Some Properties of Topological Spaces Related to the Local Density and the Local Weak Density

Beshimov, R. B.; Mamadaliev, N. K.; Mukhamadiev, F. G.

doi:10.13189/ms.2015.030404

Cited by 2 publications

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“…The local density at a point x is denoted by ld (x). The local density of a topological space X is defined as the supremum of all numbers ld (x) for x ∈ X: ld (X) = sup{ld (x) : x ∈ X} [2,6]. It is known that ld (X) ≤ d (X) for any topological space.…”

Section: Auxiliary Materialsmentioning

confidence: 99%

“…The local weak density at a point x is denoted by lwd (x). The local weak density of a topological space X is defined as the supremum of all numbers lwd(x) for x ∈ X: lwd (X) = sup{lwd (x) : x ∈ X} [2,6]. If X is a space of local density τ and f : X → Y is an open continuous "onto" mapping, then Y is a space of local density τ [12].…”

Section: Auxiliary Materialsmentioning

confidence: 99%

“…In [5], some cardinal properties of Hattori spaces and their hyperspaces were studied. In [2,6], some properties of topological spaces related to local density and local weak density in various topological spaces were studied.…”

Section: Introductionmentioning

confidence: 99%

“…In [2], it was proved that the property of local density and the property of local weak density coincide for stratifiable spaces. These cardinal numbers are preserved under open mappings and are inverse invariant of a class of closed irreducible mappings.…”

Section: Introductionmentioning

confidence: 99%

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The Local Density and the Local Weak Density in the Space of Permutation Degree and in Hattori Space

Yuldashev

Mukhamadiev

2020

UMJ

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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 \(\exp _{n} X\).

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Section: Auxiliary Materialsmentioning

confidence: 99%

Section: Auxiliary Materialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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