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D-Spaces and Covering Properties View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 146–147 (2005) 437–449 www.elsevier.com/locate/topol D-spaces and covering properties Alexander V. Arhangel’skii Arhangel’skii A.V., Department of Mathematics, 321 Morton Hall, Ohio University, Athens, OH 45701, USA Received 2 October 2002; received in revised form 20 August 2003 Abstract We study the D-space property and its generalizations, the notions of an aD-space and a weak aD-space in connection with covering properties. A brief survey on D-spaces is presented in Section 1. Among new results, it is proved that if a linearly ordered space is an aD-space, then it is paracompact. The statement further extends the list of equivalences in [Proc. Amer. Math. Soc. 125 (1997) 1237]. We also establish some sufficient conditions for the free topological group of a Tychonoff space to be a D-space. In particular, the free topological group of a semi-stratifiable space is shown to be a D-space, while it need not be semi-stratifiable. A similar result is established for the free topological group of a space with a point-countable base. Some new interesting open problems on D-spaces and on spaces close to them are formulated. In particular, we discuss several such questions in connection with the sum theorems. 2004 Elsevier B.V. All rights reserved. MSC: 54D20; 54J99 Keywords: D-space; Semi-stratifiable space; Point-countable base; Free topological group; aD-space 1. D-spaces and their generalizations This paper is a continuation of [3]. We study the D-space property and its general- izations, the notions of an aD-space and a weak D-space, especially we discuss them in connection with various popular covering properties. We also establish a certain sum the- orem which provides sufficient conditions for the free topological group of a space to be a D-space. For undefined notions from the theory of covering properties see [11]. We define the extent e(X) of a space X as follows. A subset A of a space X is discrete in X(locally finite E-mail address: [email protected] (A.V. Arhangel’skii). 0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2003.08.029 438 A.V. Arhangel’skii / Topology and its Applications 146–147 (2005) 437–449 in X) if every point x ∈ X has an open neighbourhood Ox containing not more than one element (only finitely many elements) of A.Theextent e(X) of a space X is the smallest infinite cardinal number τ such that |A| τ , for every subset A of X which is locally finite in X. This definition obviously coincides with the usual definition of the extent of X for all T1-spaces. A neighbourhood assignment on a topological space X is a mapping φ of X into the topology T of X such that x ∈ φ(x), for each x ∈ X. A space X is called a D-space if, for every neighbourhood assignment φ on X, there exists a locally finite in X subset A of X such that the family φ(A) covers X. A principal property of D-spaces is that the extent coincides with the Lindelöf number in such spaces. In particular, every countably compact D-space is compact and every D-space with the countable extent is Lindelöf. These facts make the notion of a D-space useful in studying covering properties. It is well known that many covering properties of topological spaces combined with countable compactness imply compactness. This naturally leads to the question: how is the D property related to these covering properties? It is still an open problem (van Douwen [17]) whether every regular Lindelöf space is a D-space. It is even unknown whether every hereditarily Lindelöf regular T1-space is a D-space. Van Douwen also asked whether there exists a subparacompact or metacompact space which is not a D-space. These questions are still open. Recall that a space X is said to be subparacompact if every open covering of X can be refined by a σ -discrete closed covering [11]. It is known for some time that all metrizable spaces, and, more generally, all Moore spaces and all semi-stratifiable spaces, are D-spaces [9]. An interesting general result was recently obtained by R.Z. Buzyakova: every strong Σ-space is a D-space [12]. It follows from her theorem that all Tychonoff spaces with a countable network, all σ -spaces, and all Lindelöf Σ-spaces (that is, all Tychonoff continuous images of Lindelöf p-spaces) are D-spaces. Recall that a σ -space is a space with a σ -discrete network. On the other hand, there exists a Hausdorff, locally compact, locally countable, separable, first countable, submetrizable, σ -discrete, realcompact space with a Gδ-dia- gonal which is not a D-space: the space Γ constructed by Wicke and van Douwen in [18] has all these properties. Thus, there exists a locally compact σ -metrizable Tychonoff space with a Gδ-diagonal which is not a D-space. Note that we call a space Xσ-discrete (σ -metrizable) if it is the union of a countable family of not necessarily closed discrete (metrizable) subspaces. Observe, in connection with the space Γ , that if a regular T1-space X is the union of a finite collection of metrizable subspaces, then X is a D-space [3]. However, the next question remains open (though it is highly improbable, in my opinion, that the answer to it will be “yes”). Problem 1.1. Suppose that a (regular, Hausdorff, Tychonoff) T1-space X is the union of two subspaces which are both D-spaces. Is then X a D-space as well? At the end of the article we formulate an interesting version of this problem, which might have a positive answer. A.V. Arhangel’skii / Topology and its Applications 146–147 (2005) 437–449 439 Often it is not easy to verify whether a space is a D-space. We will now consider a formally weaker property which is much easier to verify for large classes of spaces and is still strong enough to imply compactness for countably compact spaces. A space X is an aD-space [3] if for each closed subset F of X and each open covering γ of X there exist a locally finite in F subset A of F and a mapping φ of A into γ such that a ∈ φ(a), for each a ∈ A, and the family φ(A) ={φ(a): a ∈ A} covers F .Anysuch mapping φ as above will be called a pointer (from A to γ ). The next statement is proved in [9]. Theorem 1.2 (Borges and Wehrly). Every subparacompact space is an aD-space. Here are some important, though obvious, properties of aD-spaces. Proposition 1.3. Every closed subspace of an aD-space is an aD-space. Lemma 1.4. If X = Y ∪Z,whereY is an aD-space (a D-space) and Y is closed in X, and every closed in X subspace of Z is an aD-space (a D-space),thenX is also an aD-space (a D-space, respectively). Proposition 1.5 [3]. If X = Y ∪ Z,whereY and Z are aD-spaces (D-spaces) and Y is closed in X,thenX is also an aD-space (a D-space, respectively). Recall that a space X is said to be isocompact if every closed countably compact subset of X is compact (see [28] about this notion and its applications). Clearly, every countably compact aD-space is compact. Therefore, we have: Proposition 1.6. Every aD-space is isocompact. Let us say that a space X is aweakaD-space if for each open covering γ of X there exist a locally finite in X subset A of X and a mapping φ of A into the set of all countable subfamilies of γ such that a ∈ V , for each V ∈ φ(a), and the family {φ(a): a ∈ A} covers X.Ifφ(a) can be selected to contain exactly one element of γ , we will call X a bD-space. It was observed by Borges and Wehrly in [9] that all subparacompact spaces are bD-spaces. The next statement is obvious. Proposition 1.7. A space X is an aD-space if and only if every closed subspace of X is a bD-space. On the other hand, the notion of a bD-space is closely related to the following notion, introduced much earlier by Arens and Dugundji [1]. A covering η of a space X is said to be minimal if η does not contain any proper subcovering of X. A space X is minimal cover-refinable (or irreducible) if every open cover of X can be refined by a minimal open covering η. It is not difficult to show that a T1-space X is minimal cover-refinable if and only if X is a bD-space (see [8,9]). Therefore, we have: 440 A.V. Arhangel’skii / Topology and its Applications 146–147 (2005) 437–449 Theorem 1.8. A T1-space X is an aD-space if and only if every closed subspace of X is minimal cover-refinable. The assumption that X is T1 is essential. Example 1.9. Let N be the set of all positive natural numbers and Un ={k ∈ ω: k<n}, for n ∈ N. The family γ ={Un: n ∈ N} is a base of the topology TN = γ ∪{N} on N.
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