The closed finite-to-one mappings and their applications.

In this paper, we discuss the closed finite-to-one mapping theorems on generalized metric spaces and their applications. It is proved that point-G_δ properties, ℵ₀-snf-countability and csf-countability are invariants and inverse invariants under closed finite-to-one mappings. By the relationships between the weak first-countabilities, we obtain the closed finite-to-one mapping theorems of weak quasi-first-countability, quasi-first-countability, snf-countability, gf-countability and sof-countability. Furthermore, these results are applied to the study of symmetric products of topological spaces. [ABSTRACT FROM AUTHOR]