Some classes of topological spaces and the space of G-permutation degree

Abstract In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G -permutation degree 𝖲𝖯 G n {\operatorname{\sf SP}^{n}_{G}} . We prove: (a) if a space X is an r -space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (b) if X is a cosmic space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (c) if a space X is a C ⁢ ( κ ) {C(\kappa)} -cosmic, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} , (d) if a space X is an α-space, then so is 𝖲𝖯 G n ⁡ X {\operatorname{\sf SP}_{G}^{n}X} .