F (x) is cofinal in ω1 , then x is not the first element of X, x− does not exist and x is not the supremum of any countable subset of (←, x). (b) If L(x) is cofinal in ω1 , then x is not the last element of X, x+ does not exist and x is not the infimum of any countable subset of (x, →). Proof. (a) Clearly x cannot be the first element of X, for F (x) is cofinal. Now suppose that C is a countable subset of (←, x). For each y ∈ C, there is δy < ω1 such that for every α < δy , yα = xα and yδy < xδy . Since
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F (x) is cofinal and C is countable, there is some β ∈ F (x) with β > sup{δy : y ∈ C}. Since β ∈ F (x), we may choose zβ ∈ Xβ with zβ < xβ . Define zα = xα for each α ∈ ω1 \ {β}. Then y < z < x for every y ∈ C, so that x cannot be the supremum of the countable set C. A similar proof shows that x cannot have an immediate predecessor in X. (b) The proof is analogous. 2 We also need the following lemma in the sequel. Lemma 3.2. Suppose that for every n ∈ N, Xn is a LOTS with at least two points. Let X be the lexicographic ∞ product n=1 Xn and for each x ∈ X, define F (x) = {n ∈ N: xn is not the first element of Xn } and L(x) = {n ∈ N: xn is not the last element of Xn }. If F (x) (L(x)) is an infinite subset of N, then x is the limit of an increasing (a decreasing) sequence. Proof. Let L(x) be an infinite subset of N. Regard L(x) as an increasing subsequence {nk }∞ k=1 of integers ∞ and define a sequence f : N → n=1 Xn as follows: (f (k))n = xn , if n = nk and (f (k))n = ynk , if n = nk , where ynk ∈ Xnk and ynk > xnk . It is easy to see that f is a decreasing sequence and f → x. Whenever F (x) is infinite, the proof is similar. 2 Corollary 3.3. For each α < ω1 , let Xα be a LOTS with at least two points. Then every point of the lexicographic product α<ω1 Xα is an almost P -point. Proof. For any x ∈ X, at least one of the sets F (x) and L(x) defined in Lemma 3.1, is cofinal in ω1 . Hence each x ∈ X is an almost P -point by Lemmas 2.1 and 3.1. 2 Since linearly ordered quasi F -spaces and linearly ordered almost P -spaces coincide, and a point in a LOTS is an F -point if and only if it is a P -point, the following remark also gives an example of a quasi F -space in which both the set of F -points and the set of non-F -points are dense. Remark 3.4. If Xα is a LOTS, ∀α < ω1 , then the lexicographic product α<ω1 Xα is not necessarily a P -space. For example α<ω1 [0, 1] is an almost P -space by Corollary 3.3, but not every point of ω1 [0, 1] is a P -point. In fact both the set of P -points and the set of non-P -points of α<ω1 [0, 1] are dense. First we show that the set of P -points of α<ω1 [0, 1] is dense in α<ω1 [0, 1]. Let S=
(xα )α<ω1 ∈
[0, 1]: ∀β < ω1 , ∃δ > β such that xδ = 0, 1 .
α<ω1
For each x ∈ S, both F (x) and L(x) defined in Lemma 3.1 are cofinal. Hence by this lemma, x is neither the limit of an increasing nor the limit of a decreasing sequence. Now by 5O in [6], x is a P -point. On the other hand S is dense in α<ω1 [0, 1], for if a, b ∈ α<ω1 [0, 1], a < b, then there is smallest σ such that aσ = bσ , i.e., aσ < bσ . Now set x ∈ S such that aσ < xσ < bσ , xα = aα , ∀α < σ and xα ∈ (0, 1), ∀α > σ. Clearly x ∈ S ∩ (a, b), i.e., S is dense in α<ω1 [0, 1]. Next we consider the set T = (xα )α<ω1 ∈ [0, 1]: xδ ∈ (0, 1), and xβ = 0, ∀β > δ, for some δ < ω1 . α<ω1
We observe that every element of T is a non-P -point and T is dense in α<ω1 [0, 1]. It is easy to see that if x ∈ T , where xδ ∈ (0, 1) and xα = 0, ∀α > δ, then the sequence {xn } in α<ω1 [0, 1] such that xnδ = n−1 n xδ n and xα = xα , ∀α = δ converges to x. This means that x is not a P -point by 5O in [6]. Moreover, T is
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dense in α<ω1 [0, 1], for if x < y and β is the smallest index such that xβ = yβ (xβ < yβ ), then we choose z ∈ α<ω1 [0, 1], where zα = xα , ∀α < β, xβ < zβ < yβ and zα = 0, ∀α > β. Clearly z ∈ T ∩ (x, y), i.e., T is dense in α<ω1 [0, 1]. The following proposition shows that
α<ω1
Xα may be a P -space.
Proposition 3.5. Let Xα be a LOTS with at least two points, ∀α < ω1 and F (L) be the set of all α < ω1 such that Xα does not have first (last) element. If F and L are cofinal, then α<ω1 Xα is a P -space without isolated points. Proof. Clearly F ⊆ F (x) and L ⊆ L(x), ∀x ∈ X. Thus by Lemma 3.1 and 5O in [6], X is a P -space. Now it remains to show that α<ω1 Xα has no consecutive elements. In fact if x, y ∈ α<ω1 Xα and x < y, then there exists a smallest index β < ω1 such that xβ = yβ and hence xβ < yβ . Since F is cofinal, there is λ ∈ F such that λ > β and yλ is not the first element of Xλ . Consider z ∈ α<ω1 Xα , so that zλ < yλ and zα = yα , ∀α < λ. Clearly x < z < y and this implies that α<ω1 Xα has no consecutive elements, whence it has no any isolated point. 2 The following corollary shows that
α<ω1
R is a P -space without isolated points.
Corollary 3.6. Let Xα be a LOTS, ∀α < ω1 and each Xα does not have first and last elements. Then α<ω1 Xα is a P -space without isolated points. Using a proof similar to that of Proposition 3.5, we obtain a more general result as follows: Corollary 3.7. Let Xα be a LOTS, ∀α < ω1 and F (L) be the set of all α < ω1 such that Xα does not have first (last) element. (a) If F is cofinal, then α<ω1 Xα is a P − -space without isolated points. (b) If L is cofinal, then α<ω1 Xα is a P + -space without isolated points. It is well-known that a finite product of (Tychonoff) P -spaces is a P -space. But no infinite product of (Tychonoff) spaces of more than one point is a P -space, see [6, 4K]. In Proposition 3.5, we observed that the lexicographic product α<ω1 Xα may be a P -space. But in the following proposition, we show that ∞ the lexicographic product n=1 Xn of linearly ordered spaces of more than one point does not have any P -point. Proposition 3.8. If Xn is a LOTS with at least two elements, ∀n ∈ N, then the lexicographic product ∞ X = n=1 Xn never has an almost P -point (a P -point). Proof. For each x ∈ X, clearly one of the sets F (x) and L(x) defined in Lemma 3.2 is infinite. Now by Lemma 3.2, x is the limit of a sequence and this means that x is not an almost P -point (a P -point). 2 ∞ Proposition 3.8 states that the lexicographic product n=1 Xn does not have any P -point even each Xn ∞ is a P -space (discrete). By part (a) of the following proposition, the lexicographic product n=1 Xn may not contain any almost P -point. Parts (b) and (c) of the following proposition show that whenever each factor space Xn is a P -space with first (last) element and without consecutive elements, then the set of ∞ almost P -points of the lexicographic product n=1 Xn is dense. Part (a) of this proposition also shows that the existence of first and last elements of the factor spaces is essential for existence of almost P -points of ∞ n=1 Xn .
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Proposition 3.9. Let Xn be a LOTS with at least two points, ∀n ∈ N. (a) If F = {n ∈ N: Xn does not have first element} and L = {n ∈ N: Xn does not have last element} are ∞ infinite sets, then the lexicographic product n=1 Xn does not have any almost P -point. (b) If each Xn is a P + -space with last element and no element of Xn has immediate successor, then the ∞ set of almost P -points of the lexicographic product n=1 Xn is dense. (c) If each Xn is a P − -space with first element and no element of Xn has immediate predecessor, then the ∞ set of almost P -points of the lexicographic product n=1 Xn is dense. ∞ Proof. (a) For each x ∈ n=1 Xn , we have F ⊆ F (x) and L ⊆ L(x), hence F (x) and L(x) are infinite sets. Now by Lemma 3.2, x is the limit of increasing and decreasing sequences. This implies that x is not almost P -point by Lemma 2.1. ∞ (b) Let ln be the last element of Xn , ∀n ∈ N and consider A = {x ∈ n=1 Xn : xk = lk for some k ∈ ∞ N and xn = ln , ∀n > k}. We show that every point of A is an almost P -point and A is dense in n=1 Xn . + Let a ∈ A, where an = ln , ∀n > k and ak = lk . Since a+ k does not exist, a does not exist. On the other hand a is not the limit of a decreasing sequence. To see this let {y n } be a decreasing sequence and a < y n , n n ∀n ∈ N. Suppose mn is the smallest integer such that amn = ym (so, amn < ym ). Clearly mn k, n n ∀n ∈ N and this implies that there exists t k which is the smallest index such that at < ytnk for an infinite nk + sequence {nk }∞ at and hence there exists yt ∈ Xt such that for each k=1 . Since Xt is a P -space, yt ∞ nk n0 ∈ N, nk > n0 exists so that yt > yt > at . If we consider z ∈ n=1 Xn such that zt = yt and zn = an , ∀n = t, then a < z and for every n0 ∈ N, y nk > z for some nk > n0 , i.e., y n a. Now it remains to prove ∞ ∞ that A is dense in n=1 Xn . Let (b, c) be an interval in n=1 Xn and k be the smallest index such that ∞ bk = ck . So bk < ck and bi = ci , ∀i < k. We take x ∈ n=1 Xn , where xn = bn , ∀n < k, bk < xk < ck (note that no element of Xk has immediate successor) and xn = ln , ∀n > k. Thus x ∈ (b, c) ∩ A, i.e., A is dense ∞ in n=1 Xn . (c) The proof is similar. 2 4. Sequential properties of lexicographic products Theorem 2.2 in [7] shows that the countable Tychonoff product of sequentially connected spaces is sequentially connected. Theorem 4.1.1 in [5] also investigates the connectedness of lexicographic products. In this section, using characterization of a sequentially connected LOTS in terms of almost P -points in Corollary 2.5, we investigate the sequential connectedness of the lexicographic product spaces. First we need the following proposition. Although this proposition may be proved by Theorem 4.1.1 in [5], we give a direct proof. Proposition 4.1. Let Xn be a LOTS with first and last elements for each n ∈ N. Then the lexicographic ∞ product n=1 Xn is connected if and only if each Xn is. ∞ Proof. First suppose that n=1 Xn is connected and An is a nonempty subset of Xn . Let A be a subset of ∞ n=1 Xn containing the elements of the form (l1 , . . . , ln−1 , an , ln+1 , . . .), where an ∈ An and lm is the last element of Xm , ∀m = n. Since A is bounded above, it has a supremum. Let α = sup A, since α a, ∀a ∈ A, αi = li , ∀i < n and an αn , ∀an ∈ An . This implies that αn = sup An , i.e., Xn is Dedekind complete. Next, each Xn has no consecutive elements, for if an , bn ∈ Xn are consecutive elements of Xn and an < bn , ∞ then we may take x, y ∈ n=1 Xn such that xn = an , yn = bn , xm = ym , ∀m < n and xm = lm , ym = fm , ∞ ∀m > n, where fm is the first element of Xm . Then x and y are two consecutive elements of n=1 Xn which is a contradiction.
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∞ Conversely, let each Xn be connected and A ⊆ n=1 Xn . Suppose that A1 is the set of all a1 ∈ X1 such that a1 is the first component of an element of A. Since A1 is bounded above and X1 is Dedekind complete, α1 = sup A1 exists. By induction, let Ak be the set of ak ∈ Xk such that α1 , α2 , . . . , αk−1 and ak be the first k components of an element of A and take αk = sup Ak . If Ak = ∅, ∀k ∈ N, then α = (α1 , . . . , αk , . . .) = sup A. In fact if β < α and n is the smallest index such that βn < αn (so βi = αi , ∀i < n), then we take a ∈ A such that α1 , . . . , αn are the first n components of a. Therefore β < a α, i.e., α = sup A. Now let Ak = ∅ for some integer k and n be the smallest integer such that An = ∅ (so, Am = ∅, ∀m < n). Consider α = (α1 , . . . , αn−1 , fn , fn+1 , . . .), where fm is the first element of Xm , ∀m n. We show that α = sup A. First, if a ∈ A, then a α, for otherwise, a > α implies that there is k < n (note that An = ∅) such that ak > αk and ai = αi , ∀i < k which contradicts αk = sup Ak . Next, let β < α and k be the smallest index such that βk = αk , so βi = αi , ∀i < k and βk < αk . Now either k = n − 1 or k < n − 1 for βm ≮ fm , ∀m n. If k = n − 1, then for all an−1 ∈ An−1 , we have an−1 < αn−1 and since αn−1 = sup An−1 , there exists a∗n−1 ∈ An−1 such that βn−1 < a∗n−1 < αn−1 (note that αn−1 ∈ / An−1 for An = ∅). Since An−1 = ∅, we may take a ∈ A whose first n − 1 components are α1 , . . . , αn−2 and a∗n−1 . Clearly β < a α, i.e., α = sup A. If k < n − 1, we can consider a ∈ A such that am = αm , ∀m < n − 1, for An−1 = ∅. Clearly β < a < α, whence we have α = sup A. 2 Now we prove our main theorem in this section. Theorem 4.2. Suppose that Xn is a LOTS with first and last elements for each n ∈ N. Then the lexicographic ∞ product n=1 Xn is sequentially connected if and only if each Xn is. ∞ Proof. By Proposition 4.1 and Corollary 2.5, it is enough to show that n=1 Xn contains a non-isolated almost P -point if and only if for some k ∈ N, Xk contains a non-isolated almost P -point. First suppose that for some k ∈ N, Xk contains an almost P -point xk ∈ Xk which is not isolated point. Then one of the conditions (a) and (b) of Lemma 2.1 holds, say (a). Hence x− k does not exist and xk is not the limit of an increasing sequence. Whenever fm is the first element of Xm , ∀m ∈ N, then x = (f1 , . . . , fk−1 , xk , fk+1 , . . .) ∞ is an almost P -point of n=1 Xn . In fact if y < x, and m is the smallest integer for which ym = xm , i.e., ym < xm , then m should be k. Hence yk < xk and this implies that x− does not exist for x− k does not exist. Now if {xn } is an increasing sequence which converges to x, then xn < x, ∀n ∈ N implies that xnk < xk , ∀n ∈ N and xnm = fm , ∀m < k. Thus {xnk } is an increasing sequence in Xk which converges to xk , a contradiction. In case part (b) of Lemma 2.1 holds for xk , then we take x = (l1 , . . . , lk−1 , xk , lk+1 , . . .), where lm is the last element of Xm , ∀m = k and the remainder of the proof is similar. ∞ Next, suppose that n=1 Xn has an almost P -point x = (x1 , . . . , xk , . . .) and let part (a) of Lemma 2.1 holds. Since x is not the limit of an increasing sequence, there exists k ∈ N such that xm = fm , ∀m > k. For otherwise if there is a subsequence {nk } of integers such that xnk is not the first element of Xnk , then ∞ we may define a sequence h : N → n=1 Xn , with (h(k))nk = fnk and (h(k))m = xm , ∀m = nk . Clearly h is increasing and h → x. To see h → x, let y < x and m be the smallest index such that ym = xm (i.e., ym < xm and yn = xn , ∀n < m). Then for nk > m, we have h(k) ∈ (y, x]. This contradicts our hypothesis and hence x is of the form x = (x1 , . . . , xk , fk+1 , . . .), where fm is the first element of Xm , ∀m > k. Now it is not hard to see that xk is an almost P -point of Xk . In case part (b) of Lemma 2.1 holds for x, then x will be of the form x = (x1 , . . . , xk , lk+1 , . . .) for some k ∈ N, where lm is the last element of Xm , ∀m > k and similarly xk will be an almost P -point of Xk . 2 In the proof of Theorem 4.2, we have shown that whenever each Xn is a LOTS with first and last elements, ∞ then the lexicographic product n=1 Xn has a non-isolated almost P -point if and only if some factor space ∞ Xk has. This indicates that the lexicographic product n=1 Xn is not sequential if and only if some factor ∞ space Xk is not sequential, by Proposition 2.4. Equivalently, the lexicographic product n=1 Xn is not first
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countable if and only if some factor space is not first countable. Now as an immediate consequence of this fact, we have the following corollary. Corollary 4.3. Let Xn be a LOTS with first and last elements. Then the lexicographic product a sequential space if and only if each factor space Xn is.
∞ n=1
Xn is
∞ By Corollary 2.6, if the lexicographic product n=1 Xn is sequentially connected, then it is sequential. Whenever each factor space Xn is connected, the converse is also true. In fact, if the lexicographic product ∞ ∞ will be an isolated n=1 Xn is sequential, then every almost P -point of the lexicographic product n=1 Xn ∞ point by Proposition 2.4. But Proposition 3.8 states that the lexicographic product n=1 Xn does not ∞ contain any isolated point and this means that the lexicographic product n=1 Xn is sequentially connected by Corollary 2.5. Corollary 4.4. If each factor space Xn is a connected LOTS, then the lexicographic product sequential if and only if it is sequentially connected.
∞ n=1
Xn is
By Proposition 3.9(a) and Proposition 2.4, the following corollary is evident. Corollary 4.5. Let each Xn be a LOTS without first and last elements. Then the lexicographic product ∞ n=1 Xn is a sequential space. Corollary 4.6. The lexicographic product
α<ω1
Xα is never a sequential space.
Proof. By Corollary 3.3, every point of α<ω1 Xα is an almost P -point. Now consider x ∈ α<ω1 Xα such that for some λ < ω1 , xα is not the last element of Xα , ∀α > λ. If y ∈ α<ω1 Xα and x < y, then there is β < ω1 so that xβ < yβ and xα = yα , ∀α < β. Take σ < ω1 as the smallest element of the set {α < ω1 : α > β and xα is not the last element of Xα }. Now we consider z ∈ α<ω1 Xα such that zα = xα , ∀α < σ and zσ > xσ . Clearly x < z < y and this shows that x is not an isolated point of α<ω1 Xα which means by Proposition 2.4 that α<ω1 Xα is not sequential. 2 Finally, by Corollaries 2.6 and 4.6, the following result is evident. Corollary 4.7. The lexicographic product
α<ω1
Xα is never sequentially connected.
Acknowledgement We would like to thank the referee. His/her careful reading of the manuscript combined with some very useful comments have hopefully improved the quality of the paper. References [1] F. Azarpanah, O.A.S. Karamzadeh, A. Rezai Aliabad, z ◦ -ideals in C(X), Fundam. Math. 160 (1999) 15–25. [2] F. Azarpanah, M. Etebar, Linearly ordered P ± -spaces, JP J. Geom. Topol. 10 (2) (2010) 99–112. [3] F. Dashiell, A. Hager, M. Henriksen, Order-Cauchy completion of rings and vector lattices of continuous functions, Can. J. Math. XXXII (3) (1980) 657–685. [4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [5] M.J. Faber, Metrizability in generalized ordered spaces, PhD dissertation, Vrije University te Amsterdam, Tract 53, Math. Centre Tracts, Amsterdam, 1974. [6] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [7] Q. Huang, S. Lin, Notes on sequentially connected spaces, Acta Math. Hung. 110 (1–2) (2006) 159–164. [8] R. Levy, Almost P -spaces, Can. J. Math. 2 (1977) 284–288. [9] P.R. Meyer, Sequential properties of ordered topological spaces, Compos. Math. 21 (1969) 102–106.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Sequential properties of lexicographic products F. Azarpanah, M. Etebar Department of Mathematics, Chamran University, Ahvaz, Iran
a r t i c l e
i n f o
MSC: 54F05 Keywords: Lexicographic product Almost P -point P + (P − )-point Sequential space Sequentially connected space Quasi F -space
a b s t r a c t In this article, using the characterization of almost P -points of a linearly ordered topological space (LOTS) in terms of sequences, we observe that in the category of linearly ordered topological spaces, quasi F -spaces and almost P -spaces coincide. This coincidence gives examples of quasi F -spaces with no F -points. We also use the characterization of sequentially connected LOTS in terms of almost P -points to show that whenever each LOTS Xn has first and last elements, the lexicographic product ∞ each n=1 Xn is sequentially connected if and only if each Xn is. Whenever X Xn is a LOTS without first and last elements, thenit is shown that ∞ n is n=1 always a sequential space. The lexicographic product α<ω1 Xα , where ω1 is the first uncountable ordinal, isalso investigated and it is shown that if each Xα contains at least two points, then α<ω1 Xα is always an almost P -space (a quasi F -space) but it is neither sequential nor sequentially connected. Using this lexicographic product, we give an example of a quasi F -space in which the set of F -points and the set of ω1 , does not have first and last non-F -points are dense. Whenever each Xα , α < elements, we show that the lexicographic product α<ω1 Xα is a P -space without isolated points. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For each f in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff (Tychonoff) space X, Z(f ) = {x ∈ X: f (x) = 0} is called a zeroset and X \ Z(f ) is called a cozeroset. A point x in a completely regular Hausdorff space X is said to be a P -point (an almost P -point) if every Gδ -set or every zeroset containing x is a neighborhood of x (has a nonempty interior) and X is called a P -space (an almost P -space) if every point of X is a P -point (an almost P -point), see [1,6,8] for more details and properties of these spaces. A point x in a completely regular Hausdorff space X is said to be an F -point if the ideal Ox = {f ∈ C(X): x ∈ intβX clβX Z(f )} is prime, where βX is the Stone–Čech compactification of X. A space X is called an F -space if every point of βX is an F -point. It is well-known that a space X is an F -space if and only if every cozeroset is C ∗ -embedded in X, see Theorem 14.25 in [6]. A quasi F -space E-mail addresses: [email protected] (F. Azarpanah), [email protected] (M. Etebar). http://dx.doi.org/10.1016/j.topol.2014.02.011 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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is a space in which every dense cozeroset is C ∗ -embedded, see [3] for more properties and characterizations of quasi F -spaces. Throughout this paper, topological spaces under consideration are linearly ordered spaces and the reader is referred to [4] and [6] for undefined terms and notations. A linearly ordered topological space (LOTS) is a triple (X, τ, <), where < is a linear ordering of the set X and τ is the usual open interval topology defined by <. For every x in a LOTS X, x+ (x− ) denotes the immediate successor (predecessor) of x, if it exists and the set of all points y ∈ X satisfying y > x (y < x) is denoted by (x, →) ((←, x)). A subset S of a LOTS X is said to be cofinal if, for every x ∈ X, there exists s ∈ S such that s x. It is well-known that every LOTS is a normal Hausdorff space, see [4] and a LOTS is connected if and only if it is Dedekind-complete (i.e., every nonempty subset with an upper bound has a supremum or equivalently, every nonempty subset with a lower bound has an infimum) and it does not have consecutive elements, see [6, 3O]. Whenever (W, ) is a well-ordered set and for every α ∈ W , (Xα , α ) is a linearly ordered set, then the lexicographic product of the family {(Xα , α ): α ∈ W } is the set of all points x = (xα )α∈W with the order < defined as follows: if x = (xα )α∈W and y = (yα )α∈W have x = y, let σ be the first element of the set {α ∈ W : xα = yα } in the ordering ≺ and then define x < y provided xσ <σ yσ . In what follows, we shall omit putting subscripts on the various orderings because context will make clear which ordering is meant. The lexicographic product will be denoted by α∈W Xα . In this paper we consider two familiar well-ordered sets as index sets: ω1 , the set of all countable ordinals and N, the set of all natural numbers. We also assume that each factor space Xα in the lexicographic product α∈W Xα contains at least two elements. In a LOTS X, we call a point x a P + -point if every Gδ -set containing x contains an interval [x, y) for some x < y ∈ X. P − -points will be defined similarly. If we define O+ (x) = {f ∈ C(X): [x, y) ⊆ Z(f ), for some y ∈ X, x < y} and similarly O− (x) = {f ∈ C(X): (y, x] ⊆ Z(f ), for some y ∈ X, x > y}, then clearly x is a P + -point (P − -point) if and only if O+ (x) = Mx (O− (x) = Mx ), where Mx = {f ∈ C(X): f (x) = 0}. A LOTS X is said to be P + -space (P − -space) if every point of X is a P + -point (P − -point) and it is clear that a LOTS X is a P -space if and only if it is both a P + -space and a P − -space. The space of countable ordinals is an example of a P + -space which is not P − -space, whence it is not a P -space, see [4] and [6] for more details of the space of ordinals. A subset A of a topological space X is said to be sequentially open if whenever {xn } is a sequence in X that converges to a point in A, then {xn } is eventually in A. Similarly a subset B of X is called sequentially closed if whenever there is a sequence in B that converges to some point x ∈ X, then x ∈ B. We recall that a space X is a sequential space if every sequentially closed (open) subset of X is closed (open). It is well-known that a LOTS is sequential if and only if it is first countable, see [4] and [9]. A space X is said to be sequentially connected if X cannot be expressed as the union of two nonempty disjoint sequentially closed (open) sets. Our aim in this article is to reveal the importance of almost P -points and their role in characterizing of some sequential properties of lexicographic products. In Section 2, we give some preliminary results and cite some facts from [2]. By the characterization of almost P -points of a LOTS in terms of sequences, we observe that quasi F -spaces and almost P -spaces coincide in the category of linearly ordered spaces. Using the characterization of a sequential LOTS and a sequentially connected LOTS in terms of almost P -points, we observe that a LOTS is sequentially connected if and only if it is connected without any almost P -points. In Sections 3 and 4 we study the lexicographic product α∈W Xα , where the well-ordered set W is either the set of natural numbers N or the set of countable ordinals ω1 . We observe that the existence of almost P -points of the lexicographic product n∈N Xn depends on first and last elements of the factor spaces. Using this lexicographic product, we give examples of P + -spaces and P − -spaces in which the set of almost P -points is dense. Whenever the set of countable ordinals is considered as an index set, we will show that the lexicographic product is always an almost P -space and whenever each factor space does not have first and last elements, then this lexicographic product will be a P -space without isolated points. Sequential
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connectedness of the lexicographic countable products is also investigated and it is shown that if each Xn ∞ has first and last elements, then n=1 Xn is sequentially connected if and only if each Xn is. Whenever ∞ each Xn is a LOTS without first and last elements, we show that the lexicographic product n=1 Xn is always a sequential space. Finally, using lexicographic products and the coincidence of quasi F -spaces and almost P -spaces in the category of linearly ordered spaces, we give an example of a quasi F -space (an almost P -space) without F -points (P -points) and an example of a quasi F -space (an almost P -space) in which the set of F -points (P -points) and the set of non-F -points (non-P -points) are dense. 2. Preliminaries In this section, using the characterization of almost P -points in terms of sequences, we observe that a LOTS is an almost P -space if and only if it is a quasi F -space. By the characterization of almost P -points in terms of sequences and the characterization of a sequentially connected LOTS in terms of almost P -points in this section together with the results in Section 3, enable us to prove the main results of the article in Section 4. In [6, 5O], it is shown that a point x of a LOTS is a P -point if and only if it is neither the limit of an increasing sequence nor the limit of a decreasing sequence. By the following lemma, we also characterize almost P -points of a LOTS in terms of sequences. We note that whenever x is not the first (last) point of a LOTS X and x is not the limit of an increasing (a decreasing) sequence in X and moreover x− (x+ ) does not exist, then every Gδ -set containing x, contains a nonempty interval (α, x] ([x, α)) for some α < x (α > x). On the other hand, for a point x ∈ X, whenever x− or x+ exists and x is both the limit of an increasing sequence and the limit of a decreasing sequence in X, then either x is an isolated point or the singleton {x} is a Gδ -set with empty interior. Using this argument the proof of the following result is evident. Lemma 2.1. A non-isolated point x in a LOTS X is an almost P -point if and only if one of the following statements holds. (a) x− does not exist and x is not the limit of an increasing sequence. (b) x+ does not exist and x is not the limit of a decreasing sequence. We note that whenever x is the first (last) element of X, then the condition (b) ( (a)) holds. We also cite the following lemma from [3] which is needed in the proof of the next theorem. Lemma 2.2. Let X be a completely regular Hausdorff quasi F -space and x ∈ X be a Gδ -point. Then x is not the limit of a distinct sequence in X. In particular, if X is a quasi F -space satisfying the first axiom of countability, then it is discrete. Now we prove the coincidence of quasi F -spaces and almost P -spaces in the category of linearly ordered spaces; for a different proof, see [2]. Theorem 2.3. A LOTS is an almost P -space if and only if it is a quasi F -space. Proof. Since the only dense cozeroset in an almost P -space X is X itself, clearly every almost P -space is a quasi F -space. Now let X be a quasi F -space and x ∈ X. If x+ and x− exist, then x is an isolated point and hence it is an almost P -point. Therefore suppose that one of x+ and x− does not exist, say x+ . If x is the limit of a decreasing sequence, then x is a Gδ -point and by Lemma 2.2, x is again an isolated point. Whenever x is not the limit of a decreasing sequence, then x is an almost P -point, by Lemma 2.1. Now
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suppose that both x+ and x− do not exist. If x is the limit of a decreasing sequence and it is also the limit of an increasing sequence, then x is again a Gδ -point and hence x is an isolated point, by Lemma 2.2. Whenever one of these cases does not happen, x should be an almost P -point, by Lemma 2.1. Therefore X is an almost P -space and we are through. 2 Now using Theorem 2.3, examples of linearly ordered almost P -spaces with no P -points (see Example 2.9 in [2] and the example given in the first paragraph of Section 4 in [8]) are also examples of quasi F -spaces with no F -points. It is well-known that every first countable space is a Fréchet space (a space X is called a Fréchet space if for every A ⊆ X and every x ∈ clX A, there exists a sequence of points of A converging to x) and every Fréchet space is a sequential space, see [4, Theorem 1.6.14]. The converse is also true for any LOTS, i.e., every sequential LOTS is first countable, see [9]. It is also easy to see that a LOTS X is sequential if and only if every point of X is a Gδ -point. Now using these well-known facts and Lemma 2.1, the following proposition is evident. Proposition 2.4. A LOTS X is a sequential space if and only if every almost P -point of X is an isolated point. Clearly every disconnected space is sequentially disconnected, but not conversely. In fact a connected space may be sequentially disconnected. The following lemma characterizes connected sequentially disconnected linearly ordered spaces in terms of almost P -points, see [2] for the proof. Corollary 2.5. A LOTS X is sequentially connected if and only if X is connected without any almost P -point. Using Proposition 2.4 and Corollary 2.5, the following corollary is evident. Corollary 2.6. If a LOTS is sequentially connected, then it is sequential. 3. P -points and almost P -points in lexicographic products In this section we investigate almost P -points and P -points of the lexicographic products. We note that n the existence of almost P -points and P -points of the lexicographic finite product i=1 Xi only depends on n the last factor space Xn . In fact if x = (x1 , x2 , . . . , xn ) ∈ i=1 Xi , then x− (x+ ) exists if and only if x− n (x+ P -point (P -point) if and only if xn is an almost P -point (P -point). n ) exists in Xn and x is an almost ∞ But for a lexicographic product n=1 Xn , we will observe that the existence of almost P -points depends on first and last elements of the factor spaces. For lexicographic products α<ω1 Xα , we show that its every point is an almost P -point. First we need the following lemma. Lemma 3.1. Suppose that for every α < ω1 , Xα is a LOTS with at least two points. Let X be the lexicographic product α<ω1 Xα and for each x ∈ X, define F (x) = {α < ω1 : xα is not the first element of Xα } and L(x) = {α < ω1 : xα is not the last element of Xα }. (a) If F (x) is cofinal in ω1 , then x is not the first element of X, x− does not exist and x is not the supremum of any countable subset of (←, x). (b) If L(x) is cofinal in ω1 , then x is not the last element of X, x+ does not exist and x is not the infimum of any countable subset of (x, →). Proof. (a) Clearly x cannot be the first element of X, for F (x) is cofinal. Now suppose that C is a countable subset of (←, x). For each y ∈ C, there is δy < ω1 such that for every α < δy , yα = xα and yδy < xδy . Since
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F (x) is cofinal and C is countable, there is some β ∈ F (x) with β > sup{δy : y ∈ C}. Since β ∈ F (x), we may choose zβ ∈ Xβ with zβ < xβ . Define zα = xα for each α ∈ ω1 \ {β}. Then y < z < x for every y ∈ C, so that x cannot be the supremum of the countable set C. A similar proof shows that x cannot have an immediate predecessor in X. (b) The proof is analogous. 2 We also need the following lemma in the sequel. Lemma 3.2. Suppose that for every n ∈ N, Xn is a LOTS with at least two points. Let X be the lexicographic ∞ product n=1 Xn and for each x ∈ X, define F (x) = {n ∈ N: xn is not the first element of Xn } and L(x) = {n ∈ N: xn is not the last element of Xn }. If F (x) (L(x)) is an infinite subset of N, then x is the limit of an increasing (a decreasing) sequence. Proof. Let L(x) be an infinite subset of N. Regard L(x) as an increasing subsequence {nk }∞ k=1 of integers ∞ and define a sequence f : N → n=1 Xn as follows: (f (k))n = xn , if n = nk and (f (k))n = ynk , if n = nk , where ynk ∈ Xnk and ynk > xnk . It is easy to see that f is a decreasing sequence and f → x. Whenever F (x) is infinite, the proof is similar. 2 Corollary 3.3. For each α < ω1 , let Xα be a LOTS with at least two points. Then every point of the lexicographic product α<ω1 Xα is an almost P -point. Proof. For any x ∈ X, at least one of the sets F (x) and L(x) defined in Lemma 3.1, is cofinal in ω1 . Hence each x ∈ X is an almost P -point by Lemmas 2.1 and 3.1. 2 Since linearly ordered quasi F -spaces and linearly ordered almost P -spaces coincide, and a point in a LOTS is an F -point if and only if it is a P -point, the following remark also gives an example of a quasi F -space in which both the set of F -points and the set of non-F -points are dense. Remark 3.4. If Xα is a LOTS, ∀α < ω1 , then the lexicographic product α<ω1 Xα is not necessarily a P -space. For example α<ω1 [0, 1] is an almost P -space by Corollary 3.3, but not every point of ω1 [0, 1] is a P -point. In fact both the set of P -points and the set of non-P -points of α<ω1 [0, 1] are dense. First we show that the set of P -points of α<ω1 [0, 1] is dense in α<ω1 [0, 1]. Let S=
(xα )α<ω1 ∈
[0, 1]: ∀β < ω1 , ∃δ > β such that xδ = 0, 1 .
α<ω1
For each x ∈ S, both F (x) and L(x) defined in Lemma 3.1 are cofinal. Hence by this lemma, x is neither the limit of an increasing nor the limit of a decreasing sequence. Now by 5O in [6], x is a P -point. On the other hand S is dense in α<ω1 [0, 1], for if a, b ∈ α<ω1 [0, 1], a < b, then there is smallest σ such that aσ = bσ , i.e., aσ < bσ . Now set x ∈ S such that aσ < xσ < bσ , xα = aα , ∀α < σ and xα ∈ (0, 1), ∀α > σ. Clearly x ∈ S ∩ (a, b), i.e., S is dense in α<ω1 [0, 1]. Next we consider the set T = (xα )α<ω1 ∈ [0, 1]: xδ ∈ (0, 1), and xβ = 0, ∀β > δ, for some δ < ω1 . α<ω1
We observe that every element of T is a non-P -point and T is dense in α<ω1 [0, 1]. It is easy to see that if x ∈ T , where xδ ∈ (0, 1) and xα = 0, ∀α > δ, then the sequence {xn } in α<ω1 [0, 1] such that xnδ = n−1 n xδ n and xα = xα , ∀α = δ converges to x. This means that x is not a P -point by 5O in [6]. Moreover, T is
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dense in α<ω1 [0, 1], for if x < y and β is the smallest index such that xβ = yβ (xβ < yβ ), then we choose z ∈ α<ω1 [0, 1], where zα = xα , ∀α < β, xβ < zβ < yβ and zα = 0, ∀α > β. Clearly z ∈ T ∩ (x, y), i.e., T is dense in α<ω1 [0, 1]. The following proposition shows that
α<ω1
Xα may be a P -space.
Proposition 3.5. Let Xα be a LOTS with at least two points, ∀α < ω1 and F (L) be the set of all α < ω1 such that Xα does not have first (last) element. If F and L are cofinal, then α<ω1 Xα is a P -space without isolated points. Proof. Clearly F ⊆ F (x) and L ⊆ L(x), ∀x ∈ X. Thus by Lemma 3.1 and 5O in [6], X is a P -space. Now it remains to show that α<ω1 Xα has no consecutive elements. In fact if x, y ∈ α<ω1 Xα and x < y, then there exists a smallest index β < ω1 such that xβ = yβ and hence xβ < yβ . Since F is cofinal, there is λ ∈ F such that λ > β and yλ is not the first element of Xλ . Consider z ∈ α<ω1 Xα , so that zλ < yλ and zα = yα , ∀α < λ. Clearly x < z < y and this implies that α<ω1 Xα has no consecutive elements, whence it has no any isolated point. 2 The following corollary shows that
α<ω1
R is a P -space without isolated points.
Corollary 3.6. Let Xα be a LOTS, ∀α < ω1 and each Xα does not have first and last elements. Then α<ω1 Xα is a P -space without isolated points. Using a proof similar to that of Proposition 3.5, we obtain a more general result as follows: Corollary 3.7. Let Xα be a LOTS, ∀α < ω1 and F (L) be the set of all α < ω1 such that Xα does not have first (last) element. (a) If F is cofinal, then α<ω1 Xα is a P − -space without isolated points. (b) If L is cofinal, then α<ω1 Xα is a P + -space without isolated points. It is well-known that a finite product of (Tychonoff) P -spaces is a P -space. But no infinite product of (Tychonoff) spaces of more than one point is a P -space, see [6, 4K]. In Proposition 3.5, we observed that the lexicographic product α<ω1 Xα may be a P -space. But in the following proposition, we show that ∞ the lexicographic product n=1 Xn of linearly ordered spaces of more than one point does not have any P -point. Proposition 3.8. If Xn is a LOTS with at least two elements, ∀n ∈ N, then the lexicographic product ∞ X = n=1 Xn never has an almost P -point (a P -point). Proof. For each x ∈ X, clearly one of the sets F (x) and L(x) defined in Lemma 3.2 is infinite. Now by Lemma 3.2, x is the limit of a sequence and this means that x is not an almost P -point (a P -point). 2 ∞ Proposition 3.8 states that the lexicographic product n=1 Xn does not have any P -point even each Xn ∞ is a P -space (discrete). By part (a) of the following proposition, the lexicographic product n=1 Xn may not contain any almost P -point. Parts (b) and (c) of the following proposition show that whenever each factor space Xn is a P -space with first (last) element and without consecutive elements, then the set of ∞ almost P -points of the lexicographic product n=1 Xn is dense. Part (a) of this proposition also shows that the existence of first and last elements of the factor spaces is essential for existence of almost P -points of ∞ n=1 Xn .
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Proposition 3.9. Let Xn be a LOTS with at least two points, ∀n ∈ N. (a) If F = {n ∈ N: Xn does not have first element} and L = {n ∈ N: Xn does not have last element} are ∞ infinite sets, then the lexicographic product n=1 Xn does not have any almost P -point. (b) If each Xn is a P + -space with last element and no element of Xn has immediate successor, then the ∞ set of almost P -points of the lexicographic product n=1 Xn is dense. (c) If each Xn is a P − -space with first element and no element of Xn has immediate predecessor, then the ∞ set of almost P -points of the lexicographic product n=1 Xn is dense. ∞ Proof. (a) For each x ∈ n=1 Xn , we have F ⊆ F (x) and L ⊆ L(x), hence F (x) and L(x) are infinite sets. Now by Lemma 3.2, x is the limit of increasing and decreasing sequences. This implies that x is not almost P -point by Lemma 2.1. ∞ (b) Let ln be the last element of Xn , ∀n ∈ N and consider A = {x ∈ n=1 Xn : xk = lk for some k ∈ ∞ N and xn = ln , ∀n > k}. We show that every point of A is an almost P -point and A is dense in n=1 Xn . + Let a ∈ A, where an = ln , ∀n > k and ak = lk . Since a+ k does not exist, a does not exist. On the other hand a is not the limit of a decreasing sequence. To see this let {y n } be a decreasing sequence and a < y n , n n ∀n ∈ N. Suppose mn is the smallest integer such that amn = ym (so, amn < ym ). Clearly mn k, n n ∀n ∈ N and this implies that there exists t k which is the smallest index such that at < ytnk for an infinite nk + sequence {nk }∞ at and hence there exists yt ∈ Xt such that for each k=1 . Since Xt is a P -space, yt ∞ nk n0 ∈ N, nk > n0 exists so that yt > yt > at . If we consider z ∈ n=1 Xn such that zt = yt and zn = an , ∀n = t, then a < z and for every n0 ∈ N, y nk > z for some nk > n0 , i.e., y n a. Now it remains to prove ∞ ∞ that A is dense in n=1 Xn . Let (b, c) be an interval in n=1 Xn and k be the smallest index such that ∞ bk = ck . So bk < ck and bi = ci , ∀i < k. We take x ∈ n=1 Xn , where xn = bn , ∀n < k, bk < xk < ck (note that no element of Xk has immediate successor) and xn = ln , ∀n > k. Thus x ∈ (b, c) ∩ A, i.e., A is dense ∞ in n=1 Xn . (c) The proof is similar. 2 4. Sequential properties of lexicographic products Theorem 2.2 in [7] shows that the countable Tychonoff product of sequentially connected spaces is sequentially connected. Theorem 4.1.1 in [5] also investigates the connectedness of lexicographic products. In this section, using characterization of a sequentially connected LOTS in terms of almost P -points in Corollary 2.5, we investigate the sequential connectedness of the lexicographic product spaces. First we need the following proposition. Although this proposition may be proved by Theorem 4.1.1 in [5], we give a direct proof. Proposition 4.1. Let Xn be a LOTS with first and last elements for each n ∈ N. Then the lexicographic ∞ product n=1 Xn is connected if and only if each Xn is. ∞ Proof. First suppose that n=1 Xn is connected and An is a nonempty subset of Xn . Let A be a subset of ∞ n=1 Xn containing the elements of the form (l1 , . . . , ln−1 , an , ln+1 , . . .), where an ∈ An and lm is the last element of Xm , ∀m = n. Since A is bounded above, it has a supremum. Let α = sup A, since α a, ∀a ∈ A, αi = li , ∀i < n and an αn , ∀an ∈ An . This implies that αn = sup An , i.e., Xn is Dedekind complete. Next, each Xn has no consecutive elements, for if an , bn ∈ Xn are consecutive elements of Xn and an < bn , ∞ then we may take x, y ∈ n=1 Xn such that xn = an , yn = bn , xm = ym , ∀m < n and xm = lm , ym = fm , ∞ ∀m > n, where fm is the first element of Xm . Then x and y are two consecutive elements of n=1 Xn which is a contradiction.
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∞ Conversely, let each Xn be connected and A ⊆ n=1 Xn . Suppose that A1 is the set of all a1 ∈ X1 such that a1 is the first component of an element of A. Since A1 is bounded above and X1 is Dedekind complete, α1 = sup A1 exists. By induction, let Ak be the set of ak ∈ Xk such that α1 , α2 , . . . , αk−1 and ak be the first k components of an element of A and take αk = sup Ak . If Ak = ∅, ∀k ∈ N, then α = (α1 , . . . , αk , . . .) = sup A. In fact if β < α and n is the smallest index such that βn < αn (so βi = αi , ∀i < n), then we take a ∈ A such that α1 , . . . , αn are the first n components of a. Therefore β < a α, i.e., α = sup A. Now let Ak = ∅ for some integer k and n be the smallest integer such that An = ∅ (so, Am = ∅, ∀m < n). Consider α = (α1 , . . . , αn−1 , fn , fn+1 , . . .), where fm is the first element of Xm , ∀m n. We show that α = sup A. First, if a ∈ A, then a α, for otherwise, a > α implies that there is k < n (note that An = ∅) such that ak > αk and ai = αi , ∀i < k which contradicts αk = sup Ak . Next, let β < α and k be the smallest index such that βk = αk , so βi = αi , ∀i < k and βk < αk . Now either k = n − 1 or k < n − 1 for βm ≮ fm , ∀m n. If k = n − 1, then for all an−1 ∈ An−1 , we have an−1 < αn−1 and since αn−1 = sup An−1 , there exists a∗n−1 ∈ An−1 such that βn−1 < a∗n−1 < αn−1 (note that αn−1 ∈ / An−1 for An = ∅). Since An−1 = ∅, we may take a ∈ A whose first n − 1 components are α1 , . . . , αn−2 and a∗n−1 . Clearly β < a α, i.e., α = sup A. If k < n − 1, we can consider a ∈ A such that am = αm , ∀m < n − 1, for An−1 = ∅. Clearly β < a < α, whence we have α = sup A. 2 Now we prove our main theorem in this section. Theorem 4.2. Suppose that Xn is a LOTS with first and last elements for each n ∈ N. Then the lexicographic ∞ product n=1 Xn is sequentially connected if and only if each Xn is. ∞ Proof. By Proposition 4.1 and Corollary 2.5, it is enough to show that n=1 Xn contains a non-isolated almost P -point if and only if for some k ∈ N, Xk contains a non-isolated almost P -point. First suppose that for some k ∈ N, Xk contains an almost P -point xk ∈ Xk which is not isolated point. Then one of the conditions (a) and (b) of Lemma 2.1 holds, say (a). Hence x− k does not exist and xk is not the limit of an increasing sequence. Whenever fm is the first element of Xm , ∀m ∈ N, then x = (f1 , . . . , fk−1 , xk , fk+1 , . . .) ∞ is an almost P -point of n=1 Xn . In fact if y < x, and m is the smallest integer for which ym = xm , i.e., ym < xm , then m should be k. Hence yk < xk and this implies that x− does not exist for x− k does not exist. Now if {xn } is an increasing sequence which converges to x, then xn < x, ∀n ∈ N implies that xnk < xk , ∀n ∈ N and xnm = fm , ∀m < k. Thus {xnk } is an increasing sequence in Xk which converges to xk , a contradiction. In case part (b) of Lemma 2.1 holds for xk , then we take x = (l1 , . . . , lk−1 , xk , lk+1 , . . .), where lm is the last element of Xm , ∀m = k and the remainder of the proof is similar. ∞ Next, suppose that n=1 Xn has an almost P -point x = (x1 , . . . , xk , . . .) and let part (a) of Lemma 2.1 holds. Since x is not the limit of an increasing sequence, there exists k ∈ N such that xm = fm , ∀m > k. For otherwise if there is a subsequence {nk } of integers such that xnk is not the first element of Xnk , then ∞ we may define a sequence h : N → n=1 Xn , with (h(k))nk = fnk and (h(k))m = xm , ∀m = nk . Clearly h is increasing and h → x. To see h → x, let y < x and m be the smallest index such that ym = xm (i.e., ym < xm and yn = xn , ∀n < m). Then for nk > m, we have h(k) ∈ (y, x]. This contradicts our hypothesis and hence x is of the form x = (x1 , . . . , xk , fk+1 , . . .), where fm is the first element of Xm , ∀m > k. Now it is not hard to see that xk is an almost P -point of Xk . In case part (b) of Lemma 2.1 holds for x, then x will be of the form x = (x1 , . . . , xk , lk+1 , . . .) for some k ∈ N, where lm is the last element of Xm , ∀m > k and similarly xk will be an almost P -point of Xk . 2 In the proof of Theorem 4.2, we have shown that whenever each Xn is a LOTS with first and last elements, ∞ then the lexicographic product n=1 Xn has a non-isolated almost P -point if and only if some factor space ∞ Xk has. This indicates that the lexicographic product n=1 Xn is not sequential if and only if some factor ∞ space Xk is not sequential, by Proposition 2.4. Equivalently, the lexicographic product n=1 Xn is not first
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countable if and only if some factor space is not first countable. Now as an immediate consequence of this fact, we have the following corollary. Corollary 4.3. Let Xn be a LOTS with first and last elements. Then the lexicographic product a sequential space if and only if each factor space Xn is.
∞ n=1
Xn is
∞ By Corollary 2.6, if the lexicographic product n=1 Xn is sequentially connected, then it is sequential. Whenever each factor space Xn is connected, the converse is also true. In fact, if the lexicographic product ∞ ∞ will be an isolated n=1 Xn is sequential, then every almost P -point of the lexicographic product n=1 Xn ∞ point by Proposition 2.4. But Proposition 3.8 states that the lexicographic product n=1 Xn does not ∞ contain any isolated point and this means that the lexicographic product n=1 Xn is sequentially connected by Corollary 2.5. Corollary 4.4. If each factor space Xn is a connected LOTS, then the lexicographic product sequential if and only if it is sequentially connected.
∞ n=1
Xn is
By Proposition 3.9(a) and Proposition 2.4, the following corollary is evident. Corollary 4.5. Let each Xn be a LOTS without first and last elements. Then the lexicographic product ∞ n=1 Xn is a sequential space. Corollary 4.6. The lexicographic product
α<ω1
Xα is never a sequential space.
Proof. By Corollary 3.3, every point of α<ω1 Xα is an almost P -point. Now consider x ∈ α<ω1 Xα such that for some λ < ω1 , xα is not the last element of Xα , ∀α > λ. If y ∈ α<ω1 Xα and x < y, then there is β < ω1 so that xβ < yβ and xα = yα , ∀α < β. Take σ < ω1 as the smallest element of the set {α < ω1 : α > β and xα is not the last element of Xα }. Now we consider z ∈ α<ω1 Xα such that zα = xα , ∀α < σ and zσ > xσ . Clearly x < z < y and this shows that x is not an isolated point of α<ω1 Xα which means by Proposition 2.4 that α<ω1 Xα is not sequential. 2 Finally, by Corollaries 2.6 and 4.6, the following result is evident. Corollary 4.7. The lexicographic product
α<ω1
Xα is never sequentially connected.
Acknowledgement We would like to thank the referee. His/her careful reading of the manuscript combined with some very useful comments have hopefully improved the quality of the paper. References [1] F. Azarpanah, O.A.S. Karamzadeh, A. Rezai Aliabad, z ◦ -ideals in C(X), Fundam. Math. 160 (1999) 15–25. [2] F. Azarpanah, M. Etebar, Linearly ordered P ± -spaces, JP J. Geom. Topol. 10 (2) (2010) 99–112. [3] F. Dashiell, A. Hager, M. Henriksen, Order-Cauchy completion of rings and vector lattices of continuous functions, Can. J. Math. XXXII (3) (1980) 657–685. [4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [5] M.J. Faber, Metrizability in generalized ordered spaces, PhD dissertation, Vrije University te Amsterdam, Tract 53, Math. Centre Tracts, Amsterdam, 1974. [6] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [7] Q. Huang, S. Lin, Notes on sequentially connected spaces, Acta Math. Hung. 110 (1–2) (2006) 159–164. [8] R. Levy, Almost P -spaces, Can. J. Math. 2 (1977) 284–288. [9] P.R. Meyer, Sequential properties of ordered topological spaces, Compos. Math. 21 (1969) 102–106.