Compact Space Of Countable Tightness Research Articles

Discrete selectivity, shrinking properties, and disjoint local π-bases

Using binary function spaces, we give an example of a pseudocompact discretely selective topological group. We show that, under PFA, every compact space of countable tightness has a countable disjoint local π-base at every point. If X is a compact space of countable tightness and all non-empty open subsets of X are non-separable, then it is proved in ZFC that X possesses a countable disjoint local π-base at every point. We also establish that a Lindelöf Σ-space has the discrete shrinking property if and only if the outer π-character of any compact subset of X is uncountable. As a consequence, any non-metrizable topological group with a countable network has the discrete shrinking property.

Generalized side-conditions and Moore–Mrówka

We prove that it is consistent (even with Martin's Axiom) that there is first-countable initially ω1-compact space with cardinality greater than the continuum. We also prove that it is consistent with Martin's Axiom and c=ω2 that there is a compact space of countable tightness which is not sequential. It is known that neither statement is consistent with the Proper Forcing Axiom. We use an innovative new method of constructing proper posets with elementary submodels as side conditions introduced by Neeman.

Function spaces jointly metrizable on compacta

If Cp(X) is jointly metrizable on compacta, then p(X)≤ω but ω1 need not be a caliber of X. If X is either submetrizable or a P-space, then Cp(Cp(X)) is jointly metrizable on compacta and, in particular, all compact subsets of Cp(Cp(X)) are metrizable. We show that for any dyadic compact X, the space Cp(X) is jointly metrizable on compacta. Therefore, the JCM property of Cp(X) for a compact space X does not imply that X is separable. If X is a compact space of countable tightness and Cp(X) is jointly metrizable on compacta, then it is independent of ZFC whether X must be separable.

Maximal countably compact spaces and embeddings in MP-spaces

We study embeddings in maximal pseudocompact spaces together with maximal countable compactness in the class of Tychonoff spaces. It is proved that under MA $${+\neg}$$ CH any compact space of weight $${\kappa < \mathfrak{c}}$$ is a retract of a compact maximal pseudocompact space. If κ is strictly smaller than the first weakly inaccessible cardinal, then the Tychonoff cube [0, 1]κ is maximal countably compact. However, for a measurable cardinal κ, the Tychonoff cube of weight κ is not even embeddable in a maximal countably compact space. We also show that if X is a maximal countably compact space, then the functional tightness of X is countable. It is independent of ZFC whether every compact space of countable tightness must be maximal countably compact. On the other hand, any countably compact space X with the Mazur property ( $${\equiv}$$ every real-valued sequentially continuous function on X is continuous) must be maximal countably compact. We prove that for any ω-monolithic compact space X, if C p (X) has the Mazur property, then it is a Frechet–Urysohn space.

Every $$k$$ k -separable Čech-complete space is subcompact

We establish that a Cech-complete space $$X$$ must be subcompact if it has a dense subspace representable as the countable union of closed subcompact subspaces of $$X$$ . In particular, if a Cech-complete space contains a dense $$\sigma $$ -compact subspace then it is subcompact. This result is new even for separable Cech-complete spaces. Furthermore, if $$X$$ is a compact space of countable tightness then $$X\backslash A$$ is subcompact for any countable set $$A\subset X$$ . We also show that any $$G_\delta $$ -subset of a dyadic compact space is subcompact and give a comparatively simple proof of the fact that $$X\backslash A$$ is subcompact for any linearly ordered compact space $$X$$ and any countable set $$A\subset X$$ .

Domination by second countable spaces and Lindelöf Σ-property

Given a space M, a family of sets A of a space X is ordered by M if A = { A K : K is a compact subset of M} and K ⊂ L implies A K ⊂ A L . We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space C p ( X ) belongs to M if and only if it is a Lindelöf Σ-space. Under MA ( ω 1 ) , if X is compact and ( X × X ) \\ Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ = { ( x , x ) : x ∈ X } is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X 2 \\ Δ belongs to M then X is metrizable in ZFC. We also consider the class M ⁎ of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P ⊂ X there exists F ∈ F with P ⊂ F . It is a ZFC result that if X is a compact space and ( X × X ) \\ Δ belongs to M ⁎ then X is metrizable. We also establish that, under CH, if X is compact and C p ( X ) belongs to M ⁎ then X is countable.

Embeddings and frames of function spaces

For every Tychonoff space X we denote by C p ( X ) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of R X . A set P is a frame for the space C p ( X ) if C p ( X ) ⊂ P ⊂ R X . We prove that if C p ( X ) embeds in a σ -compact space of countable tightness then X is countable. This shows that it is natural to study when C p ( X ) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space C p ( X ) has a Lindelöf frame of countable tightness then t ( X ) ⩽ ω . We give some generalizations of this result for the case of frames as well as for embeddings of C p ( X ) in arbitrary spaces.

Closures of discrete sets in compact spaces

We consider the question of whether a compact space will always have a discrete subset whose closure has the same cardinality as the whole space. We obtain many positive results for compact spaces of countable tightness and a consistent negative result for a space of tightness and density ?1.

Point-countable π-bases in first countable and similar spaces

It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable $ \pi $-base. We look at general spaces with point-countable $\pi $-bases and prove, in particular, that, under the Continuum Hypothesis, any Linde

Sequential compactness versus pseudo-radiality in compact spaces

Strengthening a result of Šapirovskiǐ we prove that if 2 ω ⩽ ω 2 then sequential compactness implies pseudo-radiality in compact T 2 spaces, moreover we show that the failure of this implication is consistent with 2 ω = ω 3. We also prove that if a conpact T 2 space X is not pseudo-radial then there is a Y⊂ X with | Y|< c such that Ȳ is not pseudo-radial. Finally, sequential compactness implies pseudo-radiality in compact spaces of countable tightness if either MA ( σ-centered) holds or ω 1 Cohen reals are added to any ground model.

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.

Murray G. Bell. Spaces of ideals of partial functions. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 1–4. - Alan Dow. Compact spaces of countable tightness in the Cohen model. Set theory and

Murray G. Bell. Spaces of ideals of partial functions. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 1–4. - Alan Dow. Compact spaces of countable tightness in the Cohen model. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 55–67. - Peter J. Nyikos. Classes of compact sequential spaces. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 135–159. - Franklin D. Tall. Topological problems for set-theorists. Set theory and its applications, Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21,1987, edited by J. Streprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 194–200. - Volume 56 Issue 2