Countably Compact Research Articles

Pseudocomplete and weakly pseudocompact spaces

We prove that if X is a quasiregular, countably compact space with a pseudobase consisting of closed Gδ-sets, then every Gδ-dense subspace of X is pseudocomplete in the sense of Todd. In particular, every weakly pseudocompact space is pseudocomplete in the sense of Todd. Some sufficient conditions are found that guarantee that a weakly pseudocompact space is pseudocomplete in the sense of Oxtoby. It is shown that every weakly pseudocompact space without isolated points has cardinality at least continuum. An example is given of a weakly pseudocompact space with one non-isolated point that contains the one-point lindelöfication of an uncountable discrete space. We apply this example to show that weak pseudocompactness is not preserved by the relation of M-equivalence.

Selections and countable compactness

Abstract The present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

Topological classification of scattered IFS-attractors

We study countable compact spaces as potential attractors of iterated function systems. We give an example of a convergent sequence in the real line which is not an IFS-attractor and for each countable ordinal δ we show that a countable compact space of height δ+1 can be embedded in the real line so that it becomes the attractor of an IFS. On the other hand, we show that a scattered compact metric space of limit height is never an IFS-attractor.

Remarks on selectively (a)-spaces

In this paper, we continue to study the versions of property (a), and show the following statements:(1)There exists a Tychonoff countably compact space X such that AD(X) is selectively (a), but X is not selectively (a);(2)There exists a Tychonoff selectively (a)-space X having a regular-closed subspace which is not selectively (a);(3)Assuming 2ℵ0=2ℵ1, there exists a normal space X that is not selectively (a);(4)If X is a normal selectively (a)-space with e(X)<ω1, then AD(X) is selectively (a).The statement (1) answers negatively Caserta, Di Maio and Kočinac (2011) [1, Question 2.11].

The classical subspaces of the projective tensor products of lpand C(α) spaces, α< ω1

We completely determine the lq and C(K) spaces which are isomorphic to a subspace of lp⊗ˆπC(α), the projective tensor product of the classical lp space, 1≤p<∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α<ω1. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from lp to l1, 1≤p<∞. The first main theorem is an extension of a result of E. Oja and states that the only lq space which is isomorphic to a subspace of lp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is lp. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pelczynski which allows us to classify, up to isomorphism, the separable spaces (X,Y) of nuclear operators, where X and Y are direct sums of lp and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K1 and K3 are finite or countable compact metric spaces of the same cardinality and $1 (a) (lp⊕C(K1),lq⊕C(K2)) and (lp⊕C(K3),lq⊕C(K4)) are isomorphic.

Calibers, $$\omega $$ -continuous maps and function spaces

The operation of extending functions from \(\scriptstyle X\) to \(\scriptstyle \upsilon X\) is \(\scriptstyle \omega \)-continuous, so it is natural to study \(\scriptstyle \omega \)-continuous maps systematically if we want to find out which properties of \(\scriptstyle C_p(X)\) “lift” to \(\scriptstyle C_p(\upsilon X)\). We study the properties preserved by \(\scriptstyle \omega \)-continuous maps and bijections both in general spaces and in \(\scriptstyle C_p(X)\). We show that \(\scriptstyle \omega \)-continuous maps preserve primary \(\scriptstyle \Sigma \)-property as well as countable compactness. On the other hand, existence of an \(\scriptstyle \omega \)-continuous injection of a space \(\scriptstyle X\) to a second countable space does not imply \(\scriptstyle G_\delta \)-diagonal in \(\scriptstyle X\); however, existence of such an injection for a countably compact \(\scriptstyle X\) implies metrizability of \(\scriptstyle X\). We also establish that \(\scriptstyle \omega \)-continuous injections can destroy caliber \(\scriptstyle \omega _1\) in pseudocompact spaces. In the context of relating the properties of \(\scriptstyle C_p(X)\) and \(\scriptstyle C_p(\upsilon X)\), a countably compact subspace of \(\scriptstyle C_p(X)\) remains countably compact in the topology of \(\scriptstyle C_p(\upsilon X)\); however, compactness, pseudocompactness, Lindelof property and Lindelof \(\scriptstyle \Sigma \)-property can be destroyed by strengthening the topology of \(\scriptstyle C_p(X)\) to obtain the space \(\scriptstyle C_p(\upsilon X)\). We show that Lindelof\(\scriptstyle \Sigma \)-property of \(\scriptstyle C_p(X)\) together with \(\scriptstyle \omega _1\) being a caliber of \(\scriptstyle C_p(X)\) implies that \(\scriptstyle X\) is cosmic.

On the relative strength of forms of compactness of metric spaces and their countable productivity in ZF

We show in ZF that:(i)A countably compact metric space need not be limit point compact or totally bounded and, a limit point compact metric space need not be totally bounded.(ii)A complete, totally bounded metric space need not be limit point compact or Cantor complete.(iii)A Cantor complete, totally bounded metric space need not be limit point compact.(iv)A second countable, limit point compact metric space need not be totally bounded or Cantor complete.(v)A sequentially compact, selective metric space (the family of all non-empty open subsets of the space has a choice function) is compact.(vi)A countable product of sequentially compact (resp. compete and totally bounded) metric spaces is sequentially compact (resp. compete and totally bounded).

Discrete reflexivity and complements of the diagonal

We prove that, for a countably compact space X of weight at most ω1, if \({\mathcal {P}}\) ∈{countable tightness, Frechet–Urysohn property, sequentiality, first countability} and the closure of every discrete subspace of X has \({\mathcal {P}}\) then X has \({\mathcal {P}}\). Given a countably compact space X, if X2∖Δ is discretely Lindelof then X is metrizable. We establish that a Lindelof Σ-space X is cosmic whenever X2∖Δ is paracompact; this generalizes the respective result of Gruenhage for compact X. Furthermore, a countably compact space X is metrizable if the closure of every discrete subspace of X2 is metrizable. It turns out that many non-discretely reflexive properties behave much better in finite powers of spaces. In particular, if X is compact and \(\overline{D}\) is Corson (Eberlein) compact for any discrete D⊂X3 then X is a Corson (Eberlein) compact.

Ultracompleteness of hyperspaces of compact sets

A space X is called ultracomplete if it has countable character in its Stone–Cech compactification βX. A space X is called almost locally compact if the set of all points at which X is not locally compact is contained in a compact set of countable outer character. For a given Tychonoff space X let 2X be the hyperspace of all nonempty compact subsets of X endowed with the Vietoris topology. We prove that 2X is almost locally compact if and only if X is locally compact. We also prove that for a countably compact ultracomplete space X the hyperspace Fn(X)={K∈2X∣K has at most n points} is also countably compact ultracomplete for every natural number n. We also analyse ultracompleteness of Fn(X) and 2X.

REMARKS ON CS-STARCOMPACT SPACES

Abstract. A space X is cs-starcompact if for every open cover U of X ,there exists a convergent sequence S of X such that St ( S;U ) = X , where St ( S;U ) =∪ fU 2 U : U S = ∅ . In this paper, we prove the followingstatements:(1) There exists a Tycff cs-starcompact space having a regular-closed subset which is not cs-starcompact;(2) There exists a ff cs-starcompact space with arbitrary largeextent;(3) Every ff centered-Lindelof space can be embedded in aff cs-starcompact space as a closed subspace. 1. IntroductionBy a space, we mean a topological space. Let us recall that a space X is countably compact if every countable open cover of X has a nite subcover.Fleischman [5] de ned a space X to be starcompact if for every open cover U of X∪ , there exists a nite subset F of X such that St ( F;U ) = X , where St ( F;U ) = fU 2 U : U F = ∅ ; and he proved that every countably compact space isstarcompact. Conversely, van Douwen-Reed-Roscoe-Tree [3] proved that everyff starcompact space is countably compact, but this does not hold for

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly ⁎ subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that, 1) on weakly subparacompact spaces, countable compactness = compactness, ω 1 -compactness = Lindelöfness; 2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and 3) on weakly subparacompact regular T 1 -spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness. The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wickeʼs factorization of paracompactness (over Hausdorff spaces) along with Krajewskiʼs are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.

The [formula omitted] spaces for compact metric spaces K and X with a uniformly convex maximal factor

In this paper, we prove that if a Banach space X contains some uniformly convex subspace in certain geometric position, then the C ( K , X ) spaces of all X-valued continuous functions defined on the compact metric spaces K have exactly the same isomorphism classes that the C ( K ) spaces. This provides a vector-valued extension of classical results of Bessaga and Pełczyński (1960) [2] and Milutin (1966) [13] on the isomorphic classification of the separable C ( K ) spaces. As a consequence, we show that if 1 < p < q < ∞ then for every infinite countable compact metric spaces K 1 , K 2 , K 3 and K 4 are equivalent: (a) C ( K 1 , l p ) ⊕ C ( K 2 , l q ) is isomorphic to C ( K 3 , l p ) ⊕ C ( K 4 , l q ) . (b) C ( K 1 ) is isomorphic to C ( K 3 ) and C ( K 2 ) is isomorphic to C ( K 4 ) .

COUNTABLE COMPACTNESS AND THE LINDELOF PROPERTY IN L-FUZZY TOPOLOGICAL SPACES

In this paper, the concepts of L-fuzzy countable compactness and the L-fuzzy Lindelof property are introduced in L-fuzzy topological spaces, where L is a completely distributive DeMorgan algebra. An L-fuzzy compact L-fuzzy set is L-fuzzy countably compact and has the L-fuzzy Lindelof property. An L-fuzzy set having the L-fuzzy Lindelof property is L-fuzzy countably compact if and only if it is L-fuzzy compact. Many characterizations of L-fuzzy countable compactness and the L-fuzzy Lindelof property are presented.

ON SPACES IN WHICH THE THREE MAIN KINDS OF COMPACTNESS ARE EQUIVALENT

In this paper, we introduce a new property (*) of a topological space and prove that if X satisfies one of the following conditions (1) and (2), then compactness, countable compactness and sequential compactness are equivalent in X; (1) Each countably compact subspace of X with (*) is a sequential or AP space. (2) X is a sequential or AP space with (*).

The anti-Specker property, uniform sequential continuity, and a countable compactness property

It is shown constructively that, on a metric space that is dense in itself, if every pointwise continuous, real-valued function is uniformly sequentially continuous, then the space has the anti-Specker property. The converse is also discussed. Finally, we show that the anti-Specker property implies a restricted form of countable compactness.

An algebraic version of Tamano's theorem for countably compact spaces

We show that if X is countably compact but not compact then one can find a compact space K such that X ⊕ K does not embed closedly into any normal topological group.

Normality in products with a countable factor

The class of normal spaces that have normal product with every countable space is considered. A countably compact normal space X and a countable Y such that X × Y is not normal is constructed assuming CH. Also, ⋄ is used to construct a perfectly normal countably compact X and a countable Y such that X × Y is not normal. The question whether a Dowker space can have normal product with itself is considered. It is shown that if X is Dowker and contains any countable non-discrete subspace, then X 2 is not normal. It follows that a product of a Dowker space and a countable space is normal if and only if the countable space is discrete. If X is Rudin's ZFC Dowker space, then X 2 is normal. An example of a Dowker space of cardinality ℵ 2 with normal square is constructed assuming ⋄ ω 2 ∗ .

More on Relatively almost Countably Compact Subsets

More on Relatively almost Countably Compact Subsets

Sequential compactness vs. countable compactness

Sequential compactness vs. countable compactness

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product 2R

On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product 2^R