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The o-Malykhin property for spaces Ck(X)
Topology and its Applications 160 (2013) 143–148
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The o-Malykhin property for spaces C k ( X ) A. Bareche Département de Mathématiques, Université de Rouen, UMR CNRS 6085, Avenue de l’Université, BP.12, F76801 Saint-Etienne du Rouvray, France
a r t i c l e
i n f o
Article history: Received 28 September 2011 Received in revised form 8 October 2012 Accepted 9 October 2012 Keywords: o-Malykhin property Moving off property Moving off collection Compact-finite Topological game
a b s t r a c t Let X be a Tychonoff space and C k ( X ) be the vector topological space of continuous realvalued functions on X with the compact open topology. The problem of characterizing C k ( X ) in terms of X has interested several authors; in particular, Gruenhage and Ma for the baireness property and recently Sakai with the κ -Fréchet Urysohn property. Motivated by their works, we are interested in the o-Malykhin property for C k ( X ). In this note, we will show that C k ( X ) is o-Malykhin if and only if every moving off collection of non-empty compacts of X contains an infinite compact-finite collection. We will also characterize the o-Malykhin property for C k ( X ) by a topological game defined on X, and we will give some related results. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Let X be a Tychonoff space. We denote by C k ( X ) (respectively, C p ( X )) the space of real-valued continuous functions on X , endowed with the compact-open topology (respectively, the pointwise convergence topology). A collection K of nonempty compact subsets of X is said to be a moving off collection if, for any compact L ⊂ X , there is some K ∈ K such that K ∩ L = ∅. Note that a moving off collection is always infinite. A collection K of non-empty subsets of X is said to be compact-finite if, for each compact L ⊂ X , the set { K ∈ K: L ∩ K = ∅} is finite. Problems of characterizing C k ( X ) (or C p ( X )) in terms of topological properties of X , have received the attention of many authors. In particular, Gruenhage and Ma [6] have shown that for a q-space X , the space C k ( X ) is Baire if and only if, every moving off collection K of non-empty compacts of X contains an infinite collection { K n : n ∈ N} which has a discrete open expansion in X (means that there are open subsets U n of X , such that U n ⊃ K n for all n ∈ N and the collection {U n : n ∈ N} is discrete). Recently, Sakai [12] has shown that the space C k ( X ) is κ -Fréchet Urysohn iff every moving off collection of non-empty compacts of X admits an infinite strongly compact-finite subcollection. The collection { K n : n ∈ N} of non-empty compacts of X is said to be strongly compact-finite if there exist open subsets U n of X such that U n ⊃ K n for all n ∈ N and the collection {U n : n ∈ N} is compact-finite. The space X is κ -Fréchet Urysohn if for every open U ⊂ X and each point x ∈ U , there is a sequence (xn )n∈N ⊂ U converging to x in X . For more results about these problems, the reader is referred to [6,12] and the references therein. In this work, we will study the property said o-Malykhin, for the spaces C k ( X ). Definition 1.1. The space X is said to be o-Malykhin if for each family ( O i )i ∈ I of open subsets of X and each point x ∈ X satisfying
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x∈
Oi \
i∈I
O i,
i∈I
there exists an infinite J ⊂ I such that the set { j ∈ J : O j ∩ V = ∅} is finite for any neighborhood V of x in X . The o-Malykhin property has been introduced by Bouziad and Troallic [4], as an extension of the Malykhin notion which has been studied by A.V. Arhangel’skiˇı [1]; it plays an important role in the study of Itzkowitz’s problem for topological groups (for more about this problem see [2]). On the other hand, we can consider it in some sense, as a generalization of Reznichenko’s property (weak Fréchet Urysohn property) see [10]. It is clear that every κ -Fréchet Urysohn space is o-Malykhin; but the converse does not hold in general. Indeed, the space C p ( X ) is o-Malykhin if X is completely regular; this is because C p ( X ) is a dense subspace in the product space R X which is known to be Malykhin (thus o-Malykhin), see [9]. However, it is well-known that the space C p (R) is not κ -Fréchet Urysohn. In the next section, we shall prove that the space C k ( X ) is o-Malykhin if and only if every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. We shall be also characterizing the o-Malykhin property for the space C k ( X ) by a topological game defined on the space X and we shall propose a class of topological spaces X for which the space C k ( X ) is not o-Malykhin. Finally in the last section, we shall compare the o-Malykhin property for the space C k ( X ) with other topological properties, particularly, the κ -Fréchet Urysohn property for the space C k ( X ) and the Prohorov property on the space X . 2. The o-Malykhin property for spaces C k ( X ) In the following, X is assumed to be a completely regular topological space and c 0 is the real-valued constant function defined on X , by c 0 (x) = 0 for all x ∈ X . Theorem 2.1. The following statements are equivalent: (1) The space C k ( X ) is o-Malykhin. (2) Every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. (3) For any sequence (Kn )n∈N of moving off collections of non-empty compacts of X , there exist K ni ∈ Kni such that the sequence ( K ni )i ∈N is compact-finite. Proof. (1) → (2). Let K be a moving off collection of non-empty compacts of X . For K ∈ K, we mean by O K = [ K ; (1, ∞)] the open set of C k ( X ), where [ K ; (1, ∞)] = { f ∈ C k ( X ): | f (x)| > 1 ∀x ∈ K }. It is easy to check that
c0 ∈
OK \
K ∈K
OK,
K ∈K
because X is completely regular. Since C k ( X ) is o-Malykhin, there exists an infinite collection L ⊂ K such that the set { L ∈ L: O L ∩ O = ∅} is finite for every neighborhood O of c 0 in C k ( X ). We fix a non-empty compact K ⊂ X and a positive integer n. Then, considering the neighborhood [ K ; (−1/n, 1/n)] of c 0 , we can see that the set M = { L ∈ L: O L ∩ [ K ; (−1/n, 1/n)] = ∅} is finite. Consequently, for each L ∈ L \ M, the set [ L ; (1, ∞)] ∩ [ K ; (−1/n, 1/n)] is not empty, thus L ∩ K = ∅. As a result, the collection L is compact-finite. (2) → (3). Let (Kn )n∈N be a sequence of moving off collections of non-empty compacts of X . By (2), to each n ∈ N, we associate an infinite countable compact-finite subcollection of Kn that we denote Ln , where Ln = { K nj : j ∈ N}. For n ∈ N, let L n = K 0j ∪ K 1j ∪ · · · ∪ K nj , with j i ∈ N, K ij ∈ Ln , 0 i n; hence L n is a non-empty compact subset of X . It is not difficult n 0 1 i to observe that T = { L n : n ∈ N} is a moving off collection of non-empty compacts of X ; using (2) again, we can obtain a n subcollection of T , which is infinite and compact-finite. This allows us to extract a subsequence K j i ∈ Kni , i ∈ N, such that ni
n
K = ( K jni )i ∈N is an infinite compact-finite sequence. i (3) → (1). Since the space C k ( X ) is homogeneous (as a vector topological space), in order to prove that it is o-Malykhin, it suffices to check that the o-Malykhin property holds at the origin c 0 . Let ( O i )i ∈ I be a family of non-empty open subsets of C k ( X ) with
c0 ∈
i∈I
Oi \
O i.
i∈I
For each i ∈ I and each f i ∈ O i , there is an open set O i ( f i ) = [ K i ( f i ), f i , εi ( f i )] ⊂ O i , where K i ( f i ) is a non-empty compact subset of X , εi ( f i ) is a positive real number and [ K i ( f i ), f i , εi ( f i )] = { g ∈ C k ( X ): | f i (x) − g (x)| < εi ( f i ) ∀x ∈ K i ( f i )}. For k ∈ N, with k > 0, let C i ( f i , k) = K i ( f i ) ∩ ( X \ f i−1 (−1/k, 1/k)) (with i ∈ I , f i ∈ O i ). We put
I 0 = k ∈ N \ {0}: there are i ∈ I , f i ∈ O i with C i ( f i , k) = ∅
A. Bareche / Topology and its Applications 160 (2013) 143–148
and
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I 1 = k ∈ N \ {0}: C i ( f i , k) = ∅ ∀i ∈ I and f i ∈ O i . We are studying the two following cases: (a) I 0 is infinite. Let g be a mapping defined over I 0 into I as follows: for each k ∈ I 0 , g (k) = ik ∈ I with C ik ( f ik , k) = ∅ and f ik ∈ O ik . Now, we consider ik ∈ g ( I 0 ). As the space X is completely regular and the set K ik ( f ik ) is compact, we can find
a continuous function g ik : X → [0, 1] such that g ik (x) = 1 for all x ∈ K ik ( f ik ) and g ik (x) = 0 for all x ∈ X \ f i−1 (−1/k, 1/k). k We define the function h ik by h ik = f ik · g ik . Clearly, h ik is continuous on X , moreover, h ik ( X ) ⊂ (−1/k, 1/k) and h ik ∈ O ik . It follows that the sequence (h ik )k∈ I 0 converges uniformly to c 0 , thus c 0 ∈ {h ik : k ∈ I 0 }.
The subset J = g ( I 0 ) of I is infinite, because we have at the same time c 0 ∈ / O j for each j ∈ J and c 0 ∈ j ∈ J O j . Now, we fix a non-empty compact K ⊂ X and a positive integer n. Then, there is kn ∈ I 0 such that for every k ∈ I 0 with k kn , we have h ik ∈ [ K ; (−1/n, 1/n)] (means that [ K ; (−1/n, 1/n)] ∩ O ik = ∅). As a consequence, the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is finite. (b) I 1 is infinite. Wetake k ∈ I 1 and we consider the collection Lk = {C i ( f i , k): i ∈ I , f i ∈ O i }. Let K ⊂ X be a non-empty compact set. As c 0 ∈ i ∈ I O i , there are ik ∈ I and f ik ∈ O ik with f ik ( K ) ⊂ (−1/k, 1/k). This gives C ik ( f ik , k) ∩ K = ∅. Then, for all k ∈ I 1 , Lk is a moving off collection of non-empty compacts of X . By (3), there is some infinite J 1 ⊂ I 1 such that the collection K = {C ik ( f ik , k): k ∈ J 1 } is infinite and compact-finite. We suppose without loss of generality, that for all k, l ∈ J 1 , we will have C ik ( f ik , k) = C il ( f il , l) if k = l or ik = il . Define a mapping h over J 1 into I , which associates to each k ∈ J 1 an index h(k) = ik ∈ I such that C ik ( f ik , k) ∈ K. Let J = h( J 1 ). We show first that J is infinite. For every non-empty compact L ⊂ X and every positive integer m, we put S ( L ) = {k ∈ J 1 : L ∩ C ik ( f ik , k) = ∅} and S m ( L ) = {ik ∈ J : k max(m, S ( L ))}. The set S ( L ) is finite since the collection K is compactfinite, and S m ( L ) is obviously finite. For k ∈ J 1 with k > max(m, S ( L )), we have ik ∈ J \ S m ( L ), hence L ∩ C ik ( f ik , k) = ∅. If f ik ( L ) ⊂ (−1/k, 1/k), then f ik ∈ [ L ; (−1/k, 1/k)] ∩ O ik . If not, we consider the compact L k = L ∩ ( X \ f i−1 (−1/k, 1/k)) k which is disjoint from K ik ( f ik ). By this, we will get a continuous function g ik : X → [0, 1] such that g ik ( L k ) ⊂ {0} and g ik ( K ik ( f ik )) ⊂ {1}. Let h ik = g ik · f ik . It is evident that h ik is a continuous function belonging to O ik ∩ [ L ; (−1/k, 1/k)]. We conclude in the two cases that [ L ; (−1/m, 1/m)] ∩ O ik = ∅, thus
c0 ∈
O j.
j∈ J
Therefore, J is infinite because c 0 ∈ / O j for all j ∈ J . At this moment, we take a fixed non-empty compact K of X and a fixed integer n > 0. We will show that the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is finite. As above, the sets S ( K ) = {k ∈ J 1 : K ∩ C ik ( f ik , k) = ∅} and S n ( K ) = {ik ∈ J : k max(n, S ( K ))} are finite; so J \ S n ( K ) is infinite. Let ik ∈ J \ S n ( K ); if f ik ( K ) ⊂ (−1/k, 1/k), it will have
f ik ∈ [ K ; (−1/k, 1/k)] ∩ O ik ⊂ [ K ; (−1/n, 1/n)] ∩ O ik . If not, the compact K ∩ ( X \ f i−1 (−1/k, 1/k)) will be denoted by K k k and then we will get a continuous function g ik : X → [0, 1] such that g ik ( K k ) ⊂ {0} and g ik ( K ik ( f ik )) ⊂ {1}. The function h ik = g ik · f ik belongs to the intersection [ K ; (−1/n, 1/n)] ∩ O ik . Finally, we conclude that for each j ∈ J \ S n ( K ), [ K ; (−1/n, 1/n)] ∩ O j = ∅. As a result, the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is included in the finite set S n ( K ), which completes the proof. 2
In order to characterize the o-Malykhin property for C k ( X ) in terms of topological games, we take the game introduced by Gruenhage and Ma in [6] with some modification. The modification will concern only the winning condition and our game will be presented as follows: The infinite game G ( X ) is defined on the space X , its players are respectively denoted by P1 and P2. The play runs alternatively between P1 and P2, where at each step n, P1 chooses a non-empty compact K n ⊂ X while P2 responds by giving a non-empty compact L n ⊂ X \ K n . Player P1 wins the play ( K n , L n )n∈N if the sequence ( L n )n∈N chosen by P2 is compact-finite; otherwise, P2 is the winner of this play. (For more informations about these games and their applications see [5] and [8].) Theorem 2.2. The space C k ( X ) is o-Malykhin if and only if Player P2 has no winner strategy in the game G ( X ). Proof. Since the space C k ( X ) is o-Malykhin, the condition (3) of Theorem 2.1 is satisfied. As a consequence, the fact that Player P2 has no winning strategy in the game G ( X ) can be established in the same way as the implication (ii) → (iii) in Theorem 2.3 from [6]. To prove the reverse implication, suppose the space C k ( X ) is not o-Malykhin. By Theorem 2.1, there is a moving off collection K of non-empty compacts of X , no infinite subcollection of which is compact-finite. Player P2 can win the play by choosing its compacts in K, thus (2) cannot be satisfied, which completes the proof. 2 At the end of this section, it is natural to ask if there exists a topological space X , which is completely regular and for which the space C k ( X ) is not o-Malykhin. In fact, to answer this question, it suffices to consider the class of metrizable
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spaces which are not locally compact, as Q the space of rational numbers or also P the space of irrational numbers. This result will be established by the following statement. Proposition 2.3. Let X be a topological space for which the space C k ( X ) is o-Malykhin. Suppose that each point x ∈ X admits a sequence of neighborhoods (U n )n∈N satisfying the following property: for every sequence (xn )n∈N ⊂ X with xn ∈ U n ( for all n ∈ N), there is some compact K ⊂ X which contains all terms of the sequence (xn )n∈N . Then, X is locally compact. Proof. Suppose that X is not locally compact. So we can find a point x ∈ X which has no closed compact neighborhood. Let (U n )n∈N be a sequence of neighborhoods of x in X that satisfies proposition’s property. For every n ∈ N, we take Kn = {{ y }: y ∈ U n }. Clearly, no member of (U n )n∈N has a compact closure in X . This permits us to conclude that (Kn )n∈N is a sequence of moving off collections of non-empty compacts of X . Using the condition (3) of Theorem 2.1, we can extract a sequence zni ∈ U ni such that the collection K = {{ zni }: i ∈ N} is compact-finite. However, the compact K = { zn : n ∈ N} contains all elements of the collection K, this produces a contradiction. 2 3. Some related results Let X be a topological completely regular space. This section is devoted to establish the relation between the space X such that C k ( X ) is o-Malykhin and some well-known topological spaces. According to Gruenhage and Ma [6], we say that the space X has the moving off property MOP if every moving off collection of non-empty compacts of X contains an infinite subcollection which has an open discrete expansion. Following Bouziad [3], the space X satisfies the weak moving off property WMOP if every moving off collection of non-empty compacts of X contains an infinite subcollection which is locally finite. Definition 3.1. We say that the space X has the weak weak moving off property WWMOP if every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. It is evident that the MOP property implies the WMOP property and that the WMOP property implies the WWMOP property. However, the converses do not hold in general. Indeed, in his paper [13], Shakhmatov has proved the existence of a pseudocompact space, in which all countable subsets are closed. Such a space (as mentioned in [3]) satisfies the WMOP, but it does not verify the MOP. In the other hand, the space cited in Example 3.5 (below) has the WWMOP, but it has not the WMOP. Remark 3.2. The two points of this remark will be used later. (1) Following Sakai [12], the space X verifies the WMOP since the space C k ( X ) is κ -Fréchet Urysohn. (2) By Theorem 2.1, the space C k ( X ) is o-Malykhin if and only if the space X verifies the WWMOP. Definition 3.3. The space X is said to be hemicompact if there is some sequence ( K n )n∈N of compact subsets of X , such that for every compact K ⊂ X , there exists n ∈ N with K ⊂ K n . Proposition 3.4. Let X be a hemicompact space. Then the space C k ( X ) is o-Malykhin. Proof. Let C be an arbitrary moving off collection of non-empty compacts of X . By Remark 3.2 (2), to establish the o-Malykhin property on C k ( X ), it will be sufficient to find an infinite compact-finite sequence (C n )n∈N ⊂ C . of of X satisfying the hemicompact condition. Let C 1 ∈ C with K 1 ∩ C 1 = ∅. For Let ( K n )n∈N be the sequence ncompacts n −1 n 2, we put L n = ( i =1 K i ) ∪ ( i =1 C i ) and we choose C n ∈ C such that C n ∩ L n = ∅. Then, we get a sequence (C n )n∈N with the required conditions. Indeed, for every compact K ⊂ X , there is some n ∈ N such that K ⊂ K n ; thus K ⊂ L m and C m ∩ K = ∅ for all m n. 2 As mentioned in the introduction, every κ -Fréchet Urysohn space is o-Malykhin, hence the space C k ( X ) will be o-Malykhin if it is κ -Fréchet Urysohn. This can be also concluded from Remark 3.2. Yet, we can ask if the converse is true. The following example gives a negative answer to this question.
/ M ∪ N. Example 3.5. Let N be a countable set and M be a non-countable set. Let ∗ and ∗∗ be two points such that ∗, ∗∗ ∈ We consider the two spaces Y = N ∪ {∗} and Z = M ∪ {∗∗}, endowed with their topologies as follows: All points of N are isolated in Y and the family { A ∪{∗}} (with A ⊂ N and N \ A is finite) represents a basis neighborhoods of ∗ in Y . All points of M are isolated in Z and the sets of the form B ∪ {∗∗} (with B ⊂ M and M \ B is countable) are considered as basis neighborhoods of ∗∗ in Z .
A. Bareche / Topology and its Applications 160 (2013) 143–148
Then for X = (Y × Z )\{(∗, ∗∗)} endowed with the product topology, the space C k ( X ) is o-Malykhin and it is not Urysohn.
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κ -Fréchet
Proof. According to Remark 3.2 (2) and Theorem 2.3 in [12], it suffices to show that X has the WWMOP property and that there is some moving off collection of compact subsets of X , no infinite subcollection of which is strongly compact-finite. Let C be an arbitrary moving off collection of non-empty compacts of X . Note that for every compact K ⊂ X , there exists a finite set B ⊂ Z such that K ⊂ Y × B. So we can choose a sequence (C n )n∈N ⊂ C for which there exists a pairwise disjoint sequence ( B n )n∈N ⊂ Z of finite sets, such that C n ⊂ Y × B n for all n ∈ N. It is easy to check that the sequence (C n )n∈N is compact-finite. In the other hand, the collection {{(n, ∗∗)}: n ∈ N} is moving off and does not admit any infinite subcollection being strongly compact-finite. Indeed, for every n ∈ N, there is an open setU n containing (n, ∗∗) such U n = {n} × O n , where M \ O n is countable for all n. Since M is not countable, the set O = n∈N O n is not empty. Finally, the set Y × {x} with x ∈ O is a compact which intersects all open sets U n . 2 Remark 3.6. The o-Malykhin property for the space C k ( X ) is determined intrinsically by compacts of the space X . More precisely, suppose that we have defined two topologies on a set X . If these topologies have the same compacts, then the corresponding spaces C k ( X ) will (or not) o-Malykhin simultaneously, although the two topologies may be different. In particular, the case when X is a topological space and Z is its k-modification (i.e., O ⊂ X is open in Z if its intersection with any compact of X is open in this compact); then C k ( X ) is o-Malykhin if and only if C k ( Z ) is o-Malykhin. Indeed, it is well-known that X and Z have the same compacts. By (2) of Remark 3.2, the space C k ( X ) is o-Malykhin for every space X satisfying the WMOP property. Example 3.7. Suppose that X is a pseudocompact space and all countable subsets of which are closed. Then the space C k ( X ) is o-Malykhin. Conversely, if X is a k-space and C k ( X ) is o-Malykhin, we will have the following result: Proposition 3.8. Let X be a k-space for which C k ( X ) is o-Malykhin. Then X verifies the WMOP property. Proof. It is well-known that every compact-finite collection of non-empty compacts in a k-space is locally finite. Combining this fact with (2) of Remark 3.2, we get the conclusion. 2 Let μ be a positive Borel measure over a topological space X . We say that the measure μ is τ -additive (or τ -smooth) if μ( U ) = sup{μ(U ): U ∈ U } for every upwards directed collection U of open subsets of X . We denote by P τ ( X ) the space of positive τ -additive probability measures over X , endowed with the weak topology. The weak topology on P τ ( X ) is defined by the following property: a net (μα ) ⊂ P τ ( X ) converges to μ in P τ ( X ) if and only if lim inf μα (U ) μ(U ) for any open set U ⊂ X . The subspace of P τ ( X ) formed of all Radon (or tight) probability measures on X will be denoted by P t ( X ). Recall that μ is Radon if μ is inner regular with respect to compact subsets of X in the sense that for every open set A ⊂ X , μ( A ) = sup{μ( K ): K ⊂ A }, where K is a compact subset of X . Definition 3.9. The space X is said to be Prohorov if X is regular and every (relatively) compact set A ⊂ P t ( X ) is uniformly tight, that is, for every 0 < ε < 1 there exists a compact set K ⊂ X such that μ( K ) ε for all μ ∈ A. The symbol F( X ) denotes the lattice of all closed subspaces of X . Recall that K( X ) is the set of all compact subsets of X . The co-compact topology defined on F( X ), that we denote Tco , is generated by the subbasis { O K : K ∈ K( X )}, where O K = { F ∈ F( X ): F ∩ K = ∅}. In the other hand, the upper Kuratowski topology Tu K on F( X ), is the topology associated to the upper Kuratowski convergence defined as follows: a net ( F γ )γ ∈Γ ⊂ F( X ) converges to F ∈ F( X ), with respect to the
upper Kuratowski convergence, if and only if { { F γ : γ α }: α ∈ Γ } ⊂ F . The following notion has been introduced by Nogura and Shakhmatov in [11] after the work of Dolecki, Greco and Lechicki [7]. Definition 3.10. Let H( X ) be a sublattice of F( X ). The space X is called H-trivial if the two topologies Tu K and Tco coincide on H( X ). The F-trivial spaces are exactly the consonant spaces first defined by Dolecki, Greco and Lechicki. Proposition 2.3 and Corollary 3.4 in [3] give the following:
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Corollary 3.11. Let X be a k-space for which the space C k ( X ) is o-Malykhin. Then X is a K-trivial space and thus a Prohorov space. Corollary 3.12. Let X be such that the space C k ( X ) is o-Malykhin, then the k-modification of X is a K-trivial space and consequently it is a Prohorov space. The following example shows that spaces X (with C k ( X ) is o-Malykhin) are not consonant in general.
/ A × N. We call S A the Fréchet–Urysohn fan space of Definition 3.13. Let A be an infinite set and S A = ( A × N) ∪ {∗}, with ∗ ∈ size | A |, if the set S A is endowed with the following topology. All points of A × N are isolated in S A . A basis neighborhoods of ∗ in S A is given by the family {U f : f ∈ N A }, where U f = {(a, n): n f (a)} ∪ {∗} and N A represents the set of all functions defined from A to N. Example 3.14. The Fréchet–Urysohn fan space S A , where the set A is not countable, cannot be consonant; see [11]. However, the space C k ( S A ) is always o-Malykhin since the space S A has the MOP property (see Proposition 2.4 [3]). In what follows, we denote by Γ ( X ) the set of all real-valued functions which are bounded on every compact of X . By equipping Γ ( X ) with the uniform convergence topology of compact subsets of X . We get a topological vector space that we denote Γk ( X ). The uniform convergence topology of the compacts of X is defined on Γ ( X ) by the subbasis formed of the following sets: [ K , I ] = { f ∈ Γ ( X ): f ( K ) ⊂ I }, where K ⊂ X is a compact set and I is bounded open interval in R. Note that C ( X ) ⊂ Γ ( X ) and C k ( X ) is dense in Γk ( X ) when X is completely regular. In the following, we establish a link between the o-Malykhin property for the space C k ( X ) and the baireness property of the space Γk ( X ). Proposition 3.15. The space C k ( X ) is o-Malykhin since the space Γk ( X ) is Baire. Proof. Let C be a moving off collection of non-empty compacts of X . For each n ∈ N, we denote U n the open set of Γk ( X ) formed of functions f ∈ Γk ( X ) for which there is some compact K ⊂ X with f ( K ) ⊂ (n, ∞[ . Since X is completely regular and C is moving off, then immediately, the set U n is dense in Γk ( X ). As a result, the set n∈N U n is dense in Γk ( X ). Let
f ∈ n∈N U n . For each n ∈ N, there is K n ∈ C with f ( K n ) ⊂ (n, ∞[. The sequence ( K n )n∈N is of course infinite; so all what remains is to prove that it is compact-finite. Let K ⊂ X be a non-empty compact subset of X . As f ( K ) is bounded in R, K will intersect only a finite number of compacts K n . 2 The following question is still open.
Question 1. Suppose that X is a topological space for which C k ( X ) is o-Malykhin. Then, is Γk ( X ) a Baire space? References [1] A.V. Arhangel’skiˇı, On power homogeneous spaces, Topology Appl. 125 (2002) 419–446. [2] A. Bareche, A. Bouziad, Itzkowitz’s problem for groups of finite exponent, Topology Proc. 36 (2010) 375–383. [3] A. Bouziad, Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets and the Prohorov property, Topology Appl. 120 (3) (2002) 283–299. [4] A. Bouziad, J.P. Troallic, Left and right uniform structures on functionally balanced groups, Topology Appl. 153 (2006) 2351–2361. [5] G. Gruenhage, The story of a topological game, Rocky Mountain J. Math. 36 (2006) 1885–1914. [6] G. Gruenhage, D.K. Ma, Baireness of C k ( X ) for locally compact X , Topology Appl. 80 (1997) 131–139. [7] S. Dolecki, G.H. Greco, A. Lechicki, When do the upper topology (homeomorphically, Scott topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc. 347 (1995) 2869–2884. [8] Jiling Cao, W.B. Moors, A survey on topological games and their applications in analysis, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 100 (2006) 39–49. [9] V.I. Malychin, On homogeneous spaces, in: Proceedings of Leningrad International Topology Conference, August 1982, Nauka, Leningrad, 1983, pp. 50– 61. [10] V.I. Malykhin, G. Tironi, Weakly Fréchet–Urysohn and Pytkeev spaces, Topology Appl. 104 (2000) 181–190. [11] T. Nogura, D. Shakhmatov, When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?, Topology Appl. 70 (1996) 213–243. [12] M. Sakai, κ -Fréchet Urysohn property of C k ( X ), Topology Appl. 154 (2007) 1516–1520. [13] D.B. Shakhmatov, A pseudocompact space all countable of which are closed and C ∗ -embedded, Topology Appl. 22 (1986).
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The o-Malykhin property for spaces C k ( X ) A. Bareche Département de Mathématiques, Université de Rouen, UMR CNRS 6085, Avenue de l’Université, BP.12, F76801 Saint-Etienne du Rouvray, France
a r t i c l e
i n f o
Article history: Received 28 September 2011 Received in revised form 8 October 2012 Accepted 9 October 2012 Keywords: o-Malykhin property Moving off property Moving off collection Compact-finite Topological game
a b s t r a c t Let X be a Tychonoff space and C k ( X ) be the vector topological space of continuous realvalued functions on X with the compact open topology. The problem of characterizing C k ( X ) in terms of X has interested several authors; in particular, Gruenhage and Ma for the baireness property and recently Sakai with the κ -Fréchet Urysohn property. Motivated by their works, we are interested in the o-Malykhin property for C k ( X ). In this note, we will show that C k ( X ) is o-Malykhin if and only if every moving off collection of non-empty compacts of X contains an infinite compact-finite collection. We will also characterize the o-Malykhin property for C k ( X ) by a topological game defined on X, and we will give some related results. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Let X be a Tychonoff space. We denote by C k ( X ) (respectively, C p ( X )) the space of real-valued continuous functions on X , endowed with the compact-open topology (respectively, the pointwise convergence topology). A collection K of nonempty compact subsets of X is said to be a moving off collection if, for any compact L ⊂ X , there is some K ∈ K such that K ∩ L = ∅. Note that a moving off collection is always infinite. A collection K of non-empty subsets of X is said to be compact-finite if, for each compact L ⊂ X , the set { K ∈ K: L ∩ K = ∅} is finite. Problems of characterizing C k ( X ) (or C p ( X )) in terms of topological properties of X , have received the attention of many authors. In particular, Gruenhage and Ma [6] have shown that for a q-space X , the space C k ( X ) is Baire if and only if, every moving off collection K of non-empty compacts of X contains an infinite collection { K n : n ∈ N} which has a discrete open expansion in X (means that there are open subsets U n of X , such that U n ⊃ K n for all n ∈ N and the collection {U n : n ∈ N} is discrete). Recently, Sakai [12] has shown that the space C k ( X ) is κ -Fréchet Urysohn iff every moving off collection of non-empty compacts of X admits an infinite strongly compact-finite subcollection. The collection { K n : n ∈ N} of non-empty compacts of X is said to be strongly compact-finite if there exist open subsets U n of X such that U n ⊃ K n for all n ∈ N and the collection {U n : n ∈ N} is compact-finite. The space X is κ -Fréchet Urysohn if for every open U ⊂ X and each point x ∈ U , there is a sequence (xn )n∈N ⊂ U converging to x in X . For more results about these problems, the reader is referred to [6,12] and the references therein. In this work, we will study the property said o-Malykhin, for the spaces C k ( X ). Definition 1.1. The space X is said to be o-Malykhin if for each family ( O i )i ∈ I of open subsets of X and each point x ∈ X satisfying
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x∈
Oi \
i∈I
O i,
i∈I
there exists an infinite J ⊂ I such that the set { j ∈ J : O j ∩ V = ∅} is finite for any neighborhood V of x in X . The o-Malykhin property has been introduced by Bouziad and Troallic [4], as an extension of the Malykhin notion which has been studied by A.V. Arhangel’skiˇı [1]; it plays an important role in the study of Itzkowitz’s problem for topological groups (for more about this problem see [2]). On the other hand, we can consider it in some sense, as a generalization of Reznichenko’s property (weak Fréchet Urysohn property) see [10]. It is clear that every κ -Fréchet Urysohn space is o-Malykhin; but the converse does not hold in general. Indeed, the space C p ( X ) is o-Malykhin if X is completely regular; this is because C p ( X ) is a dense subspace in the product space R X which is known to be Malykhin (thus o-Malykhin), see [9]. However, it is well-known that the space C p (R) is not κ -Fréchet Urysohn. In the next section, we shall prove that the space C k ( X ) is o-Malykhin if and only if every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. We shall be also characterizing the o-Malykhin property for the space C k ( X ) by a topological game defined on the space X and we shall propose a class of topological spaces X for which the space C k ( X ) is not o-Malykhin. Finally in the last section, we shall compare the o-Malykhin property for the space C k ( X ) with other topological properties, particularly, the κ -Fréchet Urysohn property for the space C k ( X ) and the Prohorov property on the space X . 2. The o-Malykhin property for spaces C k ( X ) In the following, X is assumed to be a completely regular topological space and c 0 is the real-valued constant function defined on X , by c 0 (x) = 0 for all x ∈ X . Theorem 2.1. The following statements are equivalent: (1) The space C k ( X ) is o-Malykhin. (2) Every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. (3) For any sequence (Kn )n∈N of moving off collections of non-empty compacts of X , there exist K ni ∈ Kni such that the sequence ( K ni )i ∈N is compact-finite. Proof. (1) → (2). Let K be a moving off collection of non-empty compacts of X . For K ∈ K, we mean by O K = [ K ; (1, ∞)] the open set of C k ( X ), where [ K ; (1, ∞)] = { f ∈ C k ( X ): | f (x)| > 1 ∀x ∈ K }. It is easy to check that
c0 ∈
OK \
K ∈K
OK,
K ∈K
because X is completely regular. Since C k ( X ) is o-Malykhin, there exists an infinite collection L ⊂ K such that the set { L ∈ L: O L ∩ O = ∅} is finite for every neighborhood O of c 0 in C k ( X ). We fix a non-empty compact K ⊂ X and a positive integer n. Then, considering the neighborhood [ K ; (−1/n, 1/n)] of c 0 , we can see that the set M = { L ∈ L: O L ∩ [ K ; (−1/n, 1/n)] = ∅} is finite. Consequently, for each L ∈ L \ M, the set [ L ; (1, ∞)] ∩ [ K ; (−1/n, 1/n)] is not empty, thus L ∩ K = ∅. As a result, the collection L is compact-finite. (2) → (3). Let (Kn )n∈N be a sequence of moving off collections of non-empty compacts of X . By (2), to each n ∈ N, we associate an infinite countable compact-finite subcollection of Kn that we denote Ln , where Ln = { K nj : j ∈ N}. For n ∈ N, let L n = K 0j ∪ K 1j ∪ · · · ∪ K nj , with j i ∈ N, K ij ∈ Ln , 0 i n; hence L n is a non-empty compact subset of X . It is not difficult n 0 1 i to observe that T = { L n : n ∈ N} is a moving off collection of non-empty compacts of X ; using (2) again, we can obtain a n subcollection of T , which is infinite and compact-finite. This allows us to extract a subsequence K j i ∈ Kni , i ∈ N, such that ni
n
K = ( K jni )i ∈N is an infinite compact-finite sequence. i (3) → (1). Since the space C k ( X ) is homogeneous (as a vector topological space), in order to prove that it is o-Malykhin, it suffices to check that the o-Malykhin property holds at the origin c 0 . Let ( O i )i ∈ I be a family of non-empty open subsets of C k ( X ) with
c0 ∈
i∈I
Oi \
O i.
i∈I
For each i ∈ I and each f i ∈ O i , there is an open set O i ( f i ) = [ K i ( f i ), f i , εi ( f i )] ⊂ O i , where K i ( f i ) is a non-empty compact subset of X , εi ( f i ) is a positive real number and [ K i ( f i ), f i , εi ( f i )] = { g ∈ C k ( X ): | f i (x) − g (x)| < εi ( f i ) ∀x ∈ K i ( f i )}. For k ∈ N, with k > 0, let C i ( f i , k) = K i ( f i ) ∩ ( X \ f i−1 (−1/k, 1/k)) (with i ∈ I , f i ∈ O i ). We put
I 0 = k ∈ N \ {0}: there are i ∈ I , f i ∈ O i with C i ( f i , k) = ∅
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and
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I 1 = k ∈ N \ {0}: C i ( f i , k) = ∅ ∀i ∈ I and f i ∈ O i . We are studying the two following cases: (a) I 0 is infinite. Let g be a mapping defined over I 0 into I as follows: for each k ∈ I 0 , g (k) = ik ∈ I with C ik ( f ik , k) = ∅ and f ik ∈ O ik . Now, we consider ik ∈ g ( I 0 ). As the space X is completely regular and the set K ik ( f ik ) is compact, we can find
a continuous function g ik : X → [0, 1] such that g ik (x) = 1 for all x ∈ K ik ( f ik ) and g ik (x) = 0 for all x ∈ X \ f i−1 (−1/k, 1/k). k We define the function h ik by h ik = f ik · g ik . Clearly, h ik is continuous on X , moreover, h ik ( X ) ⊂ (−1/k, 1/k) and h ik ∈ O ik . It follows that the sequence (h ik )k∈ I 0 converges uniformly to c 0 , thus c 0 ∈ {h ik : k ∈ I 0 }.
The subset J = g ( I 0 ) of I is infinite, because we have at the same time c 0 ∈ / O j for each j ∈ J and c 0 ∈ j ∈ J O j . Now, we fix a non-empty compact K ⊂ X and a positive integer n. Then, there is kn ∈ I 0 such that for every k ∈ I 0 with k kn , we have h ik ∈ [ K ; (−1/n, 1/n)] (means that [ K ; (−1/n, 1/n)] ∩ O ik = ∅). As a consequence, the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is finite. (b) I 1 is infinite. Wetake k ∈ I 1 and we consider the collection Lk = {C i ( f i , k): i ∈ I , f i ∈ O i }. Let K ⊂ X be a non-empty compact set. As c 0 ∈ i ∈ I O i , there are ik ∈ I and f ik ∈ O ik with f ik ( K ) ⊂ (−1/k, 1/k). This gives C ik ( f ik , k) ∩ K = ∅. Then, for all k ∈ I 1 , Lk is a moving off collection of non-empty compacts of X . By (3), there is some infinite J 1 ⊂ I 1 such that the collection K = {C ik ( f ik , k): k ∈ J 1 } is infinite and compact-finite. We suppose without loss of generality, that for all k, l ∈ J 1 , we will have C ik ( f ik , k) = C il ( f il , l) if k = l or ik = il . Define a mapping h over J 1 into I , which associates to each k ∈ J 1 an index h(k) = ik ∈ I such that C ik ( f ik , k) ∈ K. Let J = h( J 1 ). We show first that J is infinite. For every non-empty compact L ⊂ X and every positive integer m, we put S ( L ) = {k ∈ J 1 : L ∩ C ik ( f ik , k) = ∅} and S m ( L ) = {ik ∈ J : k max(m, S ( L ))}. The set S ( L ) is finite since the collection K is compactfinite, and S m ( L ) is obviously finite. For k ∈ J 1 with k > max(m, S ( L )), we have ik ∈ J \ S m ( L ), hence L ∩ C ik ( f ik , k) = ∅. If f ik ( L ) ⊂ (−1/k, 1/k), then f ik ∈ [ L ; (−1/k, 1/k)] ∩ O ik . If not, we consider the compact L k = L ∩ ( X \ f i−1 (−1/k, 1/k)) k which is disjoint from K ik ( f ik ). By this, we will get a continuous function g ik : X → [0, 1] such that g ik ( L k ) ⊂ {0} and g ik ( K ik ( f ik )) ⊂ {1}. Let h ik = g ik · f ik . It is evident that h ik is a continuous function belonging to O ik ∩ [ L ; (−1/k, 1/k)]. We conclude in the two cases that [ L ; (−1/m, 1/m)] ∩ O ik = ∅, thus
c0 ∈
O j.
j∈ J
Therefore, J is infinite because c 0 ∈ / O j for all j ∈ J . At this moment, we take a fixed non-empty compact K of X and a fixed integer n > 0. We will show that the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is finite. As above, the sets S ( K ) = {k ∈ J 1 : K ∩ C ik ( f ik , k) = ∅} and S n ( K ) = {ik ∈ J : k max(n, S ( K ))} are finite; so J \ S n ( K ) is infinite. Let ik ∈ J \ S n ( K ); if f ik ( K ) ⊂ (−1/k, 1/k), it will have
f ik ∈ [ K ; (−1/k, 1/k)] ∩ O ik ⊂ [ K ; (−1/n, 1/n)] ∩ O ik . If not, the compact K ∩ ( X \ f i−1 (−1/k, 1/k)) will be denoted by K k k and then we will get a continuous function g ik : X → [0, 1] such that g ik ( K k ) ⊂ {0} and g ik ( K ik ( f ik )) ⊂ {1}. The function h ik = g ik · f ik belongs to the intersection [ K ; (−1/n, 1/n)] ∩ O ik . Finally, we conclude that for each j ∈ J \ S n ( K ), [ K ; (−1/n, 1/n)] ∩ O j = ∅. As a result, the set { j ∈ J : [ K ; (−1/n, 1/n)] ∩ O j = ∅} is included in the finite set S n ( K ), which completes the proof. 2
In order to characterize the o-Malykhin property for C k ( X ) in terms of topological games, we take the game introduced by Gruenhage and Ma in [6] with some modification. The modification will concern only the winning condition and our game will be presented as follows: The infinite game G ( X ) is defined on the space X , its players are respectively denoted by P1 and P2. The play runs alternatively between P1 and P2, where at each step n, P1 chooses a non-empty compact K n ⊂ X while P2 responds by giving a non-empty compact L n ⊂ X \ K n . Player P1 wins the play ( K n , L n )n∈N if the sequence ( L n )n∈N chosen by P2 is compact-finite; otherwise, P2 is the winner of this play. (For more informations about these games and their applications see [5] and [8].) Theorem 2.2. The space C k ( X ) is o-Malykhin if and only if Player P2 has no winner strategy in the game G ( X ). Proof. Since the space C k ( X ) is o-Malykhin, the condition (3) of Theorem 2.1 is satisfied. As a consequence, the fact that Player P2 has no winning strategy in the game G ( X ) can be established in the same way as the implication (ii) → (iii) in Theorem 2.3 from [6]. To prove the reverse implication, suppose the space C k ( X ) is not o-Malykhin. By Theorem 2.1, there is a moving off collection K of non-empty compacts of X , no infinite subcollection of which is compact-finite. Player P2 can win the play by choosing its compacts in K, thus (2) cannot be satisfied, which completes the proof. 2 At the end of this section, it is natural to ask if there exists a topological space X , which is completely regular and for which the space C k ( X ) is not o-Malykhin. In fact, to answer this question, it suffices to consider the class of metrizable
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spaces which are not locally compact, as Q the space of rational numbers or also P the space of irrational numbers. This result will be established by the following statement. Proposition 2.3. Let X be a topological space for which the space C k ( X ) is o-Malykhin. Suppose that each point x ∈ X admits a sequence of neighborhoods (U n )n∈N satisfying the following property: for every sequence (xn )n∈N ⊂ X with xn ∈ U n ( for all n ∈ N), there is some compact K ⊂ X which contains all terms of the sequence (xn )n∈N . Then, X is locally compact. Proof. Suppose that X is not locally compact. So we can find a point x ∈ X which has no closed compact neighborhood. Let (U n )n∈N be a sequence of neighborhoods of x in X that satisfies proposition’s property. For every n ∈ N, we take Kn = {{ y }: y ∈ U n }. Clearly, no member of (U n )n∈N has a compact closure in X . This permits us to conclude that (Kn )n∈N is a sequence of moving off collections of non-empty compacts of X . Using the condition (3) of Theorem 2.1, we can extract a sequence zni ∈ U ni such that the collection K = {{ zni }: i ∈ N} is compact-finite. However, the compact K = { zn : n ∈ N} contains all elements of the collection K, this produces a contradiction. 2 3. Some related results Let X be a topological completely regular space. This section is devoted to establish the relation between the space X such that C k ( X ) is o-Malykhin and some well-known topological spaces. According to Gruenhage and Ma [6], we say that the space X has the moving off property MOP if every moving off collection of non-empty compacts of X contains an infinite subcollection which has an open discrete expansion. Following Bouziad [3], the space X satisfies the weak moving off property WMOP if every moving off collection of non-empty compacts of X contains an infinite subcollection which is locally finite. Definition 3.1. We say that the space X has the weak weak moving off property WWMOP if every moving off collection of non-empty compacts of X admits an infinite subcollection which is compact-finite. It is evident that the MOP property implies the WMOP property and that the WMOP property implies the WWMOP property. However, the converses do not hold in general. Indeed, in his paper [13], Shakhmatov has proved the existence of a pseudocompact space, in which all countable subsets are closed. Such a space (as mentioned in [3]) satisfies the WMOP, but it does not verify the MOP. In the other hand, the space cited in Example 3.5 (below) has the WWMOP, but it has not the WMOP. Remark 3.2. The two points of this remark will be used later. (1) Following Sakai [12], the space X verifies the WMOP since the space C k ( X ) is κ -Fréchet Urysohn. (2) By Theorem 2.1, the space C k ( X ) is o-Malykhin if and only if the space X verifies the WWMOP. Definition 3.3. The space X is said to be hemicompact if there is some sequence ( K n )n∈N of compact subsets of X , such that for every compact K ⊂ X , there exists n ∈ N with K ⊂ K n . Proposition 3.4. Let X be a hemicompact space. Then the space C k ( X ) is o-Malykhin. Proof. Let C be an arbitrary moving off collection of non-empty compacts of X . By Remark 3.2 (2), to establish the o-Malykhin property on C k ( X ), it will be sufficient to find an infinite compact-finite sequence (C n )n∈N ⊂ C . of of X satisfying the hemicompact condition. Let C 1 ∈ C with K 1 ∩ C 1 = ∅. For Let ( K n )n∈N be the sequence ncompacts n −1 n 2, we put L n = ( i =1 K i ) ∪ ( i =1 C i ) and we choose C n ∈ C such that C n ∩ L n = ∅. Then, we get a sequence (C n )n∈N with the required conditions. Indeed, for every compact K ⊂ X , there is some n ∈ N such that K ⊂ K n ; thus K ⊂ L m and C m ∩ K = ∅ for all m n. 2 As mentioned in the introduction, every κ -Fréchet Urysohn space is o-Malykhin, hence the space C k ( X ) will be o-Malykhin if it is κ -Fréchet Urysohn. This can be also concluded from Remark 3.2. Yet, we can ask if the converse is true. The following example gives a negative answer to this question.
/ M ∪ N. Example 3.5. Let N be a countable set and M be a non-countable set. Let ∗ and ∗∗ be two points such that ∗, ∗∗ ∈ We consider the two spaces Y = N ∪ {∗} and Z = M ∪ {∗∗}, endowed with their topologies as follows: All points of N are isolated in Y and the family { A ∪{∗}} (with A ⊂ N and N \ A is finite) represents a basis neighborhoods of ∗ in Y . All points of M are isolated in Z and the sets of the form B ∪ {∗∗} (with B ⊂ M and M \ B is countable) are considered as basis neighborhoods of ∗∗ in Z .
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Then for X = (Y × Z )\{(∗, ∗∗)} endowed with the product topology, the space C k ( X ) is o-Malykhin and it is not Urysohn.
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κ -Fréchet
Proof. According to Remark 3.2 (2) and Theorem 2.3 in [12], it suffices to show that X has the WWMOP property and that there is some moving off collection of compact subsets of X , no infinite subcollection of which is strongly compact-finite. Let C be an arbitrary moving off collection of non-empty compacts of X . Note that for every compact K ⊂ X , there exists a finite set B ⊂ Z such that K ⊂ Y × B. So we can choose a sequence (C n )n∈N ⊂ C for which there exists a pairwise disjoint sequence ( B n )n∈N ⊂ Z of finite sets, such that C n ⊂ Y × B n for all n ∈ N. It is easy to check that the sequence (C n )n∈N is compact-finite. In the other hand, the collection {{(n, ∗∗)}: n ∈ N} is moving off and does not admit any infinite subcollection being strongly compact-finite. Indeed, for every n ∈ N, there is an open setU n containing (n, ∗∗) such U n = {n} × O n , where M \ O n is countable for all n. Since M is not countable, the set O = n∈N O n is not empty. Finally, the set Y × {x} with x ∈ O is a compact which intersects all open sets U n . 2 Remark 3.6. The o-Malykhin property for the space C k ( X ) is determined intrinsically by compacts of the space X . More precisely, suppose that we have defined two topologies on a set X . If these topologies have the same compacts, then the corresponding spaces C k ( X ) will (or not) o-Malykhin simultaneously, although the two topologies may be different. In particular, the case when X is a topological space and Z is its k-modification (i.e., O ⊂ X is open in Z if its intersection with any compact of X is open in this compact); then C k ( X ) is o-Malykhin if and only if C k ( Z ) is o-Malykhin. Indeed, it is well-known that X and Z have the same compacts. By (2) of Remark 3.2, the space C k ( X ) is o-Malykhin for every space X satisfying the WMOP property. Example 3.7. Suppose that X is a pseudocompact space and all countable subsets of which are closed. Then the space C k ( X ) is o-Malykhin. Conversely, if X is a k-space and C k ( X ) is o-Malykhin, we will have the following result: Proposition 3.8. Let X be a k-space for which C k ( X ) is o-Malykhin. Then X verifies the WMOP property. Proof. It is well-known that every compact-finite collection of non-empty compacts in a k-space is locally finite. Combining this fact with (2) of Remark 3.2, we get the conclusion. 2 Let μ be a positive Borel measure over a topological space X . We say that the measure μ is τ -additive (or τ -smooth) if μ( U ) = sup{μ(U ): U ∈ U } for every upwards directed collection U of open subsets of X . We denote by P τ ( X ) the space of positive τ -additive probability measures over X , endowed with the weak topology. The weak topology on P τ ( X ) is defined by the following property: a net (μα ) ⊂ P τ ( X ) converges to μ in P τ ( X ) if and only if lim inf μα (U ) μ(U ) for any open set U ⊂ X . The subspace of P τ ( X ) formed of all Radon (or tight) probability measures on X will be denoted by P t ( X ). Recall that μ is Radon if μ is inner regular with respect to compact subsets of X in the sense that for every open set A ⊂ X , μ( A ) = sup{μ( K ): K ⊂ A }, where K is a compact subset of X . Definition 3.9. The space X is said to be Prohorov if X is regular and every (relatively) compact set A ⊂ P t ( X ) is uniformly tight, that is, for every 0 < ε < 1 there exists a compact set K ⊂ X such that μ( K ) ε for all μ ∈ A. The symbol F( X ) denotes the lattice of all closed subspaces of X . Recall that K( X ) is the set of all compact subsets of X . The co-compact topology defined on F( X ), that we denote Tco , is generated by the subbasis { O K : K ∈ K( X )}, where O K = { F ∈ F( X ): F ∩ K = ∅}. In the other hand, the upper Kuratowski topology Tu K on F( X ), is the topology associated to the upper Kuratowski convergence defined as follows: a net ( F γ )γ ∈Γ ⊂ F( X ) converges to F ∈ F( X ), with respect to the
upper Kuratowski convergence, if and only if { { F γ : γ α }: α ∈ Γ } ⊂ F . The following notion has been introduced by Nogura and Shakhmatov in [11] after the work of Dolecki, Greco and Lechicki [7]. Definition 3.10. Let H( X ) be a sublattice of F( X ). The space X is called H-trivial if the two topologies Tu K and Tco coincide on H( X ). The F-trivial spaces are exactly the consonant spaces first defined by Dolecki, Greco and Lechicki. Proposition 2.3 and Corollary 3.4 in [3] give the following:
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Corollary 3.11. Let X be a k-space for which the space C k ( X ) is o-Malykhin. Then X is a K-trivial space and thus a Prohorov space. Corollary 3.12. Let X be such that the space C k ( X ) is o-Malykhin, then the k-modification of X is a K-trivial space and consequently it is a Prohorov space. The following example shows that spaces X (with C k ( X ) is o-Malykhin) are not consonant in general.
/ A × N. We call S A the Fréchet–Urysohn fan space of Definition 3.13. Let A be an infinite set and S A = ( A × N) ∪ {∗}, with ∗ ∈ size | A |, if the set S A is endowed with the following topology. All points of A × N are isolated in S A . A basis neighborhoods of ∗ in S A is given by the family {U f : f ∈ N A }, where U f = {(a, n): n f (a)} ∪ {∗} and N A represents the set of all functions defined from A to N. Example 3.14. The Fréchet–Urysohn fan space S A , where the set A is not countable, cannot be consonant; see [11]. However, the space C k ( S A ) is always o-Malykhin since the space S A has the MOP property (see Proposition 2.4 [3]). In what follows, we denote by Γ ( X ) the set of all real-valued functions which are bounded on every compact of X . By equipping Γ ( X ) with the uniform convergence topology of compact subsets of X . We get a topological vector space that we denote Γk ( X ). The uniform convergence topology of the compacts of X is defined on Γ ( X ) by the subbasis formed of the following sets: [ K , I ] = { f ∈ Γ ( X ): f ( K ) ⊂ I }, where K ⊂ X is a compact set and I is bounded open interval in R. Note that C ( X ) ⊂ Γ ( X ) and C k ( X ) is dense in Γk ( X ) when X is completely regular. In the following, we establish a link between the o-Malykhin property for the space C k ( X ) and the baireness property of the space Γk ( X ). Proposition 3.15. The space C k ( X ) is o-Malykhin since the space Γk ( X ) is Baire. Proof. Let C be a moving off collection of non-empty compacts of X . For each n ∈ N, we denote U n the open set of Γk ( X ) formed of functions f ∈ Γk ( X ) for which there is some compact K ⊂ X with f ( K ) ⊂ (n, ∞[ . Since X is completely regular and C is moving off, then immediately, the set U n is dense in Γk ( X ). As a result, the set n∈N U n is dense in Γk ( X ). Let
f ∈ n∈N U n . For each n ∈ N, there is K n ∈ C with f ( K n ) ⊂ (n, ∞[. The sequence ( K n )n∈N is of course infinite; so all what remains is to prove that it is compact-finite. Let K ⊂ X be a non-empty compact subset of X . As f ( K ) is bounded in R, K will intersect only a finite number of compacts K n . 2 The following question is still open.
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