A note on compact-continuous mappings and k-separation axioms

Abstract

We study some properties of compact-continuous mappings. We prove that if f:X→Y is a compact-continuous surjection and X is a Lindelöf Σ-space, then Y is a Lindelöf Σ-space. A space X is called a weakly Tychonoff space, if one-point sets are closed in X and for each point x0 and each closed set A not containing x0, there is a compact-continuous function f:X→[0,1] such that f(x0)=1 and f(A)⊂{0}. Some properties of weakly Tychonoff spaces are discussed in this note. We show that if X is a weakly Tychonoff Lindelöf Σ-space and f:X→Y is a compact-continuous bijection such that iw(Y)≤ω, then X has a countable network.Let (X,T) be a topological space. The topological space (X,Tk) is called a k-leader of (X,T), where Tk={U:U∩C is open in C for each compact subspace C of X}. In the last part of this note we get some conclusions on separation axioms of (X,Tk).