The local density and the local weak density in the space of permutation degree and in Hattori space

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.

• 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 $l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)$.
• 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 $w d\, \left(S P^{n} Y\right)$ $=w d\, \left(S P^{n} X\right)$.
• 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.
• 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 $\exp _{n} X$.