Pseudocompact Research Articles

A generalization of ϰ-metrizable spaces

We introduce a new class of ϰ-metrizable spaces, namely countably ϰ-metrizable spaces. We show that the class of all ϰ-metrizable spaces is a proper subclass of countably ϰ-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with ϰ-metrizable spaces. We prove a generalization of Chigogidze's result that the Čech-Stone compactification of a pseudocompact countably ϰ-metrizable space is ϰ-metrizable.

Josefson\u2013Nissenzweig property for C_{p}-spaces

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or ell _{2}. The aim of the paper is to study a natural variant of this result for the space C_{p}(X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C_{p}(X)-spaces. We prove: for a Tychonoff space X the space C_p(X) satisfies the JNP if and only if C_p(X) has a quotient isomorphic to c_{0}:={(x_n)_{nin mathbb N}in mathbb {R}^mathbb {N}:x_nrightarrow 0} (with the product topology of mathbb {R}^mathbb {N}) if and only if C_{p}(X) contains a complemented subspace isomorphic to c_0. The last statement provides a C_{p}-version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c_{0} with the sup-norm topology. For a pseudocompact space X the space C_p(X) has the JNP if and only if C_p(X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C_{p}(K) containing a complemented subspace isomorphic to c_{0} is given.

Hausdorff compactifications in ZF

For a compactification αX of a Tychonoff space X, the algebra of all functions f∈C(X) that are continuously extendable over αX is denoted by Cα(X). It is shown that, in a model of ZF, it may happen that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that Cα(X)=Cγ(X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX×βY is the Čech-Stone compactification of X×Y when X×Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C⁎-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a maximal functional compactification is investigated in the absence of the axiom of choice.

On linearly H-closed spaces

We study a subclass of the class of pseudocompact spaces, the linearly H-closed spaces, recently introduced by M. Baillif. We give two new characterizations of these spaces in terms of filters and complete accumulation points of families of open sets. We show that a monotonically normal linearly H-closed space is compact and that linear H-closedness is preserved under a product with an H-closed space. Additionally, a condition is given in order that the property be preserved under countable products.

Coloring Cantor sets and resolvability of pseudocompact spaces

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].

Small diagonal of X and calibers of Cp(X)

Given a Lindelöf space X and a regular uncountable cardinal κ, we establish that κ is a caliber of Cp(X) if and only if the diagonal of the space X is κ-small. We show that the Lindelöf property of X cannot be omitted in this result because there exists a pseudocompact space X with a Gδ-diagonal such that ω1 is not a caliber of Cp(X). We also construct an example of a pseudocompact space X such that ω1 is a caliber of Cp(X) but not a caliber of Cp(βX)=Cp(υX); this solves two published open questions.

Finite powers of selectively pseudocompact groups

A space X is called selectively pseudocompact if for each sequence (Un)n<ω of pairwise disjoint nonempty open subsets of X there is a sequence (xn)n<ω of points in X such that xn∈Un, for each n<ω, and clX({xn:n<ω})∖(⋃n<ωUn)≠∅. Countably compact spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of CH, that for every positive integer k>2 there exists a topological group whose k-th power is countably compact but its (k+1)-st power is not selectively pseudocompact. This provides a positive answer to a question posed in [10] in any model of ZFC+CH.

Few remarks on maximal pseudocompactness

&lt;p align="LEFT"&gt;A pseudocompact space is maximal pseudocompact if every strictly finer topology is no longer pseudocompact. The main result here is a counterexample which answers a question rised by Alas, Sanchis and Wilson.&lt;/p&gt;

On c-realcompact spaces

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ0X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ0X, which is the smallest c-realcompact space between X and β0X, plays the same role (with respect to Cc(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ0X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if Cc(X) = C*c(X), or equivalently if and only if υ0X = β0 X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = Cc(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.

On pseudocompact spaces with a weak selection

We study properties of the pseudocompact spaces X with a weak selection, and we dedicate a particular attention to the weak selection topologies on X. In case when X is also locally compact, we obtain a convenient decomposition of X into a finite union of clopen sets, which are either almost compact or connected with a remainder of size two in their Stone–Čech compactification.

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

A space X is selectively sequentially pseudocompact if for every family {Un:n∈N} of non-empty open subsets of X, one can choose a point xn∈Un for every n∈N in such a way that the sequence {xn:n∈N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.

Generalization of Fixed Point Theorems in Pseudocompact Tichonov Space

In this paper we study some fixed point results in pseudocompact Tichnovo space using Edelstein type contractive conditions. The results presented in this paper include the generalization of some fixed point theorems established by Fisher and Pathak.

Selective sequential pseudocompactness

We say that a topological space X is selectively sequentially pseudocompact if for every family {Un:n∈N} of non-empty open subsets of X, one can choose a point xn∈Un for every n∈N in such a way that the sequence {xn:n∈N} has a convergent subsequence. We show that the class of selectively sequentially pseudocompact spaces is closed under arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every ω-bounded (= the closure of each countable set is compact) group is selectively sequentially pseudocompact, while compact spaces need not be selectively sequentially pseudocompact. Finally, we construct selectively sequentially pseudocompact group topologies on both the free group and the free Abelian group with continuum-many generators.

Completeness type properties on Cp(X,Y) spaces

In this paper we improve several results presented in [7] and in [9] related to the characterization of several kinds of pseudocompleteness and compactness properties in spaces of continuous functions of the form Cp(X,Y). In particular, we prove that for every Tychonoff space X and every separable metrizable topological group G for which Cp(X,G) is dense in GX, Cp(X,G) is weakly α-favorable if and only if X is uG-discrete. This result helps us to obtain two generalizations of a theorem due to V.V. Tkachuk in [24]:Theorem I5.5Let G be a separable completely metrizable topological group and X a set. If H is a dense subgroup ofGXand H is homeomorphic toGYfor some set Y, thenH=GX.A particular and interesting case is when X is a Tychonoff space and H=Cp(X,G) is dense in GX.Theorem II6.9Let G be a realcompact Čech-complete weakly α-favorable topological group with countable pseudocharacter and let X be regularC<ωG-discrete. Then,Cp(X,G)≅Gκif and only if X is a discrete space of cardinality κ.Besides, we obtain several applications to weakly pseudocompact spaces.

Spaces splittable over the class of Eberlein and descriptive compact spaces

We study splittability over some classes of compact spaces which are useful in functional analysis and general topology. Among other things we show that a scattered pseudocompact space splittable over the class of Eberlein compact spaces is Eberlein compact. We also prove that a compact space splittable over the class of Eberlein compact spaces is hereditarily \(\sigma \)-metacompact, and that if X is a compact space splittable over the class of Corson compact spaces, then \(d(X)=w(X)\). We also obtain several results on Rosenthal and on descriptive compact spaces. For instance: (1) a compact space is Rosenthal if and only if it is splittable over the class of Rosenthal compacta, (2) \(\mathfrak {c}\)-metrizable countably compact spaces which split over the class of descriptive compacta are descriptive compact spaces, (3) if X is a compact space splittable over the class of descriptive compact spaces, then \(hd(X)=w(X)\), (4) scattered compact spaces splittable over the class of descriptive compact spaces are \(\sigma \)-discrete descriptive compacta, and (5) it is consistent with ZFC that a compact space which splits over the class of descriptive compacta is a descriptive compact space.

The equivalence of two definitions of sequential pseudocompactness

&lt;p&gt;We show that two possible definitions of sequential pseudocompactness are equivalent, and point out some consequences.&lt;/p&gt;

Reflecting properties in continuous images of small weight

A topological property \(\mathscr {P}\) is reflected in continuous images of weight at most \(\omega _1\) if a space X has \(\mathscr {P}\) whenever every continuous image of X of weight at most \(\omega _1\) has \(\mathscr {P}\). When X is a generalized ordered space, we consider a number of topological properties including feeble Lindelofness and \(\kappa \)-monolithicity. In the final section we study small images of pseudocompact and countably compact spaces; we give a condition on continuous images of a pseudocompact space in order that it be compact and show that it is consistently true that a countably compact space of countable projective tightness is countably tight.

A pseudocompact group which is not strongly pseudocompact

A space X is called strongly pseudocompact if for each sequence (Un)n∈N of pairwise disjoint nonempty open subsets of X there is a sequence (xn)n∈N of points in X such that clX({xn:n∈N})∖(⋃n∈NUn)≠∅ and xn∈Un, for each n∈N. It is evident that every countably compact space is strongly pseudocompact and every strongly pseudocompact space is pseudocompact. In this paper, we construct a pseudocompact group that is not strongly pseudocompact answering two questions posed in [13].

The subspace of weak $P$-points of $\mathbb{N}^*$

Let $W$ be the subspace of $\mathbb N^*$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p \in \mathbb N^*$.

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.