Some topological and cardinal properties of space of the permutation degree

Abstract

In this paper, we prove facts and some cardinal properties on space of the permutation degree, which are introduced in \cite{Borsuk}. More precisely, we prove that the product $X^{n}$ is Lindel\"{o}f(resp. locally Lindel\"{o}f) space, then the space $SP^{n}X$ is also Lindel\"{o}f (resp. locally Lindel\"{o}f) space. We also prove that the product $X^{n}$ is weakly Lindel\"{o}f(resp. weakly locally Lindel\"{o}f) space, then the space $SP^{n}X$ is also weakly Lindel\"{o}f (resp. weakly locally Lindel\"{o}f) space. Moreover, we investigate the behavior of the network weight, $\pi$-character, and local density of topological spaces under the influence of a functor of $G$-permutation degree. It is proved that this functor preserves the network weight, $\pi$-character, and locally density of infinite topological $T_{1}$-spaces.