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

We study some properties of compact-continuous mappings. We prove that if f:X→Yf: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 x0x0 and each closed set A   not containing x0x0, there is a compact-continuous function f:X→[0,1]f:X→[0,1] such that f(x0)=1f(x0)=1 and f(A)⊂{0}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→Yf:X→Y is a compact-continuous bijection such that iw(Y)≤ωiw(Y)≤ω, then X has a countable network.Let (X,T)(X,T) be a topological space. The topological space (X,Tk)(X,Tk) is called a k-leader   of (X,T)(X,T), where Tk={U:U∩CTk={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)(X,Tk).