Countably Compact Research Articles

Compactness and countable compactness in weak topologies

A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on

On Weaker Forms of Paracompactness, Countable Compactness, and Lindelöfness

ABSTRACT: An ideal on a set X is a nonempty collection of subsets closed under the operations of finite union and subset. The concepts of parabounded and countably bounded subsets are defined as well as regularity with respect to an ideal I(i.e., I‐regular). The bounded subsets of a topological space are characterized as the subsets which are both parabounded and countably bounded. It is shown that if (X, τ) is a Hausdorff space and I is an ideal on X such that I∩τ={Ø}, then (X, τ) is compact modulo I iff (X, τ) is H‐closed and I‐regular. Examples are given which place e‐Lindelöfness in the hierarchy of known properties and it is shown that para‐Lindelöf e‐Lindelöf spaces are Lindelöf. It is shown that in the category of Hausdorff spaces, perfect images of e‐paracompact spaces are e‐paracompact. Posed as an open question is the following: What can be said about the closed continuous image of an e‐paracompact space?

Absolutely countably compact spaces

We define a new property acc which is stronger than countable compactness: X is acc if for every open cover γ and every dense subspace Y we have St( K, γ)= X for some finite K⊂ Y (it is a known fact that X is countably compact iff for every open cover γ we have St( K, γ)= X for some finite K⊂ X). Every countably compact space with countable tightness is acc. The product of an acc space with a first countable, compact space is acc while the product with an arbitrary compact space need not be acc. If X τ is acc for every τ then X is compact. Every countably compact space is a closed subspace of some acc space. Several counterexamples are provided.

M-mappings and the cellularity of spaces

It is shown that an image X of a product of Lindelöf Σ-spaces or of a countably compact (even countably protocompact) space Π under a continuous M-mapping is ℵ 0-cellular in the sense of A.V. Arhangel'skiĭ. The same holds if both Π and X have “good” lattices of continuous mappings. Some generalizations of earlier results of V.V. Uspenskiǐ and M.G. Tkačenko on continuous images of (dense subspaces of) topological groups are also given.

Topological spaces with properties undetermined in ZFC

Within ZFC there are constructed: (1) a compact space with undetermined Fréchet-Urysohn property; (2) a separable pseudocompact noncompact group with undetermined countable compactness; (3) a countable Hausdorff group which is extremaly disconnected under CH and Fréchet-Urysohn under [MA + ¬ CH].

On lattice-perfect measures

AbstractThe general properties of lattice-perfect measures are discussed. The relationship between countable compactness and measure perfectness, and the relationship between lattice-measure tightness and lattice-measure perfectness are investigated and several applications in topological measure theory are given.

Perfect maps in compact (countably compact) spaces

In this paper, among other results, characterizations of perfect maps in compact Hausdorff(Fréchet, countably compact, Hausdorff) spaces are obtained.

Pseudocompact and Countably Compact Abelian Groups: Cartesian Products and Minimality

Pseudocompact and Countably Compact Abelian Groups: Cartesian Products and Minimality

Regular and Purely Irregular Bounded Charges: A Decomposition Theorem

We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and $\sigma$-additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire $\sigma$-field of a countably compact space.

κ‐Structures: Cardinality Conditions, Compactness, Metrization

The notion of a κ‐structure on a set E is introduced as a generalization of a topology on E. Pospíšil's inequality |X|≤d(X)χ(X), Arhangel'skiĭi's inequality |X|≤ 2L(X)χ(X), and Gryzlov's inequality |X|≤ 2ψ(X) for X compact all extend to κ‐structures. Chaber's theorem, that every countably compact space with a Gδ‐diagonal is compact, also extends to κ‐structures. Two metrization theorems in terms of κ‐structures are given.

On spaces of Baire I functions over K-analytic spaces

Suppose thatℱ is a relatively countably compact subset of B1(X), the space of Baire I functions over a K-analytic space X equipped with the pointwise convergence topology. It is proved that (1) the closure ofℱ is a strongly countably compact Frechet-Urysohn space; (2) ifℱ is ℵ1 -compact,ℱ is a bicompactum; (3) if X is a paracompact space, the closure ofℱ is a bicompactum.

Hereditary normality versus countable tightness in countably compact spaces

A space X is said to be countably tight if, for each A ⊂ X and each point x in the closure of A, there is a countable subset B of A such that x is in the closure of B. We show that the statement, “every separable, compact, hereditarily normal space is countably tight” is independent of the usual axioms of set theory, and show that it is equivalent to “no version of γ N is hereditarily normal”, where γ N is a familiar type of space due to Franklin and Rajagopalan, and a number of other statements. We derive some consequences from the fact that PFA implies this statement, including the consistency of “every countably compact, hereditarily normal space is sequentially compact”.

A note on the point-countable base question

We show that a subspace X of a linearly ordered space has a point-countable base iff X satisfies open (G), i.e., one can assign to each x ϵ X a countable collection B ( x ) of open sets such that if { x n } n < ω converges to x ϵ X , then ∪ n<ω B (x n ) contains a base at x . We also show that countably compact Hausdorff spaces satisfying open (G) are metrizable.

On the consistency of the Moore- Mrówka solution

We show that a large cardinal is not necessary to prove the consistency of the recent PFA results of Balogh, Fremlin and Nyikos concerning countably compact spaces of countable tightness and closed preimages of ω 1. For example, we improve Balogh's result by showing that it is consistent that all compact spaces of countable tightness are sequential.

Convergent free sequences in compact spaces

We prove that if κ \kappa is an uncountable regular cardinal and a compact T 2 {T_2} space X X contains a free sequence of length κ \kappa , then X X also contains such a sequence that is convergent. This implies that under CH {\text {CH}} every nonfirst countable compact T 2 {T_2} space contains a convergent ω 1 {\omega _1} -sequence and every compact T 2 {T_2} space with a small diagonal is metrizable, thus answering old questions raised by the first author and M. Hušek, respectively.

Regular and purely irregular bounded charges: a decomposition theorem

We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and σ \sigma -additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire σ \sigma -field of a countably compact space.

Star covering properties

In this paper, the authors investigate starcompact properties between countable compactness and the discrete finite chain condition (i.e., pseudocompactness), and star-Lindelöf properties between the Lindelöf property and the discrete countable chain condition (i.e., the pseudo-Lindelöf property). This work represents a unification and extension of concepts previously studied by several authors in the literature. Theory is developed to establish connections between the various star properties and other covering conditions, and a large collection of nontrivial examples is given to make distinctions.

A Countably Compact Topological Group H Such that H × H is not Countably Compact

A Countably Compact Topological Group H Such that H × H is not Countably Compact

A first countable compact space that is not an N∗ image

Let N ∗ be the nonisolated points of the Stone-Čech compactification of the countable discrete space N. We show that, in the absence of CH, there can be a first countable compact space which is not an N ∗ image. This answers a question of van Douwen and Przymusinski. Our example is even hereditarily metacompact.

On first countable, countably compact spaces II: remainders in a van Douwen construction and P-ideals

Call a separable, locally compact, locally countable, countably compact, noncompact space a van Douwen space. Let the remainder of such a space be its set of nonisolated points. We show that it is independent of ZFC that there are van Douwen spaces of cardinality c whose remainder is ω-bounded. If we assume the consistency of a little more than a huge cardinal, we can remove the cardinality restriction, i.e., it is independent whether there are van Douwen spaces whose remainder is ω-bounded. The proofs involve special kinds of P-ideals, and some results are shown about them and the dual P-filters. In particular: in the Laver and “dominating reals” models, every P-point is a T-point.