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

We study some properties of compact-continuous mappings. We prove that if f : X -> Y is a compact-continuous surjection and X is a Lindelof Sigma-space, then Y is a Lindelof Sigma-space. A space X is called a weakly Tychonoff space, if one-point sets are closed in X and for each point x(0) and each closed set A not containing x(0), there is a compact-continuous function f : X -> [0, 1] such that f(x(0)) = 1 and f(A) subset of {0}. Some properties of weakly Tychonoff spaces are discussed in this note. We show that if X is a weakly Tychonoff Lindelof Sigma-space and f : X -> Y is a compact-continuous bijection such that iw(Y) <= w, then X has a countable network. Let (X, T) be a topological space. The topological space (X, T-k) is called a k-leader of (X,T), where T-k = {U: U boolean AND 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, T-k). (C) 2015 Elsevier B.V. All rights reserved.