Some classes of topological spaces and the space of <i>G</i>-permutation degree
Abstract
Abstract In this paper, we study the behavior of some classes of topological spaces under the influence of the functor of G -permutation degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>š²šÆ</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> {\operatorname{\sf SP}^{n}_{G}} . We prove: (a) if a space X is an r -space, then so is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>š²šÆ</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo>ā”</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {\operatorname{\sf SP}_{G}^{n}X} , (b) if X is a cosmic space, then so is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>š²šÆ</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo>ā”</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {\operatorname{\sf SP}_{G}^{n}X} , (c) if a space X is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>ā¢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>Īŗ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {C(\kappa)} -cosmic, then so is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>š²šÆ</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo>ā”</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {\operatorname{\sf SP}_{G}^{n}X} , (d) if a space X is an Ī±-space, then so is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>š²šÆ</m:mi> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:mo>ā”</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> {\operatorname{\sf SP}_{G}^{n}X} .