On the impact of the exponential functor on some types of continuous mappings

Abstract

This paper investigates the influence of the symmetric product functor $SP^{n}$ and the exponential functor $\exp$ on certain types of continuous mappings, focusing specifically on almost-open and pseudo-open mappings. The primary objective is to analyze how functors $SP^{n}$ and $exp$ interact with these mappings and alter their topological properties. Through the study, several key lemmas have been proven, providing insights into the behavior of the $SP^{n}$-induced mappings. Notably, it is demonstrated that for open sets $ U_1, U_2, \dots, U_n\subset X $, the set $[U_1, U_2, \dots, U_n]$ retains its openness in $ SP^{n}X $. These findings contribute to a deeper understanding of the topological implications of applying the $ SP^{n}$ functor on continuous mappings, offering new perspectives on its effect on almost-open and pseudo-open transformations.