Pseudocompact Research Articles

Some classes of topological spaces extending the class of Δ-spaces

A study of the class Δ \Delta consisting of topological Δ \Delta -spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes Δ 2 ⊂ Δ 1 \Delta _2 \subset \Delta _1 properly containing Δ \Delta . We observe that for every first-countable X X the following equivalences hold: X ∈ Δ 1 X\in \Delta _1 iff X ∈ Δ 2 X\in \Delta _2 iff each countable subset of X X is G δ G_{\delta } . Thus, new proposed concepts provide a natural extension of the family of all λ \lambda -sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X X belongs to the class Δ 1 \Delta _1 iff countable subsets of X X are scattered. (2) Every regular scattered space belongs to the class Δ 2 \Delta _2 . We investigate whether the classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the basic topological operations. Similarly to Δ \Delta , both classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to Δ \Delta , they are not preserved by closed continuous images. Let Y Y be l l -dominated by X X , i.e. C p ( X ) C_p(X) admits a continuous linear map onto C p ( Y ) C_p(Y) . We show that Y ∈ Δ 1 Y \in \Delta _1 whenever X ∈ Δ 1 X \in \Delta _1 . Moreover, we establish that if Y Y is l l -dominated by a compact scattered space X X , then Y Y is a pseudocompact space such that its Stone–Čech compactification β Y \beta Y is scattered.

Some generalized countably compact properties in topological groups

In this paper, firstly, we study strict q-spaces, strong q-spaces and point pseudocompact spaces in topological groups in terms of preimages of metrizable spaces. A continuous mapping f:X→Y is (strongly) sequential-perfect if f is closed and (f−1(F)) f−1(y) is a sequentially compact set for each (sequentially compact set F⊆Y) y∈Y. We give some characterizations as follows:(1)A topological group G is a strict q-space if and only if it is a sequential-perfect preimage of a metrizable space.(2)A topological group G is a strong q-space if and only if it is a strongly sequential-perfect preimage of a metrizable space.(3)A topological group G is a pointwise pseudocompact space if and only if there is a continuous open mapping f from G onto a metrizable space M such that f−1(F) is an r-pseudocompact set in G for each r-pseudocompact set F in M.Secondly, we study completeness of generalized countably compact topological groups in terms of preimages of completely metrizable spaces and obtain the following results:(4)A topological group G is strictly countably sieve complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is sequential-perfect.(5)A topological group G is countably sieve-s-complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is strongly sequential-perfect.(6)A topological group G is countably sieve-p-complete if and only if G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H.

Real-boards of ideals in C(X) and their applications to characterize some spaces

In this paper, using the concept of the real-boards of ideals in C(X), the ring of all real-valued continuous functions on a completely regular Hausdorff space X, some characterizations of nearly pseudocompact and nearly realcompact spaces are given. For an ideal I⊆C(X), the real-board of I is the largest subspace of βX, the Stone-Čech compactification of X, on which every member of I can be extended continuously. Using this concept, the subrings CKI(X) and C∞I(X) of C(X) are introduced and studied, and I-pseudocompact spaces which are generalizations of pseudocompact spaces are presented as well as some of their properties.

Polish cofinality in spaces of bounded functions

It is proved that Cp(X,[0,1]) is pseudocompact if and only if it has a dense cofinally Polish subspace. We also establish that Cp(X,[0,1]) is pseudocomplete if and only if it is pseudocompact and give a consistent example of a locally compact hereditarily separable space which is not cofinally Polish. This paper also features cofinally Baire spaces which form a wider class than cofinally Polish spaces; it is established that every locally pseudocompact space is cofinally Baire as well as every locally Polish space, every scattered space and every P-space. Our results solve several published open questions.

Extension of mappings from the product of pseudocompact spaces

Let X and Y be pseudocompact spaces and let a function Φ:X×Y→R be separately continuous. The following conditions are equivalent: (1) there is a dense Gδ subset of D⊂Y such that Φ is continuous at every point of X×D (Namioka property); (2) Φ is quasicontinuous; (3) Φ extends to a separately continuous function on βX×βY. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of separately continuous functions on the product of several pseudocompact spaces extending to separately continuous functions on products of Stone–Čech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.

On cellular-countably compact spaces

A space X is said to be cellular-countably compact if for each cellular family U in X, there is a countably compact subspace K of X such that U ? K ?? for each U ? U. The class of cellularcountably compact spaces contain the classes of countably compact spaces and cellular-compact spaces and contained in a class of pseudocompact spaces. We give an example of Tychonoff DCCC space which is not cellular-countably compact. By using Erd?s and Rad??s theorem, we establish the cardinal inequalities for cellular-countably compact spaces. We show that the cardinality of a normal cellular-countably compact space with a G?-diagonal is at most c. Finally, we study the topological behavior of cellular-countably compact spaces on subspaces and products.

Functions on products of pseudocompact spaces

For every natural number n, any continuous function on the product of X1×X2×...×Xn pseudocompact spaces extends to a separately continuous function on the product βX1×βX2×...×βXn of their Stone-Cech compactifications.

M-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS

In this article we consider the $m$-topology on linebreak $M(X,mathscr{A})$, the ring of all real measurable functions on a measurable space $(X, mathscr{A})$, and we denote it by $M_m(X,mathscr{A})$. We show that $M_m(X,mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,mathscr{A})‎cong‎ mathbb R^n$ as topological rings. Also, we show that $M_m(X,mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,mathscr{A})= {M}_{u}(X,mathscr{A})$, if and only if $ M_m(X,mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,mathscr{A})$ is a connected space, if and only if $M_m(X,mathscr{A})$ is a locally connected space, if and only if $M^*(X,mathscr{A})$ is a connected subset of $M_m(X,mathscr{A})$.

Ideal spaces

<p>Let C<sub>∞ </sub>(X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C<span style="vertical-align: sub;">∞ </span>(X) is an ideal of C(X). We define those spaces X to be ideal space where C<span style="vertical-align: sub;">∞ </span>(X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.</p>

A note on bf -spaces and on the distribution of the functor of the Dieudonné completion

Abstract A subset B of a space X is said to be bounded (in X) if the restriction to B of every real-valued continuous function on X is bounded. A real-valued function on X is called bf -continuous if its restriction to each bounded subset of X has a continuous extension to the whole space X. bf -spaces are spaces such that bf -continuous functions are continuous. We take advantage to the exponential map in the realm of bf -spaces in order to study bf -extensions of bf -continuous functions. This allows us to improve several results concerning the distribution of the functor of the Dieudonné completion. We also prove that a relative version of the classical Glicksberg’s theorem characterizing the product of two pseudocompact spaces is valid for kr- spaces. In the last section we show that bf -hemibounded groups are Moscow spaces and, consequently, they are strong-PT-groups.

Almost-normality of Isbell-Mrówka spaces

We explore almost-normality in Isbell-Mrówka spaces and some related concepts. We use forcing to provide an example of an almost-normal not normal almost disjoint family, explore the concept of semi-normality in Isbell-Mrówka spaces, define the concept of strongly (ℵ0,<c)-separated almost disjoint families and prove the generic existence of completely separable strongly (ℵ0,<c)-separated almost disjoint families assuming s=c and b=c. We also provide an example of a Tychonoff almost-normal not normal pseudocompact space which is not countably compact, answering a question from P. Szeptycki and S. Garcia-Balan.

Ascoli’s theorem for pseudocompact spaces

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of $$C_k(X)$$ is equicontinuous, where $$C_k(X)$$ denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively $$\omega $$ -bounded. The class of selectively $$\omega $$ -bounded spaces is studied.

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological R-vector space V(X) on X. We show that V(X) is a quotient of the free abelian topological group on [−1,1]×X, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of V(X) satisfy the so-called algebraic colimit property defined in the authors' previous work.

Metrizable quotients of free topological groups

We consider the following problem: For which Tychonoff topological spaces X do the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial (i.e. not finitely generated) metrizable quotient group? First, we give a complete solution of this problem for the key class of compact spaces X. Then, relying on this result, we resolve the problem for several more general important classes of spaces X, including the class of $$\sigma $$ -compact spaces, the class of pseudocompact spaces, the class of $$\omega $$ -bounded spaces, the class of Cech-complete spaces and the class of K-analytic spaces. Also we describe all absolutely analytic metric spaces X (in particular, completely metrizable spaces) such that the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial metrizable quotient group. Our results are based on an extensive use of topological properties of non-scattered spaces.

Rings and subrings of continuous functions with countable range

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings Ac (X) lying between Cc ∗ (X) and Cc (X)), where X is a Hausdorff zero-dimensional space, are studied in this article. It is shown that the structure space of each Ac (X) is homeomorphic to β 0 X, the Banaschewski compactification of X. From this a main result of [A. Veisi, ec -filters and ec -ideals in the function-ally countable subalgebra C∗ (X), Appl. Gen. Topol. 20(2) (2019), 395–405] easily follows. The countable counterpart of the m-topology and U -topology on C(X), namely mc -topology and Uc -topology, respectively, are introduced and using these, new characterizations of P -spaces and pseudocompact spaces are found out. Moreover, X is realized to be an almost P -space when and only when each maximal ideal/z-ideal in Cm(X) become a z 0-ideal. This leads to a characterization of Cc (X) among its intermediate rings for the case that X is an almost P -space. Noetherianness/Artinianness of Cc (X) and a few chosen subrings of Cc (X) are examined and finally, a complete description of z 0-ideals in a typical ring Ac (X) via z 0-ideals in Cc (X) is established.

Spaces with fragmentable open sets

The object of the paper are the regular topological spaces X in which there exists a metric d related to the topology in the following way: for every nonempty open subset U of X and for every ε>0 there exists a nonempty open subset V of U with d-diameter less than ε. It is shown that such a space X is pseudo-almost Čech complete if, and only if, it contains a dense completely metrizable subset (X is pseudo-almost Čech complete if it is a subset of some pseudocompact space Y and contains as a dense subset some Gδ-subset of Y).A large class of spaces L is described (containing all Borel subsets of compact spaces, all pseudocompact spaces and all p-spaces of Arhangel'skii) such that, if X belongs to L and admits a metric d with the above property, then X is “metrizable up to a first Baire category subset”. I.e. X is the union of two sets X1 and X2 where X1 is of the first Baire category in X and X2 is metrizable. We provide also different characterizations of the class L.

On cellular-compactness and related properties

We study three subclasses of the class of pseudocompact spaces. We answer two open questions concerning cellular-compact spaces and another concerning cellular-Lindelof spaces and introduce and study three other subclasses of the class of feebly compact spaces, namely the class of cellular-countably-compact spaces, that of the cellular-sequentially-compact spaces and that of the cellular-compact-metrizable spaces. We show that it is independent of ZFC whether or not a cellular-compact-metrizable LOTS is metrizable.

Normal images of the product and countably paracompact condensation

In 1997 Buzyakova [1] proved that for a Tychonoff, pseudocompact space X, X×(|βX|++1) can be condensed onto a normal space if and only if X can be condensed onto a compact, Hausdorff space. This elegant result shows the equivalence between condensation to a compact space and condensation of the product of a space with a compact space to a normal space. We prove the following result: If X is Tychonoff and X×(κ+1) can be condensed onto a normal space, then X can be condensed onto a normal, countably paracompact space, where κ=(22|X|)+.

A study of selectively star-ccc spaces

Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and every sequence (An:n∈ω) of maximal pairwise disjoint open families in X there exists a sequence (An:n∈ω) such that An∈An for every n∈ω and St(⋃n∈ωAn,U)=X. In this paper, we prove that:(1) A selectively star-ccc space is DCCC and a selectively star-ccc perfect space is CCC.(2) There exists a pseudocompact (hence, DCCC) space that is not selectively star-ccc.(3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent.(4) Every selectively star-ccc subspace of ω1ω has countable extent.(5) Under 2ℵ0=2ℵ1, there exists a selectively star-ccc normal space having a regular closed Gδ-subset which is not selectively star-ccc.(6) Every first countable selectively star-ccc space with a Gδ-diagonal has cardinality at most c.(7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c.

Separable Quotients of Free Topological Groups

AbstractWe study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.