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Countably compact spaces with n-in-countable weak bases are metrizable
Topology and its Applications 176 (2014) 43–50
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Topology and its Applications www.elsevier.com/locate/topol
Countably compact spaces with n-in-countable weak bases are metrizable ✩ Zuoming Yu School of Zhangjiagang, Jiangsu University of Science and Technology, Zhangjiagang 215600, PR China
a r t i c l e
i n f o
Article history: Received 26 August 2013 Received in revised form 23 July 2014 Accepted 23 July 2014 Available online xxxx MSC: 54D30 54D70 54E35
a b s t r a c t We prove that a Hausdorff compact space with an n-in-countable weak base is metrizable for each n ∈ N . This result gives a positive answer to a question of Bennett and Martin asking if a compact Hausdorff space with a 2-in-finite weak base is metrizable. We also discuss the properties of spaces with n-in-countable bases, monotonically monolithic spaces and spaces with property (G). Finally, we show that an example of Davis, Reed and Wage provides consistent negative answers to two problems raised by Tkachuk on monotonically monolithic spaces. © 2014 Elsevier B.V. All rights reserved.
Keywords: n-in-countable weak base Property (G) Monotonically monolithic
1. Introduction T. Hoshina has shown that every compact Hausdorff space with a point-countable weak base is metrizable [11]. In view of this result, Bennett and Martin raised a question in [5]: Question 1.1. Is every compact Hausdorff space with a 2-in-finite weak base metrizable? In [18], it was proved that if X is a compact space with |X| < b and X has a 2-in-finite weak base B = {Bx : x ∈ X} which consists of sequentially closed subsets of X, then X is metrizable ([18], Theorem 3.12). This result answers Question 1.1 partially. ✩ This paper is supported by Science and Technology Department of Jiangsu Province (No. BK20140503) and Young Teacher Research Funds of Jiangsu University of Science and Technology (No. 112110144). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.topol.2014.07.008 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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In this note, we answer Question 1.1 completely by proving the following result: For each positive integer n, a countably compact Hausdorff space X with an n-in-countable weak base is metrizable. Monotonically monolithic spaces and strongly monotonically monolithic spaces, introduced by Tkachuk [15], will be used in solving Question 1.1. It is easy to check that a space with a point-countable base is strongly monotonically monolithic. In this paper, we prove that spaces with n-in-countable bases for some n ∈ N , are monotonically monolithic. Spaces with property (G) is another generalization of spaces with point-countable bases. It was proved [8] that spaces with property (G) are monotonically monolithic. Tkachuk asked the following questions in [16]: Question 1.2. Is it true that every strongly monotonically monolithic space has a point-countable base? Question 1.3. Is it true that every strongly monotonically monolithic space has property (G)? It is proved in [6] that every space with property (G) is hereditarily metalindelöf. Tkachuk also asked the following question in [16]: Question 1.4. Is it true that every strongly monotonically monolithic space is metalindelöf? In [18], by using an example of Davis, Reed and Wage, we provided a consistent negative answer to Question 1.2. In this note, by using the same example, we show that negative answers to Question 1.3 and Question 1.4 are also consistent. 2. Notation and terminology All the spaces in this paper are Tychonoff and N denotes the set of positive integers. Suppose that A is a family of subsets of a space X. For a positive integer n, A is called n-in-countable if every set A ⊆ X with |A| = n is contained in at most countably many elements of A. The family A is called 2-in-finite if every two-point subset of X is contained in only finitely many elements of A. A 2-in-finite base was also called a weakly uniform base in [7] and [10]. In this note, we only use the term 2-in-finite base. The concept of 2-in-finite base is useful in resolving certain unsolved problems regarding spaces with a Gδ -diagonal since every Hausdorff space with a 2-in-finite base has a Gδ -diagonal [10]. Let P be a topological property or a class of spaces. A space X is said to be star P if for any open cover U of X there is a subspace Y ∈ P of X such that st(Y, U) = X, where st(Y, U) = {U ∈ U : U ∩ Y = ∅} [13]. For an infinite cardinal κ, a space X is called κ-monolithic [2] if nw(A) ≤ κ for any set A ⊂ X with |A| ≤ κ, and we say that a space X is monotonically κ-monolithic (strongly monotonically κ-monolithic) [17], if we can assign to each A ⊂ X with |A| ≤ κ a collection O(A) of subsets (open subsets) of X satisfying the following conditions: (a) |O(A)| ≤ κ; (b) if A ⊂ B, then O(A) ⊂ O(B); (c) if α ≤ κ is an ordinal and we have a family {Aβ : β < α} of subsets of X such that β < β implies Aβ ⊂ Aβ , then O( β<α Aβ ) = β<α O(Aβ ); (d) if U is open and x ∈ A ∩ U , then there is O ∈ O(A) with x ∈ O ⊂ U . A space X is called weakly monotonically κ-monolithic if it has an operator satisfying the above conditions but with condition (d) replaced by (d ) if A is not closed, then O(A) contains a network at some point x ∈ A \ A.
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We say that a space X is (weakly, strongly) monotonically monolithic if it is (weakly, strongly) monotonically κ-monolithic for any infinite cardinal κ. It is proved in [15] that every monotonically monolithic space is a D-space. Assume that X is a space and x ∈ X. We say that X is weakly countably tight at x if there is a countable subset A of X \ {x} such that x is in the closure of A. A space X is called weakly countably tight if X is weakly countably tight at every x ∈ X [3]. A weak base for a space X is a collection P = {Px : x ∈ X} of subsets of X such that (1) for each x, the family Px is closed under finite intersections and x ∈ Px , and (2) a subset U of X is open if and only if for each x ∈ U , there is a P ∈ Px such that x ∈ P ⊂ U . Let P = {Px : x ∈ X} be a weak base on X. A subset U of X is called a weak neighborhood of x if there is a P ∈ Px such that x ∈ P ⊆ U . A space X is said to be g-first countable if X has a weak base P such that Px is countable for each x ∈ X. A space X is a Lindelöf Σ-space if there is countable family F of subsets of X such that F is a network with respect to a compact cover C of the space X. 3. Metrizability of spaces with n-in-countable weak bases Lemma 3.1. ([18]) Suppose that a regular countably compact space X has an n-in-countable weak base for some n ∈ N . Then X is sequentially compact. Lemma 3.2. ([17]) If X is a monotonically ω-monolithic countably compact space, then X is a monotonically monolithic Corson compact space. Lemma 3.3. ([4]) Every subspace of a (strongly) monotonically κ-monolithic space is (strongly) monotonically κ-monolithic. Theorem 3.4. Suppose that a regular countably compact space X has an n-in-countable weak base B for some n ∈ N . Then X is metrizable. Proof. Let K be the set consisting of all the accumulation points of X, and K = {x ∈ X : there is a nontrivial sequence {xm : m ∈ N } of X such that xm → x}. Let B = {Bx : x ∈ X} be an n-in-countable weak base for some n ∈ N on X. Claim 1. If L is a countable subset of X, then L is metrizable. It is easy to prove that {B ∩ L : B ∈ B} is an n-in-countable weak base on L. Let D be the set consisting of all the accumulation points of L, and D = {x ∈ X: there is a nontrivial sequence {xm : m ∈ N } of L such that xm → x}. We know that neither D nor D is empty since L is sequentially compact by Lemma 3.1. We will prove that D is dense in D. For any y ∈ D \ D , let U be an open subset in D containing y. We can pick an open subset U of X containing y such that U ∩ D ⊂ U because X is regular. Since y is an accumulation point of L, we have |U ∩L| = ω, which means that |U ∩L| = ω. Therefore, there is a nontrivial sequence {xm : m ∈ N } of U ∩ L and a point z in X such that xm → z since X is sequentially compact by Lemma 3.1. The point z belongs to D by the definition of D . Now we can see that z ∈ U ∩ D ⊂ U . Hence D is dense in D. Let BD = x∈D Bx . By the definition of D , we can see that |B ∩ L| = ω for each B ∈ BD . This fact shows that BD is countable since BD ⊂ H∈L<ω ,|H|=n {B ∈ B : H ⊂ B}. Thus, {B ∩ D : B ∈ BD }
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is a countable network on D . Therefore, D is a countably compact space with a Gδ -diagonal since D is sequentially compact by Lemma 3.1. Thus, the subspace D is metrizable. So, D is compact and D is equal to D. Let NL = {{x} : x ∈ L} ∪ {B ∩ D : B ∈ BD }. We can see that NL is a countable network on L. This shows that L is metrizable. Claim 2. The space X is monotonically ω-monolithic. For each finite subset F of X, we put HF = {{y} : y ∈ F } if |F | < n. If |F | ≥ n, then let HF = {B : B ∈ B, F ⊂ B}. For each countable subset A of X, let O(A) = F ∈[A]<ω HF . Obviously, O(A) is countable since B is n-in-countable. It is not difficult to prove that O(A) satisfies the conditions (a), (b) and (c) in the definition of monotonically ω-monolithic spaces. Obviously, F ∈[A]<ω OF contains a network at each point of A. Pick any point x ∈ A \ A. There is a nontrivial sequence {xm : m ∈ N } of A such that xm → x by Claim 1. For each B ∈ Bx , we have |B ∩ {xm : m ∈ N }| = ω. Therefore, Bx ⊂ O(A). This shows that O(A) contains a network at x, and it follows that Claim 2 is true. Claim 3. K is dense in K. Let x ∈ K and let U be any open set in K with x ∈ U . Choose open sets U and V of X such that U = U ∩ K and x ∈ V ⊂ V ⊂ U . Then V is a closed countably compact subspace of X, and there is some point z ∈ V and a sequence {zm : m ∈ N } ⊂ V such that zm → z since X is sequentially compact. So z ∈ K and z ∈ U ∩ K = U , which means that K is dense in K.
Claim 4. K = K . We can see that K is countably compact since X is sequentially compact by Lemma 3.1. By Lemma 3.3, we know that K is monotonically ω-monolithic. So K is a D-space by Lemma 3.2. Thus, K is compact since K is countably compact. Therefore, K is a closed subset of X and it follows that K = K. Claim 5. X is metrizable. By Claim 4 and Lemma 3.2, X is a g-first countable Fréchet space. It is proved that a weak neighborhood of a point must be a neighborhood of this point in a g-first countable Fréchet space [1]. It follows that B = {B o ∩ K : B ∈ B} is an n-in-countable base of K. By the definition of K and Claim 4, B is also point-countable in K. Thus, K is metrizable since K is countably compact. Let S be a countable dense subset of K. We show that |H = {Bz : z ∈ K}| = ω. It is obvious that |Bz | = ω for each z ∈ S. Notice that there is some subsequence of S convergent y to for each y ∈ K \ S. It means that | {Bz : z ∈ K \ S}| = ω since B is n-in-countable. Let W = {Bz : z ∈ K} {{x} : x ∈ X \ K}. We can see that W is a point-countable base of X. Hence, X is metrizable because it is countably compact. 2 Theorem 3.4 answers Question 1.1 positively. Moreover, the following results can be taken as corollaries of Theorem 3.4: Corollary 3.5. ([18]) Every countably compact space with an n-in-countable base for some n ∈ N is metrizable. Corollary 3.6. Every star compact space X with an n-in-countable weak base B for some n ∈ N is star countable.
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Proof. Let U be an open cover on X. There is a compact subset K such that st(K, U) = X. It follows from Theorem 3.4 that K is separable. Suppose that L is a countable dense subset of K. It is easy to check that st(L, U) = X. 2 Lemma 3.7. ([8]) A space X is monotonically monolithic (weakly monotonically monolithic) iff one can assign to each finite subset F of X a countable collection O(F ) of subsets of X such that, for each A ⊂ X, F ∈[A]<ω O(F ) contains a network at each point of A (at some point of A \ A if A is not closed). Theorem 3.8. Every space X with an n-in-countable weak base B for some n ∈ N is weakly monotonically monolithic. Proof. Let B be an n-in-countable weak base for some n ∈ N on X. We only need to prove that X satisfies the condition of Lemma 3.7. For each finite subset F of X, if |F | < k, then put HF = {{x} : x ∈ F }; If |F | ≥ k, then put HF = {B : B ∈ B, F ⊂ B}. For each subset A of X, let O(A) = F ∈[A]<ω HF . Suppose that A is a non-closed subset of X. By the definition of weak base, there is some x ∈ A \ A such that |B ∩ A| ≥ ω for each B ∈ Bx . It is not difficult to prove that Bx ⊂ {B ∈ B: there is some finite subset L of A with |L| = n such that L ⊂ B}. It follows that O(A) contains a network at x. 2 It is proved that for a star countable space X with an n-in-countable base for some n ∈ N , e(X) is not greater than ω [19]. It is natural to ask whether this result holds if we replace “n-in-countable base” with “n-in-countable weak base”. However, the space constructed in Example 7 in [14] is a regular separable space, hence a star countable space, with a point-countable weak base such that it is not Lindelöf. Therefore, the answer of this question is negative. Theorem 3.9. Every star compact metalindelöf space X with an n-in-countable weak base B for some n ∈ N is metrizable. Proof. Let U be an open cover on X. Take a point-countable open cover V which refines U. Then we can take a countable subset of V which also covers X by Corollary 3.6. It follows that X is a Lindelöf space. Because normal star compact spaces are countably compact [12], X is metrizable by Theorem 3.4. 2 Although it is proved that every star compact regular space with an n-in-countable base for some n ∈ N is metrizable [18], the authors do not know the answer to the following question: Question 3.10. Is a star compact regular space with an n-in-countable weak base for some n ∈ N metrizable? 4. Spaces with n-in-countable bases or property (G) Theorem 4.1. If X is a space with an n-in-countable base for some n ∈ N , then X is monotonically monolithic. Proof. Let B be an n-in-countable base for some k ∈ N on X. For each finite subset F of X and each L ⊂ F , if |L| < n, then we put H(L) = {{y} : y ∈ L}; if |L| ≥ n, then we assign H(L) to L as {B : B ∈ O(B), there is some subset L ⊂ L with |L | = n and L ⊂ B}. It is easy to check that O(F ) = L⊂F H(L) is the operator described in Lemma 3.7. 2 A cover L of a space X is thick [9] if we can assign LH ∈ [L]<ω and LH = LH to each H ∈ [X]<ω in such a way that A ⊂ {LH : H ∈ [A]<ω } for each A ⊂ X. The space X is thickly covered if every open cover of X is thick.
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Notice that every monotonically monolithic space X is hereditarily thickly covered [9], we have the following corollary of Theorem 4.1: Corollary 4.2. If X is a space with an n-in-countable base for some n ∈ N , then X is hereditarily thickly covered. Lemma 4.3. ([19]) If X is a star countable space with an n-in-countable base for some n ∈ N , then e(X) = ω. Theorem 4.4. If X is a star σ-compact space with an n-in-countable base B for some n ∈ N , then X is Lindelöf. Proof. Let U be an open cover on X. There is a σ-compact subset F = i∈N Fi such that st(F, U) covers X. Obviously, {Fi ∩ B : B ∈ B} is an n-in-countable base on Fi . By Corollary 3.5, Fi is metrizable. Let Li be a countable dense subset of Fi . It is easy to see that st( {Li : i < ω}, U) covers X, and it follows that X is star countable. Hence, e(X) = ω by Lemma 4.3. We know that X is a D-space by Theorem 4.1, so X is Lindelöf. 2 Recall that ω1 is a caliber of a space X if every point-countable family of non-empty open subsets of X is countable. Theorem 4.5. If a space X has an n-in-countable base for some n ∈ N and ω1 is a caliber of X, then X is second countable. Proof. Let B be an n-in-countable base on X for some n ∈ N . If n = 1, then we know that X is second countable by the its caliber. If X is not second countable, then n ≥ 2. We prove that |B| = ω. Otherwise, B is not point-countable since ω1 is a caliber of X. Pick x1 ∈ X such that |{B : x1 ∈ B, B ∈ B}| = ω1 . Put Bx1 = {B\{x1 } : x1 ∈ B, B ∈ B}. The family Bx1 is not point-countable either. Hence, we can choose x2 ∈ X with |{B : x2 ∈ B, B ∈ Bx1 }| = ω1 and set Bx2 = {B \ {x1 , x2 } : x1 ∈ B, B ∈ B}. Continue in this way, at the n-th step, we can choose a subset {x1 , ..., xn } of X such that |{B : {x1 , ..., xn } ⊂ B, B ∈ Bx }| = ω1 . This contracts with the n-in-countableness of B, and which shows that |B| = ω. 2 By using a method similar to that used by Tkachuk in [17], we obtain the following result: Theorem 4.6. Let X be a Lindelöf Σ space with an n-in-countable base B for some n ∈ N . If each point in X is a Gδ set, then X is second countable. Proof. Fix a countable network F with respect to a compact cover C of X. For each x ∈ X, we choose countable open subsets Gx as a witness of the Gδ property at x. For each finite subset A of X, definite an operator O on A as follows: When |A| < n, put O(A) = x∈A Gx ; and when |A| ≥ n, put O(A) = x∈A Gx ∪ {B ∈ B : |B ∩ A| ≥ n}. Take a point x0 ∈ X and let A0 = {x0 }. For m ∈ N , assume that we have countable subsets A0 , ..., Am of X satisfying the following conditions: (1) A0 ⊂ ... ⊂ Am ; (2) for every i < m, if there is a finite family V ⊂ O(Ai ) and F ∈ F such that F \ V = ∅, then (F \ V) ∩ Ai+1 = ∅. For each finite collection V ⊂ O(Am ), if there is F ∈ F such that F \ V = ∅, then choose a point x(V, F ) ∈ F \ V. Put Am+1 = Am ∪ {x(V, F ) : V ∈ [O(Am )]<ω , F ∈ F}. It is trivial that conditions (1)
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and (2) are now satisfied for m + 1. By induction, we can construct a sequence {Ai : i ∈ N } of countable subsets of X such that conditions (1) and (2) hold for all m ∈ N . ∞ Suppose that A = i=0 Ai is not dense in X. We can pick a point x ∈ X \ A. Take a set C ∈ C with x ∈ C. The subset K = C ∩ A is compact. For each y ∈ A \ A and each open neighborhood Uy of y, |Uy ∩ A| = ω. So O(A) contain a local base at y. Thus, we can choose Vy ∈ O(A) with y ∈ Vy and x ∈ / Vy . For y ∈ A, we pick Gy ∈ Gy such that x ∈ / Gy . So {Vy : y ∈ A \ A} ∪ {Gy : y ∈ A} covers K. We can get a finite family W ⊂ O(A) such that x ∈ / W and K ⊂ W. Now we can see that H = W ∪ (X \ A) is an open neighborhood of C. Hence, there is some F ∈ F such that C ⊂ F ⊂ H. Since O(A) = m∈N O(Am ) and the family O(Am ) is increasing, we can find m ∈ N such that W ⊂ O(Am ). Observe that x ∈ F \ W, we know that the point x(W, F ) must belongs to Am+1 . However, x(W, F ) ∈ F \ W ⊂ X \ A, which is a contradiction. So A is a dense subset of X. It is not difficult for us to prove that {B ∈ B : |B ∩ A| ≥ n} ∪ {x ∈ X : x is an isolate point in X} is a countable base on X. 2 Collins and Roscoe introduced the following property in [6]: (open) Property (G): For each x ∈ X, there is an assigned countable collection G(x) of (open) subsets of X such that, whenever x ∈ U , U is open, there is an open V with x ∈ V ⊂ U such that, whenever y ∈ V , then x ∈ W ⊂ U for some W ∈ G(y). Lemma 4.7. ([6]) Every space with property (G) must be hereditarily metalindelöf. The following result follows directly from Lemma 4.7. Theorem 4.8. Every star countable space with property (G) is Lindelöf. It is easy to see that a space with a point-countable base must have property open (G). On the other hand, it is still an open problem whether every space with the property open (G) has a point-countable base. It is showed consistently in [18] that there is a strongly monotonically monolithic space which does not have a point-countable base. To the question that whether every strongly monotonically monolithic space has property (G) raised in [16] (i.e. Question 1.3), we show that a negative answer is also consistent. Lemma 4.9. [18] Suppose that X is weakly countably tight (or a k-space) with an m-in-countable base for some m ∈ N . Then X is strongly monotonically monolithic. The following result shows a negative answer to Question 1.3 consistently and it also shows that Question 1.4 has a consistent negative answer: Theorem 4.10. There is a strongly monotonically monolithic space, which is not metalindelöf, hence does not have property (G). Proof. Davis, Reed and Wage constructed a normal Moore space X with a 2-in-finite base but without a point-countable base under Martin’s Axiom and ω2 < 2ω0 [7]. Then X is strongly monotonically monolithic by Lemma 4.9. We show that X is not metalindelöf. Otherwise, let W = n∈N Wn be a development of X. For each n ∈ N , put Vn be a point-countable open refinement of Wn . It is not difficult to prove that V = n<ω Vn is a point-countable base on X, which is a contradiction. We can also see that X does not have property (G) by Lemma 4.7. 2
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Acknowledgement The authors are grateful to the referee, because he/she made valuable suggestions and helped us to improve the writing of this paper. References [1] A.V. Arhangel’skii, Mappings and spaces, Russ. Math. Surv. 21 (1966) 115–162. [2] A.V. Arhangel’skii, Continuous mappings, factorization theorems and function spaces, Tr. Mosk. Mat. Obˆs. 47 (1984) 3–21. [3] A.V. Arhangel’skii, W. Just, E.A. Reznichenko, P.J. Szeptycki, Sharp bases and weakly uniform bases versus pointcountable bases, Topol. Appl. 100 (2000) 39–46. [4] O.T. Alas, V.V. Tkachuk, R.G. Wilson, A broader context for monotonically monolithic spaces, Acta Math. Hung. 125 (2009) 369–385. [5] H.R. Bennett, W. Martin, A metrization theorem for countably compact spaces, Port. Math. 36 (1977) 247–250. [6] P.J. Collins, G.M. Reed, A.W. Roscoe, The point-countable base problem, in: J. van Mill, G.M. Reed (Eds.), Open Problems in Topology, Elsevier Science Publishers B.V., Amsterdam, 1990, pp. 238–250. [7] S.W. Davis, G.M. Reed, M.L. Wage, Further results on weakly uniform bases, Houst. J. Math. 2 (1) (1976) 57–63. [8] G. Gruenhage, Monotonically monolithic spaces, Corson compacts and D-spaces, Topol. Appl. 159 (2012) 1559–1564. [9] H. Guo, H.J.K. Junnila, D-spaces and thick covers, Topol. Appl. 158 (2011) 2111–2121. [10] R.W. Heath, R.W. Lindgren, Weakly uniform bases, Houst. J. Math. 2 (1) (1976) 85–90. [11] T. Hoshina, On quotient s-images of metric spaces, Sci. Rep. T.K.D. Sec. A 10 (1970) 43–46. [12] J. van Mill, V.V. Tkachuk, R.G. Wilson, Classes defined by stars and neighborhood assignments, Topol. Appl. 154 (2007) 2127–2134. [13] M. Matveev, A survey on star covering properties, Topology Atlas, April 15, 1998. [14] L. Peng, On weakly monotonically monolithic spaces, Comment. Math. Univ. Carol. 51 (2010) 133–142. [15] V.V. Tkachuk, Monolithic spaces and D-spaces revisited, Topol. Appl. 156 (2009) 840–846. [16] V.V. Tkachuk, The Collins–Roscoe property and its applications in the theory of function spaces, Topol. Appl. 159 (2012) 1529–1535. [17] V.V. Tkachuk, Lifting the Collins–Roscoe property by condensations, Topol. Proc. 42 (2013) 1–15. [18] Z. Yu, Z. Yun, On spaces with k-in-countable bases or weak bases, Topol. Appl. 159 (2012) 3545–3549. [19] Z. Yu, A note on the extent of two subclasses of star countable spaces, Cent. Eur. J. Math. 10 (3) (2012) 1067–1070.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Countably compact spaces with n-in-countable weak bases are metrizable ✩ Zuoming Yu School of Zhangjiagang, Jiangsu University of Science and Technology, Zhangjiagang 215600, PR China
a r t i c l e
i n f o
Article history: Received 26 August 2013 Received in revised form 23 July 2014 Accepted 23 July 2014 Available online xxxx MSC: 54D30 54D70 54E35
a b s t r a c t We prove that a Hausdorff compact space with an n-in-countable weak base is metrizable for each n ∈ N . This result gives a positive answer to a question of Bennett and Martin asking if a compact Hausdorff space with a 2-in-finite weak base is metrizable. We also discuss the properties of spaces with n-in-countable bases, monotonically monolithic spaces and spaces with property (G). Finally, we show that an example of Davis, Reed and Wage provides consistent negative answers to two problems raised by Tkachuk on monotonically monolithic spaces. © 2014 Elsevier B.V. All rights reserved.
Keywords: n-in-countable weak base Property (G) Monotonically monolithic
1. Introduction T. Hoshina has shown that every compact Hausdorff space with a point-countable weak base is metrizable [11]. In view of this result, Bennett and Martin raised a question in [5]: Question 1.1. Is every compact Hausdorff space with a 2-in-finite weak base metrizable? In [18], it was proved that if X is a compact space with |X| < b and X has a 2-in-finite weak base B = {Bx : x ∈ X} which consists of sequentially closed subsets of X, then X is metrizable ([18], Theorem 3.12). This result answers Question 1.1 partially. ✩ This paper is supported by Science and Technology Department of Jiangsu Province (No. BK20140503) and Young Teacher Research Funds of Jiangsu University of Science and Technology (No. 112110144). E-mail address: [email protected].
http://dx.doi.org/10.1016/j.topol.2014.07.008 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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In this note, we answer Question 1.1 completely by proving the following result: For each positive integer n, a countably compact Hausdorff space X with an n-in-countable weak base is metrizable. Monotonically monolithic spaces and strongly monotonically monolithic spaces, introduced by Tkachuk [15], will be used in solving Question 1.1. It is easy to check that a space with a point-countable base is strongly monotonically monolithic. In this paper, we prove that spaces with n-in-countable bases for some n ∈ N , are monotonically monolithic. Spaces with property (G) is another generalization of spaces with point-countable bases. It was proved [8] that spaces with property (G) are monotonically monolithic. Tkachuk asked the following questions in [16]: Question 1.2. Is it true that every strongly monotonically monolithic space has a point-countable base? Question 1.3. Is it true that every strongly monotonically monolithic space has property (G)? It is proved in [6] that every space with property (G) is hereditarily metalindelöf. Tkachuk also asked the following question in [16]: Question 1.4. Is it true that every strongly monotonically monolithic space is metalindelöf? In [18], by using an example of Davis, Reed and Wage, we provided a consistent negative answer to Question 1.2. In this note, by using the same example, we show that negative answers to Question 1.3 and Question 1.4 are also consistent. 2. Notation and terminology All the spaces in this paper are Tychonoff and N denotes the set of positive integers. Suppose that A is a family of subsets of a space X. For a positive integer n, A is called n-in-countable if every set A ⊆ X with |A| = n is contained in at most countably many elements of A. The family A is called 2-in-finite if every two-point subset of X is contained in only finitely many elements of A. A 2-in-finite base was also called a weakly uniform base in [7] and [10]. In this note, we only use the term 2-in-finite base. The concept of 2-in-finite base is useful in resolving certain unsolved problems regarding spaces with a Gδ -diagonal since every Hausdorff space with a 2-in-finite base has a Gδ -diagonal [10]. Let P be a topological property or a class of spaces. A space X is said to be star P if for any open cover U of X there is a subspace Y ∈ P of X such that st(Y, U) = X, where st(Y, U) = {U ∈ U : U ∩ Y = ∅} [13]. For an infinite cardinal κ, a space X is called κ-monolithic [2] if nw(A) ≤ κ for any set A ⊂ X with |A| ≤ κ, and we say that a space X is monotonically κ-monolithic (strongly monotonically κ-monolithic) [17], if we can assign to each A ⊂ X with |A| ≤ κ a collection O(A) of subsets (open subsets) of X satisfying the following conditions: (a) |O(A)| ≤ κ; (b) if A ⊂ B, then O(A) ⊂ O(B); (c) if α ≤ κ is an ordinal and we have a family {Aβ : β < α} of subsets of X such that β < β implies Aβ ⊂ Aβ , then O( β<α Aβ ) = β<α O(Aβ ); (d) if U is open and x ∈ A ∩ U , then there is O ∈ O(A) with x ∈ O ⊂ U . A space X is called weakly monotonically κ-monolithic if it has an operator satisfying the above conditions but with condition (d) replaced by (d ) if A is not closed, then O(A) contains a network at some point x ∈ A \ A.
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We say that a space X is (weakly, strongly) monotonically monolithic if it is (weakly, strongly) monotonically κ-monolithic for any infinite cardinal κ. It is proved in [15] that every monotonically monolithic space is a D-space. Assume that X is a space and x ∈ X. We say that X is weakly countably tight at x if there is a countable subset A of X \ {x} such that x is in the closure of A. A space X is called weakly countably tight if X is weakly countably tight at every x ∈ X [3]. A weak base for a space X is a collection P = {Px : x ∈ X} of subsets of X such that (1) for each x, the family Px is closed under finite intersections and x ∈ Px , and (2) a subset U of X is open if and only if for each x ∈ U , there is a P ∈ Px such that x ∈ P ⊂ U . Let P = {Px : x ∈ X} be a weak base on X. A subset U of X is called a weak neighborhood of x if there is a P ∈ Px such that x ∈ P ⊆ U . A space X is said to be g-first countable if X has a weak base P such that Px is countable for each x ∈ X. A space X is a Lindelöf Σ-space if there is countable family F of subsets of X such that F is a network with respect to a compact cover C of the space X. 3. Metrizability of spaces with n-in-countable weak bases Lemma 3.1. ([18]) Suppose that a regular countably compact space X has an n-in-countable weak base for some n ∈ N . Then X is sequentially compact. Lemma 3.2. ([17]) If X is a monotonically ω-monolithic countably compact space, then X is a monotonically monolithic Corson compact space. Lemma 3.3. ([4]) Every subspace of a (strongly) monotonically κ-monolithic space is (strongly) monotonically κ-monolithic. Theorem 3.4. Suppose that a regular countably compact space X has an n-in-countable weak base B for some n ∈ N . Then X is metrizable. Proof. Let K be the set consisting of all the accumulation points of X, and K = {x ∈ X : there is a nontrivial sequence {xm : m ∈ N } of X such that xm → x}. Let B = {Bx : x ∈ X} be an n-in-countable weak base for some n ∈ N on X. Claim 1. If L is a countable subset of X, then L is metrizable. It is easy to prove that {B ∩ L : B ∈ B} is an n-in-countable weak base on L. Let D be the set consisting of all the accumulation points of L, and D = {x ∈ X: there is a nontrivial sequence {xm : m ∈ N } of L such that xm → x}. We know that neither D nor D is empty since L is sequentially compact by Lemma 3.1. We will prove that D is dense in D. For any y ∈ D \ D , let U be an open subset in D containing y. We can pick an open subset U of X containing y such that U ∩ D ⊂ U because X is regular. Since y is an accumulation point of L, we have |U ∩L| = ω, which means that |U ∩L| = ω. Therefore, there is a nontrivial sequence {xm : m ∈ N } of U ∩ L and a point z in X such that xm → z since X is sequentially compact by Lemma 3.1. The point z belongs to D by the definition of D . Now we can see that z ∈ U ∩ D ⊂ U . Hence D is dense in D. Let BD = x∈D Bx . By the definition of D , we can see that |B ∩ L| = ω for each B ∈ BD . This fact shows that BD is countable since BD ⊂ H∈L<ω ,|H|=n {B ∈ B : H ⊂ B}. Thus, {B ∩ D : B ∈ BD }
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is a countable network on D . Therefore, D is a countably compact space with a Gδ -diagonal since D is sequentially compact by Lemma 3.1. Thus, the subspace D is metrizable. So, D is compact and D is equal to D. Let NL = {{x} : x ∈ L} ∪ {B ∩ D : B ∈ BD }. We can see that NL is a countable network on L. This shows that L is metrizable. Claim 2. The space X is monotonically ω-monolithic. For each finite subset F of X, we put HF = {{y} : y ∈ F } if |F | < n. If |F | ≥ n, then let HF = {B : B ∈ B, F ⊂ B}. For each countable subset A of X, let O(A) = F ∈[A]<ω HF . Obviously, O(A) is countable since B is n-in-countable. It is not difficult to prove that O(A) satisfies the conditions (a), (b) and (c) in the definition of monotonically ω-monolithic spaces. Obviously, F ∈[A]<ω OF contains a network at each point of A. Pick any point x ∈ A \ A. There is a nontrivial sequence {xm : m ∈ N } of A such that xm → x by Claim 1. For each B ∈ Bx , we have |B ∩ {xm : m ∈ N }| = ω. Therefore, Bx ⊂ O(A). This shows that O(A) contains a network at x, and it follows that Claim 2 is true. Claim 3. K is dense in K. Let x ∈ K and let U be any open set in K with x ∈ U . Choose open sets U and V of X such that U = U ∩ K and x ∈ V ⊂ V ⊂ U . Then V is a closed countably compact subspace of X, and there is some point z ∈ V and a sequence {zm : m ∈ N } ⊂ V such that zm → z since X is sequentially compact. So z ∈ K and z ∈ U ∩ K = U , which means that K is dense in K.
Claim 4. K = K . We can see that K is countably compact since X is sequentially compact by Lemma 3.1. By Lemma 3.3, we know that K is monotonically ω-monolithic. So K is a D-space by Lemma 3.2. Thus, K is compact since K is countably compact. Therefore, K is a closed subset of X and it follows that K = K. Claim 5. X is metrizable. By Claim 4 and Lemma 3.2, X is a g-first countable Fréchet space. It is proved that a weak neighborhood of a point must be a neighborhood of this point in a g-first countable Fréchet space [1]. It follows that B = {B o ∩ K : B ∈ B} is an n-in-countable base of K. By the definition of K and Claim 4, B is also point-countable in K. Thus, K is metrizable since K is countably compact. Let S be a countable dense subset of K. We show that |H = {Bz : z ∈ K}| = ω. It is obvious that |Bz | = ω for each z ∈ S. Notice that there is some subsequence of S convergent y to for each y ∈ K \ S. It means that | {Bz : z ∈ K \ S}| = ω since B is n-in-countable. Let W = {Bz : z ∈ K} {{x} : x ∈ X \ K}. We can see that W is a point-countable base of X. Hence, X is metrizable because it is countably compact. 2 Theorem 3.4 answers Question 1.1 positively. Moreover, the following results can be taken as corollaries of Theorem 3.4: Corollary 3.5. ([18]) Every countably compact space with an n-in-countable base for some n ∈ N is metrizable. Corollary 3.6. Every star compact space X with an n-in-countable weak base B for some n ∈ N is star countable.
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Proof. Let U be an open cover on X. There is a compact subset K such that st(K, U) = X. It follows from Theorem 3.4 that K is separable. Suppose that L is a countable dense subset of K. It is easy to check that st(L, U) = X. 2 Lemma 3.7. ([8]) A space X is monotonically monolithic (weakly monotonically monolithic) iff one can assign to each finite subset F of X a countable collection O(F ) of subsets of X such that, for each A ⊂ X, F ∈[A]<ω O(F ) contains a network at each point of A (at some point of A \ A if A is not closed). Theorem 3.8. Every space X with an n-in-countable weak base B for some n ∈ N is weakly monotonically monolithic. Proof. Let B be an n-in-countable weak base for some n ∈ N on X. We only need to prove that X satisfies the condition of Lemma 3.7. For each finite subset F of X, if |F | < k, then put HF = {{x} : x ∈ F }; If |F | ≥ k, then put HF = {B : B ∈ B, F ⊂ B}. For each subset A of X, let O(A) = F ∈[A]<ω HF . Suppose that A is a non-closed subset of X. By the definition of weak base, there is some x ∈ A \ A such that |B ∩ A| ≥ ω for each B ∈ Bx . It is not difficult to prove that Bx ⊂ {B ∈ B: there is some finite subset L of A with |L| = n such that L ⊂ B}. It follows that O(A) contains a network at x. 2 It is proved that for a star countable space X with an n-in-countable base for some n ∈ N , e(X) is not greater than ω [19]. It is natural to ask whether this result holds if we replace “n-in-countable base” with “n-in-countable weak base”. However, the space constructed in Example 7 in [14] is a regular separable space, hence a star countable space, with a point-countable weak base such that it is not Lindelöf. Therefore, the answer of this question is negative. Theorem 3.9. Every star compact metalindelöf space X with an n-in-countable weak base B for some n ∈ N is metrizable. Proof. Let U be an open cover on X. Take a point-countable open cover V which refines U. Then we can take a countable subset of V which also covers X by Corollary 3.6. It follows that X is a Lindelöf space. Because normal star compact spaces are countably compact [12], X is metrizable by Theorem 3.4. 2 Although it is proved that every star compact regular space with an n-in-countable base for some n ∈ N is metrizable [18], the authors do not know the answer to the following question: Question 3.10. Is a star compact regular space with an n-in-countable weak base for some n ∈ N metrizable? 4. Spaces with n-in-countable bases or property (G) Theorem 4.1. If X is a space with an n-in-countable base for some n ∈ N , then X is monotonically monolithic. Proof. Let B be an n-in-countable base for some k ∈ N on X. For each finite subset F of X and each L ⊂ F , if |L| < n, then we put H(L) = {{y} : y ∈ L}; if |L| ≥ n, then we assign H(L) to L as {B : B ∈ O(B), there is some subset L ⊂ L with |L | = n and L ⊂ B}. It is easy to check that O(F ) = L⊂F H(L) is the operator described in Lemma 3.7. 2 A cover L of a space X is thick [9] if we can assign LH ∈ [L]<ω and LH = LH to each H ∈ [X]<ω in such a way that A ⊂ {LH : H ∈ [A]<ω } for each A ⊂ X. The space X is thickly covered if every open cover of X is thick.
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Notice that every monotonically monolithic space X is hereditarily thickly covered [9], we have the following corollary of Theorem 4.1: Corollary 4.2. If X is a space with an n-in-countable base for some n ∈ N , then X is hereditarily thickly covered. Lemma 4.3. ([19]) If X is a star countable space with an n-in-countable base for some n ∈ N , then e(X) = ω. Theorem 4.4. If X is a star σ-compact space with an n-in-countable base B for some n ∈ N , then X is Lindelöf. Proof. Let U be an open cover on X. There is a σ-compact subset F = i∈N Fi such that st(F, U) covers X. Obviously, {Fi ∩ B : B ∈ B} is an n-in-countable base on Fi . By Corollary 3.5, Fi is metrizable. Let Li be a countable dense subset of Fi . It is easy to see that st( {Li : i < ω}, U) covers X, and it follows that X is star countable. Hence, e(X) = ω by Lemma 4.3. We know that X is a D-space by Theorem 4.1, so X is Lindelöf. 2 Recall that ω1 is a caliber of a space X if every point-countable family of non-empty open subsets of X is countable. Theorem 4.5. If a space X has an n-in-countable base for some n ∈ N and ω1 is a caliber of X, then X is second countable. Proof. Let B be an n-in-countable base on X for some n ∈ N . If n = 1, then we know that X is second countable by the its caliber. If X is not second countable, then n ≥ 2. We prove that |B| = ω. Otherwise, B is not point-countable since ω1 is a caliber of X. Pick x1 ∈ X such that |{B : x1 ∈ B, B ∈ B}| = ω1 . Put Bx1 = {B\{x1 } : x1 ∈ B, B ∈ B}. The family Bx1 is not point-countable either. Hence, we can choose x2 ∈ X with |{B : x2 ∈ B, B ∈ Bx1 }| = ω1 and set Bx2 = {B \ {x1 , x2 } : x1 ∈ B, B ∈ B}. Continue in this way, at the n-th step, we can choose a subset {x1 , ..., xn } of X such that |{B : {x1 , ..., xn } ⊂ B, B ∈ Bx }| = ω1 . This contracts with the n-in-countableness of B, and which shows that |B| = ω. 2 By using a method similar to that used by Tkachuk in [17], we obtain the following result: Theorem 4.6. Let X be a Lindelöf Σ space with an n-in-countable base B for some n ∈ N . If each point in X is a Gδ set, then X is second countable. Proof. Fix a countable network F with respect to a compact cover C of X. For each x ∈ X, we choose countable open subsets Gx as a witness of the Gδ property at x. For each finite subset A of X, definite an operator O on A as follows: When |A| < n, put O(A) = x∈A Gx ; and when |A| ≥ n, put O(A) = x∈A Gx ∪ {B ∈ B : |B ∩ A| ≥ n}. Take a point x0 ∈ X and let A0 = {x0 }. For m ∈ N , assume that we have countable subsets A0 , ..., Am of X satisfying the following conditions: (1) A0 ⊂ ... ⊂ Am ; (2) for every i < m, if there is a finite family V ⊂ O(Ai ) and F ∈ F such that F \ V = ∅, then (F \ V) ∩ Ai+1 = ∅. For each finite collection V ⊂ O(Am ), if there is F ∈ F such that F \ V = ∅, then choose a point x(V, F ) ∈ F \ V. Put Am+1 = Am ∪ {x(V, F ) : V ∈ [O(Am )]<ω , F ∈ F}. It is trivial that conditions (1)
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and (2) are now satisfied for m + 1. By induction, we can construct a sequence {Ai : i ∈ N } of countable subsets of X such that conditions (1) and (2) hold for all m ∈ N . ∞ Suppose that A = i=0 Ai is not dense in X. We can pick a point x ∈ X \ A. Take a set C ∈ C with x ∈ C. The subset K = C ∩ A is compact. For each y ∈ A \ A and each open neighborhood Uy of y, |Uy ∩ A| = ω. So O(A) contain a local base at y. Thus, we can choose Vy ∈ O(A) with y ∈ Vy and x ∈ / Vy . For y ∈ A, we pick Gy ∈ Gy such that x ∈ / Gy . So {Vy : y ∈ A \ A} ∪ {Gy : y ∈ A} covers K. We can get a finite family W ⊂ O(A) such that x ∈ / W and K ⊂ W. Now we can see that H = W ∪ (X \ A) is an open neighborhood of C. Hence, there is some F ∈ F such that C ⊂ F ⊂ H. Since O(A) = m∈N O(Am ) and the family O(Am ) is increasing, we can find m ∈ N such that W ⊂ O(Am ). Observe that x ∈ F \ W, we know that the point x(W, F ) must belongs to Am+1 . However, x(W, F ) ∈ F \ W ⊂ X \ A, which is a contradiction. So A is a dense subset of X. It is not difficult for us to prove that {B ∈ B : |B ∩ A| ≥ n} ∪ {x ∈ X : x is an isolate point in X} is a countable base on X. 2 Collins and Roscoe introduced the following property in [6]: (open) Property (G): For each x ∈ X, there is an assigned countable collection G(x) of (open) subsets of X such that, whenever x ∈ U , U is open, there is an open V with x ∈ V ⊂ U such that, whenever y ∈ V , then x ∈ W ⊂ U for some W ∈ G(y). Lemma 4.7. ([6]) Every space with property (G) must be hereditarily metalindelöf. The following result follows directly from Lemma 4.7. Theorem 4.8. Every star countable space with property (G) is Lindelöf. It is easy to see that a space with a point-countable base must have property open (G). On the other hand, it is still an open problem whether every space with the property open (G) has a point-countable base. It is showed consistently in [18] that there is a strongly monotonically monolithic space which does not have a point-countable base. To the question that whether every strongly monotonically monolithic space has property (G) raised in [16] (i.e. Question 1.3), we show that a negative answer is also consistent. Lemma 4.9. [18] Suppose that X is weakly countably tight (or a k-space) with an m-in-countable base for some m ∈ N . Then X is strongly monotonically monolithic. The following result shows a negative answer to Question 1.3 consistently and it also shows that Question 1.4 has a consistent negative answer: Theorem 4.10. There is a strongly monotonically monolithic space, which is not metalindelöf, hence does not have property (G). Proof. Davis, Reed and Wage constructed a normal Moore space X with a 2-in-finite base but without a point-countable base under Martin’s Axiom and ω2 < 2ω0 [7]. Then X is strongly monotonically monolithic by Lemma 4.9. We show that X is not metalindelöf. Otherwise, let W = n∈N Wn be a development of X. For each n ∈ N , put Vn be a point-countable open refinement of Wn . It is not difficult to prove that V = n<ω Vn is a point-countable base on X, which is a contradiction. We can also see that X does not have property (G) by Lemma 4.7. 2
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