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 (ld) and the local weak density (lwd) 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 SPn and the subfunctor of permutation degree SPnG, P is the cardinal number of topological spaces. Let X be an infinite T1-space. We prove that the following propositions hold. (1) Let Yn⊂Xn; (A) if d(Yn)=d(Xn), then d(SPnY)=d(SPnX); (B) if lwd(Yn)=lwd(Xn), then lwd(SPnY)=lwd(SPnX). (2) Let Y⊂X; (A) if ld(Y)=ld(X), then ld(SPnY)=ld(SPnX); (B) if wd(Y)=wd(X), then wd(SPnY)=wd(SPnX). (3) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is a locally compact T1-space, then SPnX,SPnGX, and expnX are k-spaces. (4) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is an infinite T1-space, then nπw(X)=nπw(SPnX)=nπw(SPnGX)=nπw(expnX). We also have studied that the functors SPn, SPnG, and expn preserve any k-space. The functors SP2 and SP3G do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite T1-space X coincides with the densities of the spaces Xn, SPnX, and expnX. It is also shown that the weak density of an infinite T1-space X coincides with the weak densities of the spaces Xn, SPnX, and expnX.