On maps: continuous, closed, perfect, and with closed graph

Abstract

ABSTRACT. This paper gives relationships between continuous maps, closed maps, perfect maps, and maps with closed graph in certain classes of topological spaces.

Characterizations of Functions with Closed Graph

“…A subset K of X is called limit point compact (see [4]) if every infinite subset of K has a limit point in K. For any space X, X * will denote the one point compactification of X and Q will denote the set of all rationals with the usual topology. If A is a subset of X, we say that X is T 1 at A (see [2]) if each point of A is closed in X. According to Wilansky X is said to be Frechet space (or closure sequential) if for each subset A of X, x ∈ cl(A) implies there exists a sequence {x n } in A converging to x while X will be called square Frechet if X × X is Frechet (see [6]).…”