Completely Regular Research Articles

On normal dense subspaces of iterated function spaces

If a Tychonoff space X has a point-countable network of cardinality ≤ℵ1 and Y is a normal dense subspace of Cp(X), then Y is collectionwise normal. If a Tychonoff space X has a point-countable network of cardinality ≤ℵ1, then Cp(Cp(X)) also has a point-countable network of cardinality ≤ℵ1. We prove these statements and obtain some corollaries on equivalence of normality and collectionwise normality for dense subspaces of Cp,n(X).

The Menger property and l-equivalence

For a Tychonoff space X, we denote by Cp(X) the space of all real valued continuous functions with the topology of pointwise convergence. In this short note, we remark that if Cp(X) and Cp(Y) are linearly homeomorphic and X is a Menger (resp., Hurewicz) space with property (⁎), then Y is also a Menger (resp., Hurewicz) space.

Descriptive topology for analysts

We collect together old and recent research on Descriptive Topology, mainly related to the linear space $$C\left( X\right) $$ of real-valued continuous functions on a Tychonoff space X, equipped with the pointwise or with the compact-open topology.

WHEN THE HEWITT REALCOMPACTIFICATION AND THE P-COREFLECTION COMMUTE

If X is a Tychonoff space then its P -coreflection Xδ is a Tychonoff space that is a dense subspace of the realcompact space (υX)δ, where υX denotes the Hewitt realcompactification of X. We investigate under what conditions Xδ is C-embedded in (υX)δ, i.e. under what conditions υ(Xδ )=( υX)δ. An example shows that this can fail for the product of a compact space and a P -space. It is possible for a von Neumann regular ring A to be isomorphic to a C(Y ) and lie between C(X) and C(Xδ) without being isomorphic to C(Xδ). This cannot occur if X is realcompact or more generally if υ(Xδ )=( υX)δ. Applications are given to the epimorphic hull of C(X).

Network Properties of Function Spaces Endowed with the Fell Hypograph Topology

For a Tychonoff space X, we study the network properties of the space $$C_{{\downarrow }{\mathsf {F}}}(X)$$ of continuous functions from X to $${\mathbb {R}}$$ , endowed with the Fell hypograph topology. We prove that the function space $$C_{{\downarrow }{\mathsf {F}}}(X)$$ is cosmic iff $$C_{{\downarrow }{\mathsf {F}}}(X)$$ is an $$\aleph _0$$ -space iff $$C_{{\downarrow }{\mathsf {F}}}(X)$$ is a $${\mathfrak {P}}_0$$ -space iff X is a weakly locally compact $$\aleph _0$$ -space.

Spectral representations of topological groups and near-openly generated groups

Near-openly generated groups are introduced. They form a topological and multiplicative subclass of -factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces of all continuous real-valued functions on a Tychonoff space with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory. Bibliography: 24 titles.

Baire category properties of some Baire type function spaces

We study the Baire type properties in the classes Bα(X,Y) of Baire-α functions and Bαst(X,Y) of stable Baire-α functions from a topological space X to a topological space Y, where α≥1 is a countable ordinal. Among others we prove the following results. If X is a normal space, then B1(X)=RX iff X is a Q-space. If X is a Tychonoff space of countable pseudocharacter, then: (i) for every ordinal α≥2, the spaces Bα(X) and Bαst(X) are Choquet and hence Baire, and (ii) B1(X) is a Choquet space iff X is a λ-space. Also we prove that for a Tychonoff space X the space B1(X) is meager if X is airy. We introduce a new class of almost K-analytic spaces which properly contains Čech-complete spaces and K-analytic spaces and show that for an almost K-analytic space X the following assertions are equivalent: (i) B1(X) is a Baire space, (ii) B1(X) is a Choquet space, and (iii) every compact subset of X is scattered. We show that the Baireness of the function space B1(X) is essentially a property of separable subspaces of X: if Y is a Polish space and X is a Y-Hausdorff space, then B1(X,Y) is Baire iff every countable subset of X is contained in a subspace Z⊆X such that the function space B1(Z,Y) is Baire and the restriction operator B1(X,Y)→B1(Z,Y), f↦f|Z, is surjective.

Countable tightness and $${\mathfrak {G}}$$-bases on free topological groups

Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. In this paper, we consider two topological properties of F(X) or A(X), namely the countable tightness and $$\mathfrak G$$-base. We provide some characterizations of the countable tightness and $$\mathfrak G$$-base of F(X) and A(X) for various special classes of spaces X. Furthermore, we also study the countable tightness and $$\mathfrak G$$-base of some $$F_{n}(X)$$ of F(X).

Realizing spaces as path-component spaces

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group is trivial.

Covering properties of Cp(X) and Ck(X)

Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when Cp (X) or Ck (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C(X).

On the isomorphic classification of the free topological groups of the non-Tychonoff spaces

In the paper we proposed the method for reducing the problem of isomorphic classi­fication of free topological groups of completely regular spaces to the problem of isomorphic classification of free topological groups of completely Tychonoff spaces. Cite as: N. M. Pyrch, “On the isomorphic classification of the free topological groups of the non-Tychonoff spaces,” Prykl. Probl. Mekh. Mat. , Issue 17, 27–37 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.27-37

Categorical and topological properties of the functor of Radon functionals

In this paper we introduce a functor OSR of semiadditive Radon functionals in the category Tych of Tychonoff spaces, which extends the functor OS:Comp→Comp of semiadditive functionals. It is proved that the functor OSR:Tych→Tych is normal in the category of Tychonoff spaces.

On Markov equivalence of the bundles of Tychonoff spaces 2: special isomorphisms

On Markov equivalence of the bundles of Tychonoff spaces 2: special isomorphisms

Spaces with an M-diagonal

In this note we prove that if X is a Tychonoff space and $$X^2 {\setminus }\Delta $$ is dominated by a second countable space then X is cosmic. This solves an open problem of Cascales et al. (Topol Appl 158:204–214). We also consider the case when X is compact and $$X^2 {\setminus }\Delta $$ is dominated by a metric space M; in this situation we show that if such domination is strong, then the tightness of X is bounded by the weight of M.

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.

Dunford–Pettis type properties and the Grothendieck property for function spaces

For a Tychonoff space X, let $$C_k(X)$$ and $$C_p(X)$$ be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that $$C_k(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space $$[0,\kappa )$$ for some ordinal $$\kappa $$ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and $$C_p(X) $$ has the Grothendieck property if and only if every functionally bounded subset of X is finite.

Strongly sequentially separable function spaces, via selection principles

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continuous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

Bounded sets structure of CpX and quasi‐(DF)‐spaces

AbstractFor wide classes of locally convex spaces, in particular, for the space of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of ‐spaces have led us to introduce quasi‐‐spaces, a class of locally convex spaces containing ‐spaces that preserves subspaces, countable direct sums and countable products. Regular ‐spaces as well as their strong duals are quasi‐‐spaces. Hence the space of distributions provides a concrete example of a quasi‐‐space not being a ‐space. We show that has a fundamental bounded resolution if and only if is a quasi‐‐space if and only if the strong dual of is a quasi‐‐space if and only if X is countable. If X is metrizable, then is a quasi‐‐space if and only if X is a σ‐compact Polish space.

Locally convex properties of free locally convex spaces

Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent: (i) L(X) is ℓ∞-barrelled, (ii) L(X) is ℓ∞-quasibarrelled, (iii) L(X) is c0-barrelled, (iv) L(X) is ℵ0-quasibarrelled, and (v) X is a P-space. If X is a non-discrete metrizable space, then L(X) is c0-quasibarrelled but it is neither c0-barrelled nor ℓ∞-quasibarrelled. We prove that L(X) is a (DF)-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF)-space but is not a c0-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df)-space but is not a quasi-(DF)-space. If X is a μ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford–Pettis property for every Tychonoff space X. If X is a sequential space and a μ-space (for example, metrizable), then L(X) has the sequential Dunford–Pettis property iff X is discrete.