Completely Regular Research Articles

Compactification of spaces of measures and pseudocompactness

We prove pseudocompactness of a Tychonoff space X and the space P(X) of Radon probability measures on it with the weak topology under the condition that the Stone–ech compactification of the space P(X) is homeomorphic to the space P(βX) of Radon probability measures on the Stone–ech compactification of the space X.

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

On a variant of extremal disconnectedness in locales and spaces

We study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor.

Velichko's notions close to sequential separability and their hereditary variants in Cp-theory

A space X is sequentially separable if there is a countable S⊂X such that every point of X is the limit of a sequence of points from S. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: σ-separability and F-separability. The aim of this paper is to study σ-separability and F-separability (and their hereditary variants) of the space Cp(X) of all real-valued continuous functions, defined on a Tychonoff space X, endowed with the pointwise convergence topology. In particular, we proved that σ-separability coincides with sequential separability. Hereditary variants (hereditarily σ-separability and hereditarily F-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.

On sequences of finitely supported measures related to the Josefson–Nissenzweig theorem

Given a Tychonoff space X, we call a sequence 〈μn:n∈ω〉 of signed Borel measures on X a finitely supported Josefson–Nissenzweig sequence (in short a JN-sequence) if: 1) for every n∈ω the measure μn is a finite combination of one-point measures and ‖μn‖=1, and 2) ∫Xfdμn→0 for every continuous function f∈C(X). Our main result asserts that if a Tychonoff space X admits a JN-sequence, then there exists a JN-sequence 〈μn:n∈ω〉 such that: i) supp(μn)∩supp(μk)=∅ for every n≠k∈ω, and ii) the union ⋃n∈ωsupp(μn) is a discrete subset of X. We also prove that if a Tychonoff space X carries a JN-sequence, then either there is a JN-sequence 〈μn:n∈ω〉 on X such that |supp(μn)|=2 for every n∈ω, or for every JN-sequence 〈μn:n∈ω〉 on X we have limn→∞⁡|supp(μn)|=∞.

Remarks on SHD spaces and more divergence properties

The class of SHD spaces was recently introduced in [12]. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space X such that F[X], the Pixley-Roy hyperspace of X, βX, the Stone-Čech compactification of X, and Cp(X), the ring of continuous functions over X equipped with the topology of pointwise convergence, are SHD.In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between X and F[X] regarding these new concepts.

Some remarks on the projective properties of Menger and Hurewicz

It is known that both the Menger and Hurewicz property of a Tychonoff space X can be described by the way X is placed in its Čech-Stone compactification βX. We provide analogous characterizations for the projective versions of the properties of Menger and Hurewicz.

One-Point Extensions of a Tychonoff Space X via Closed Ideals of $$C_{B}(X)$$

One-Point Extensions of a Tychonoff Space X via Closed Ideals of $$C_{B}(X)$$

Embedding the free topological group F(Xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X^n)$$\\end{document} into F(X)

In 1976, Nickolas showed that for each natural n, the free topological group F(Xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X^n)$$\\end{document} is topologically isomorphic to a subgroup of F(X) provided X is a compact space or, more generally, a kω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\\omega }$$\\end{document}-space. We complement the Nickolas’ embedding theorem by showing that it remains true for every topological space X such that all finite powers of X are pseudocompact. For example, all pseudocompact k-spaces enjoy this property. Also, we extend the embedding theorem to the class of NCω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NC_\\omega $$\\end{document}-spaces that includes, in particular, the kω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_\\omega $$\\end{document}-spaces and the well-ordered spaces of ordinals [0,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0, \\alpha )$$\\end{document}, for every ordinal α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Our results are quite sharp because we present a first example of a Tychonoff space Z such that F(Z) does not contain an isomorphic copy of the group F(Z2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(Z^2)$$\\end{document}. In addition, our space Z is countably compact, separable, and its square Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z^2$$\\end{document} is not pseudocompact.

The Josefson–Nissenzweig theorem and filters on ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}

For a free filter F on ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}, endow the space NF=ω∪{pF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F=\\omega \\cup \\{p_F\\}$$\\end{document}, where pF∉ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_F\ ot \\in \\omega $$\\end{document}, with the topology in which every element of ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is isolated whereas all open neighborhoods of pF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_F$$\\end{document} are of the form A∪{pF}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\cup \\{p_F\\}$$\\end{document} for A∈F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in F$$\\end{document}. Spaces of the form NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} carries a sequence ⟨μn:n∈ω⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\mu _n:n\\in \\omega \\rangle $$\\end{document} of normalized finitely supported signed measures such that μn(f)→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _n(f)\\rightarrow 0$$\\end{document} for every bounded continuous real-valued function f on NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} if and only if F∗≤KZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*\\le _K{\\mathcal {Z}}$$\\end{document}, that is, the dual ideal F∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*$$\\end{document} is Katětov below the asymptotic density ideal Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {Z}}$$\\end{document}. Consequently, we get that if F∗≤KZ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F^*\\le _K{\\mathcal {Z}}$$\\end{document}, then: (1) if X is a Tychonoff space and NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} is homeomorphic to a subspace of X, then the space Cp∗(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_p^*(X)$$\\end{document} of bounded continuous real-valued functions on X contains a complemented copy of the space c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document} endowed with the pointwise topology, (2) if K is a compact Hausdorff space and NF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_F$$\\end{document} is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

On bounded sets in [formula omitted

In this note we study some topological properties of the bounded sets of the locally convex space Ck(X) of all real-valued continuous functions defined on a Tychonoff space X equipped with the compact-open topology as well as of the bounded sets of a (DF)-space. We show that, assuming X is a Lindelöf Čech-complete space, if X is scattered then each bounded set in Ck(X) is a Fréchet-Urysohn space under the weak topology. In the opposite direction, if X is pseudocompact and each bounded set in Ck(X) is Fréchet-Urysohn under the weak topology, then X is k-scattered. We also derive a new condition, related to the closed bounded sets, for a (DF)-space to be quasibarrelled, which we apply to the bounded sets of Ck(X). Finally, we get a sort of Krein-like theorem for Ck(X) when X is a Lindelöf Σ-space.

Covering properties of $ C_{p}\left(Y|X\right) $

<abstract><p>Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.</p></abstract>

Zip rings (resp. czip rings) and some applications

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C(X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings.

Remote sublocales

We use the notion of a remote collection of a Tychonoff space to define a remote sublocale of any locale. Our definition is conservative, in the sense that, a subset is remote if and only if the sublocale it induces is remote in the locale of opens. We characterize remote sublocales and show that the Booleanization of a locale is the largest remote sublocale of the locale, a result with no pointset topological counterpart. We study localic maps that send remote sublocales back and forth. It turns out that the localic maps which preserve remote sublocales are precisely those with skeletal left adjoints.

Linear homeomorphisms of function spaces and the position of a space in its compactification

Abstract An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

Some properties involving feeble compactness, II: Generalized Σ-products and Cp-spaces

In [14], we begin analyzing some classes of topological spaces contained in the class of feebly compact spaces. In this article, which is a second part of that work, we determine, for a compact metrizable topological group G, when Cp(X,G) is selectively pseudocompact, and when it is weakly compact-bounded for a Tychonoff space X. Moreover, we prove that for some Hausdorff extensions XQ of a space X constructed using free filters on X, the space Cp(XQ,G) is selectively sequentially pseudocompact. We therefore obtain a class of spaces having sufficient conditions to affirmatively solve Problem 8.5 posed by Dorantes-Aldama and Shakmatov in [7].

On a Type of Generalized GS_β-Closed Set in Grill Topology

This work intends to propose the concept of generalized -closed set-in grill topological space. Further, some of the properties of such generalized -closed set are evaluated and provided with suitable illustrations. Many theorems based on generalized -closed set have also been proved in this work. Furthermore, the ideas of -regular space and -normal space are also studied. Also, the relationship between -regular and - normal spaces is discussed. The properties of -regular space and -normal space concepts have also been discussed.Introduction: Alo, R. A., and Shapiro, H. L defined the semi- normal spaces and considered some of their important inheritance. The authors further studied the compactness in grill topological spaces and studied some of their properties. Data granulation serves as a vital decision-making tool in numerous real-life applications. Mashhour, A. S and Abd, M. E introduced certain new topological tools employing rough set approximations for the purpose of data granulation. The introduction of -normality helps to separate closed subsets, continuous functions. It also extends the support to define the concepts of complete regularity and Tietze extension theorem. The -regular space studied in this article is regular, that is completely regular, which is a stronger condition.Objectives: I.Arockiarani and A. Karthika introduced a new concept such as generalized -closed set in grill topological space. The above paper motivates me to do the research in generalized -closed set. In this paper a new concept of “generalized -closed set in grill topological space” are introduced. In this article we are using stronger condition for proving some theorems.Methods: The present task ambitioned at formation of the extension of topological system applying the idea of grill. The importance of the theory of grill between the topological spaces have been elaborated. A set is countable collection based on elements, without integrity of form. At the same time some kinds of algebraic actions are applied on this set, indefinitely the elements of this set are correlated into a whole, so ultimately it grows into a space. So ultimately in the whole paper we are applying the concept of grill topological space.Results: In this article a new concept of generalized -closed set in a grill topological space are introduced. Using this definition we are proving many results based on operators on grill topological space. Furthermore, the implication such as “all closed set are generalized -closed sets” is proved and the converse may not be true are also proved with proper illustration. In addition the relationship between -normal space, -normal space, -regular space and -regular space are also studied. Further many equivalent conditions are also proved using -normal space and -regular space.Conclusions: The work proposes and investigates the concept of generalized -closed sets within the framework of grill topological spaces. It explores the properties of these sets, provides illustrative examples, and proves theorems related to their behavior. Additionally, the study delves into the concepts of -regular space and -normal space, discussing their properties and implications. The research contributes to the understanding of these concepts and their applications in the context of grill topological spaces.

A Banach space characterization of (sequentially) Ascoli spaces

We prove that a Tychonoff space X is an Ascoli space (resp., a sequentially Ascoli space) if and only if for each Banach space E, every k-continuous and almost k-compact (resp., almost k-sequential) map T form X into the Banach dual E′ of E is continuous.

On complemented copies of the space c0 in spaces Cp(X,E)$C_p(X,E)$

AbstractWe study the question for which Tychonoff spaces X and locally convex spaces E the space of continuous E‐valued functions on X contains a complemented copy of the space , both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Domański and Drewnowski, proved for the case of Banach and Fréchet spaces . Also, for given infinite Tychonoff spaces X and Y, we show that contains a complemented copy of if and only if any of the spaces and contains such a subspace.