Generalized metric spaces and the space of $G$-permutation degree

We study the connection between a $\sigma$-spaces, $\Sigma$-spaces, a stratifiable spaces, $\aleph$-spaces and the spaces of $G$-permutation degree. It is shown that the functor $SP^n_{G}$ preserves the class of $\sigma$-spaces, stratifiable spaces and $\aleph$-spaces.