Two improvements on Tkaÿcenko's addition theorem

We prove that (A) if a countably compact space is the union of countably many D subspaces then it is compact; (B) if a compact T 2 space is the union of fewer than N(ℝ) = cov(ℳ) left-separated subspaces then it is scattered. Both (A) and (B) improve results of M. G. Tkačenko from 1979; (A) also answers a question that was raised by A. V. Arhangel’skii and improves a result of G. Gruenhage.

Topology and its Applications, 2007

For any space X, denote by dis(X) the smallest (infinite) cardinal κ such that κ many discrete subspaces are needed to cover X. It is easy to see that if X is any crowded (i.e. dense-in-itself) compactum then dis(X) m, where m denotes the additivity of the meager ideal on the reals. It is a natural, and apparently quite difficult, question whether in this inequality m could be replaced by c. Here we show that this can be done if X is also hereditarily normal. Moreover, we prove the following mapping theorem that involves the cardinal function dis(X). If f : X → Y is a continuous surjection of a countably compact T 2 space X onto a perfect T 3 space Y then |{y ∈ Y : f −1 y is countable}| dis(X).

1981

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Topology and its Applications, 2007

Let H 0 (X) (H (X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H (X) is to characterize those X for which H (X) is countably compact. We conjecture that u-compactness of X for some u ∈ ω * (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction. We define the property R(κ): for every family {Z α : α < κ} of closed subsets of X separated by pairwise disjoint open sets and any family {k α : α < κ} of natural numbers, the product α<κ Z k α α is countably compact, and prove that if H (X) is countably compact for a T 2-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T 2 and H (X) is countably compact, then so is X n for all n < ω. We also prove that, for κ < t, if the T 3 space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then X κ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T 3 , homogeneous, and H (X) is countably compact, then so is X ω. Then we study the Frolík sum (also called "one-point countable-compactification") F (X α : α < κ) of a family {X α : α < κ}. We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product α<κ H 0 (X α) embeds into H (F (X α : α < κ)).

Topological Algebra and its Applications, 2014

We prove that a separable Hausdor_ topological space X containing a cocountable subset homeomorphic to [0, ω

Pacific Journal of Mathematics, 1987

Telgarsky calls a topological space C-scattered when each of its non-empty closed sets contains a compact set with non-empty relative interior. With respect to infinite products, hyperspaces, and the partially ordered set of compactifications, we study the class of paracompact C-scattered spaces and two of its subclasses, MacDonald and Willard's A'-spaces and Λ-spaces. 0. Introduction. All spaces are Hausdorff spaces. A space X is said to be C-scattered [16] provided that each of its non-empty closed subspaces contains a compact set with non-empty relative interior. The notion of C-scatteredness seems a simple simultaneous generalization of scattered (Ξ= each non-empty set has a relative isolated point) and of local compactness. However, the class of paracompact C-scattered spaces is most interesting because [19] it contains its perfect pre-images, it is closed under finite products, it contains all closed continuous images of paracompact locally compact spaces, and for each of its members X, X X Y is paracompact iff Y is paracompact. Presently we study this class and two of its subclasses. Section 1 is due to the third author and § §2 and 3 are due to the first two authors. In §1 of our paper, we show that each countable product of paracompact C-scattered spaces is paracompact. This result improves upon the same theorem, due to Rudin and Watson [18], for paracompact scattered spaces, and answers the question raised for Λ'-spaces by the first two authors of this paper. As a corollary, we find that each countable product of Lindelof C-scattered spaces is Lindelof, a result due to Alster [2]. In the second section, we investigate hyperspaces of paracompact C-scattered spaces-a situation so complex that we limit our attention to A '-spaces. An A'-space is a space whose set of accumulation points is compact [10]. Thus, an ^I'-space is paracompact C-scattered. It is known [12] that the compact-set hyperspace ^(X) is locally compact (metrizable) iff X is locally compact (respectively, metrizable). Here we present an example of a Lindelof scattered A '-space X such that ^(X) is neither C-scattered or normal. Further, we prove that ^(X) is an A '-space 277 278

Topology and its Applications, 2009

We show that if a space X is the union of not more than κ-many discrete subspaces, where κ is an infinite cardinal, then the same holds for any perfect image of X. It follows that a compact Hausdorff space with no isolated points can never be covered by fewer than continuum many discrete subspaces; this answers a question of I. Juhász and J. van Mill. We also consider coverings by right-separated and left-separated subspaces.

Topology Proc. 29

All spaces in this note are assumed to be separable and metriz- able. A subset of a space is called a C-set if it can be written as an intersection of clopen subsets of the space. Note that a space is zero-dimensional if and only if ...

Topology and its Applications, 1997

For an infinite cardinal a, we say that a subset B of a space X is Ca-compact in X if for every continuous function f : X -~ II~ ~, f [B] is a compact subset of II~ ~. This concept slightly generalizes the notion of a-pseudocompacmess introduced by J.F. Kennison: a space X is a-pseudocompact if X is Ca-compact in itself. If a = w, then we say C-compact instead of C~-compact and ~v-pseudocompactness agrees with pseudocompactness. We generalize Tamano's theorem on the pseudocompactness of a product of two spaces as follows: let A C_ X and B C_ Y be such that A is z-embedded in X. Then the following three conditions are equivalent: (1) A x B is C~compact in X x Y; (2) A and B are Ca-compact in X and Y, respectively, and the projection map ~r : X x Y ~ X is a za-map with respect to A x B and A; and (3) A and B are Co-compact in X and Y, respectively, and the projection map ~':X x Y -+ X is a strongly z~-map with respect to A x B and A (the z~-maps and the strongly z~-maps are natural generalizations of the z-maps and the strongly z-maps, respectively). The degree of C,~-compactuess of a C-compact subset B of a space X is defined by: p(B, X) = cx~ if B is compact, and if B is not compact, then p(B, X) = sup{a: B is Ca-compact in X}. We estimate the degree of pseudocompacmess of locally compact pseudocompact spaces, topological products and E-products. We also establish the relation between the pseudocompact degree and some other cardinal function's. In the context of uniform spaces, we show that if A is a bounded subset of a uniform space ~X,~/), then A is Ca-compact in X, where (X,~) is the completion of (X,/,/) iff f(A) is a compact subset of R ~ from every uniformly continuous function from X into ~'~; we characterize the C~-compact subsets of topological groups; and we also prove that if {Gi: i E I} is a set of topological groups and A~ is a Ca-compact subset of G,~ for all i E I, then I-L~ A~ is a Co-compact subset of 1-[iel Gi. © 1997 Elsevier Science B.V.

Glasgow Mathematical Journal, 1985