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

Abstract

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

Some Cardinal and Geometric Properties of the Space of Permutation Degree

“…For every mapping f : X → Y, the mapping [1,4,6]. The weak density of a space X, denoted by wd(X), is the smallest cardinal number τ ≥ ω such that there is a π-base B = ∪{B α : α < τ} in X, and for each α < τ, B α is a centered system of open sets in X [7,9,10].…”

“…In recent research, the interest in the theory of cardinal invariants and their behavior under the influence of various covariant functors is increasing (see, for example, [7][8][9]). In [10], the authors investigated several cardinal invariants under the influence of some seminormal and normal functors.…”

Some network-type properties of the space of G-permutation degree

“…The study of the influence of normal, weakly normal and seminormal functors to topological and geometric properties of topological spaces, in particular to the cardinal properties (density, weak density, local density, tightness, set tightness, T -tightness, functional tightness, mini-tightness), has been developed in recent investigations (see [1,2,3,4,6,7,9,10,11,12,14,15,16,17]). For example, in [2] it was proved that the exponential functor of finite degree preserves the functional tightness and minimal tightness of compact sets, while in [11] a similar investigation (related to the T −tightness, set tightness, functional tightness, mini-tightness) was done for the functor SP n G of G-permutation degree.…”

Some Tightness-Type Properties of the Space of Permutation Degree

“…In recent researches a number of authors was interested in the behaviour of certain cardinal invariants under the influence of various covariant functors. For example, in [5,11,12,14,19,20] the authors investigated several cardinal invariants under the influence of some weakly normal, seminormal and normal functors.…”