scholarly journals ON LOCAL WEAK τ-DENSITY OF TOPOLOGICAL SPACES

2021 ◽  
pp. 16-23
Author(s):  
F.G. Mukhamadiev ◽  
◽  
Keyword(s):  
Topological Space ◽  
Local Density ◽  
Cardinal Number ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Weak Separability ◽  
Weak Density ◽  
Locally Compact Spaces

A topological space X is locally weakly separable [3] at a point x∈X if x has a weakly separable neighbourhood. A topological space X is locally weakly separable if X is locally weakly separable at every point x∈X. The notion of local weak separability can be generalized for any cardinal τ ≥ℵ0 . A topological space X is locally weakly τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a weak τ-dense neighborhood in X [4]. The local weak density at a point x is denoted as lwd(x). The local weak density of a topological space X is defined in following way: lwd ( X ) = sup{ lwd ( x) : x∈ X } . A topological space X is locally τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a τ-dense neighborhood in X [4]. The local density at a point x is denoted as ld(x). The local density of a topological space X is defined in following way: ld ( X ) = sup{ ld ( x) : x∈ X } . It is known that for any topological space we have ld(X ) ≤ d(X ) . In this paper, we study questions of the local weak τ-density of topological spaces and establish sufficient conditions for the preservation of the property of a local weak τ-density of subsets of topological spaces. It is proved that a subset of a locally τ-dense space is also locally weakly τ-dense if it satisfies at least one of the following conditions: (a) the subset is open in the space; (b) the subset is everywhere dense in space; (c) the subset is canonically closed in space. A proof is given that the sum, intersection, and product of locally weakly τ-dense spaces are also locally weakly τ-dense spaces. And also questions of local τ-density and local weak τ-density are considered in locally compact spaces. It is proved that these two concepts coincide in locally compact spaces.

scholarly journals α-fuzzy compactness inI-topological spaces

2003 ◽  
Vol 2003 (41) ◽  
pp. 2609-2617
Author(s):  
Valentín Gregori ◽  
Hans-peter A. Künzi
Keyword(s):  
Topological Space ◽  
Baire Category ◽  
Unit Interval ◽  
Topological Spaces ◽  
Baire Category Theorem ◽  
Locally Compact ◽  
Compact Spaces ◽  
Category Theorem ◽  
Locally Compact Spaces

Using a gradation of openness in a (Chang fuzzy)I-topological space, we introduce degrees of compactness that we callα-fuzzy compactness (whereαbelongs to the unit interval), so extending the concept of compactness due to C. L. Chang. We obtain a Baire category theorem forα-locally compact spaces and construct a one-pointα-fuzzy compactification of anI-topological space.

scholarly journals THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

2020 ◽  
Vol 6 (2) ◽  
pp. 108
Author(s):  
Tursun K. Yuldashev ◽  
Farhod G. Mukhamadiev
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Real Line ◽  
Local Density ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact ◽  
The Real ◽  
Weak Density

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\),  \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\). (2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

Essentials of General Topology

2021 ◽  
pp. 191-244
Author(s):  
Adel N. Boules
Keyword(s):  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Section 8 ◽  
Finite Products ◽  
The One ◽  
One Point Compactification ◽  
Locally Compact Spaces ◽  
Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

Locally Compact Normal Spaces in the Constructible Universe

1982 ◽  
Vol 34 (5) ◽  
pp. 1091-1096 ◽  
Author(s):  
W. Stephen Watson
Keyword(s):  
Set Theory ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces ◽  
Perfectly Normal

Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.All topological spaces are assumed to be Hausdorff.First, a definition and three standard lemmata are needed.

On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

2005 ◽  
Vol 57 (6) ◽  
pp. 1121-1138 ◽  
Author(s):  
Michael Barr ◽  
R. Raphael ◽  
R. G. Woods
Keyword(s):  
Locally Compact ◽  
Compact Spaces ◽  
Open Questions ◽  
Tychonoff Spaces ◽  
Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

2003 ◽  
Vol 10 (2) ◽  
pp. 209-222
Author(s):  
I. Bakhia
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

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