Countably Compact Research Articles

Almost compactness in neutrosophic topological spaces

Abstract The aim of this write-up is to investigate some covering properties in neutrosophic topological spaces. We define proximate cover, almost compactness, almost countable compactness and almost Lindelöfness in connection with neutrosophic topological spaces and study some properties entangled with them.

A compact space K is Corson compact if and only if Cp(K) has a dense lc-scattered subspace

We establish that a compact space K is Corson compact if and only if it can be embedded in a topological group which is a W-space in the sense of Gruenhage. As a application of this result we show that K is Corson compact if and only if it embeds in Cp(X) for a Lindelöf scattered space X. We also prove that a zero-dimensional compact space K is Corson if and only if Cp(K,{0,1}) is a continuous image of a Lindelöf scattered space. Besides, a zero-dimensional countably compact space X is ω-monolithic if and only if Cp(X) has a dense functionally countable subspace.

Embeddings into countably compact Hausdorff spaces

UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn countably compact topological space.

The Mardešić conjecture for countably compact spaces

We shall show that if d and s are positive integers, Ki is a compact linearly ordered topological space for each i<d, X is a countably compact subspace of ∏i<dKi, Zj is an infinite Hausdorff space for each j<d+s, and f:X→∏j<d+sZj is a continuous surjection, then there exist at least s+1-many indexes j<d+s such that Zj is compact and metrizable. This improves the Mardešić Conjecture, which was proved by G. Martínez-Cervantes and G. Plebanek in [1].

On the Ideal Convergent Sequences in Fuzzy Normed Space

This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.

On the character of weakly normal off of the diagonal, countably compact spaces

On the character of weakly normal off of the diagonal, countably compact spaces

The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

Exponential separability is preserved by some products

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions.

Generalized metrizable properties and cardinal invariants in quotient spaces of quasitopological groups

In this paper, we study some generalized metrizable properties and cardinal invariants in quotient spaces of quasitopological groups. We mainly show that (1) If H is a neutral subgroup of a quasitopological group G such that G/H is Hausdorff, then G/H is weakly first-countable if and only if G/H is symmetrizable; (2) If H is a neutral subgroup of a quasitopological group G such that G/H is a Hausdorff countably compact space of countable pseudocharacter, then G/H is metrizable; (3) If H is a neutral subgroup of a quasitopological group G such that G/H is a q-space, then G/H is a β-space; (4) If H is a closed neutral subgroup of a quasitopological group G, then |G/H|≤2e(G/H)ψ(G/H).

A note on countable compact spaces

A note on countable compact spaces

Entropy pair realization

AbstractWe show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.

On the embedding of the Ω-saturation of a topological space

The countably-compactification of a topological space X is such its extension Y, that Y is a completely regular and countably-compact space, and any closed countably-compact subset of X is closed in Y. But this extension does not always exist. Due to this, the concept of a saturation of a topological space appeared, which is a generalisation of the countably-compactification: instead of the condition of the countably-compactness of Y, it is necessary that any infinite subset of X has a limit point in Y. Meanwhile, the second condition remains unchanged. Such an extension is already defined for any T1-space. In this paper we consider a specific construction of saturation named as Ω-saturation. It is proved that under some additional (necessary and sufficient) condition to the separation of the initial space X, its Ω-saturation is canonically embedded in the Stone – Čech compactification βX. An analogous result is obtained for the countably-compactification by K. Morita.

A New Approach to Vague Soft Bi-Topological Spaces

Fuzzy soft topology considers only membership value. It has nothing to do with the non-membership value. So an extension was needed in this direction. Vague soft topology addresses both membership and non-membership values simultaneously. Sometimes vague soft topology (single structure) is unable to address some complex structures. So an extension to vague soft bi-topology (double structure) was needed in this direction. To make this situation more meaningful, a new concept of vague soft bi-topological space is introduced and its structural characteristics are attempted with a new definition. In this article, new concept of vague soft bi-topological space (VSBTS) is initiated and its structural behaviors are attempted. This approach is based on generalized vague soft open sets, known as vague soft open sets. An ample of examples are given to understand the structures. For the non-validity of some results, counter examples are provided to pay the price. Pair-wise vague soft open and pair-wise vague soft close sets are also addressed with examples in (VSBTS). Vague soft separation axioms are initiated in (VSBTS) concerning soft points of the space. Other separation axioms are also addressed relative to soft points of the space. Vague soft bi-topological properties are studied with the application of vague soft open sets with respect to soft points of the spaces. The characterization of vague soft close as well as vague soft open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequential compactness and countable compactness in (VSBTS) with respect to soft open sets are addressed.

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.

The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group G. A deep result due to Ernest [13] and Hughes [22] asserts that every weakly compact subset of a locally compact (LC) group G is compact in the LC-topology, generalizing thereby a previous result of Glicksberg [19] for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countable compactness, pseudocompactness and functional boundedness.

Sequences and related closure and compactness properties in frames

The aim of this note is to introduce concepts of sequence and sequence convergence in the point-free setting. These are then used to define sequential closedness and associated compactness properties in frames. The interplay of these closure and compactness notions are investigated. Relating back to the pointed setting provides additional insight into an asymmetric Kuratowski-Mrówka type theorem for countable compactness in general topology.

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 $M_m(X,mathscr{A})$.

On the pseudouniform topology on C(X)

We denote by Cs(X) the set C(X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of Cs(X) initiated in Pseudouniform topologies onC(X)given by ideals (Pichardo-Mendoza et al. (2013) [6]). We prove that Cs(X) is a k-space if and only if Cs(X) is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of Cs(X). In addition, we study the cellularity, density, weight and character of Cs(X). We prove that (1) d((R2λ)s)=λ if λ=λω, and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies χ(Cs(X))=w(Cs(X)).

Algebraic reflexivity of diameter-preserving linear bijections between $C(X)$-spaces

We prove that if X and Y are first countable compact Hausdorff spaces, then the set of all diameter-preserving linear bijections from C(X) to C(Y ) is algebraically reflexive.

Countably compact spaces admitting full r-skeletons are proximal

We provide simple characterizations of spaces admitting full r-skeletons, c-skeletons and q-skeletons, by using ω-monotone functions. We use this characterization to prove that every countably compact space admitting a full r-skeleton is proximal; furthermore the characterizations are used to show that the class of spaces admitting full c-skeletons is invariant under subspaces, disjoint topological sums and Σ-products, in addition to prove that the class of spaces admitting full q-skeletons is closed under extensions, continuous images and one-point Lindelöf extensions of disjoint topological sums. These characterizations also yield some positive results for products.