On first countable quasitopological homotopy groups

Abstract

Abstract If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ωn(X, x0) → πn(X, x0), is the natural quotient map from the nth loop space of (X, x0), Ωn(X, x0), with compact-open topology to the quasitopological nth homotopy group πn(X, x0). Ultimately, using these results, we found some properties of first countable homotopy groups.