C-Tychonoff and L-Tychonoff Topological Spaces

Abstract

A topological space X is called C-Tychonoff if there exist a one-to-one function f from X onto a Tychonoff space Y such that f restriction K from K onto f(K) is a homeomorphism for each compact subspace K of X. We discuss this property and illustrate the relationships between C-Tychonoffness and some other properties like submetrizability, local compactness, L-Tychononess, C-normality, C-regularity, epinormality, sigma-compactness, pseudocompactness and zero-dimensional.

Ps-normal and Ps-Tychonoff spaces

“…Let f : X → Y be witness of C-Tychonoff. Since X is Ccompact and Frechet, then by Theorem 5 [1], f is continuous. Since f is continuous and A is compact, then f(A) is compact in Y .…”

“…We now show that P s-normality and P s-Tychonoffness are independent properties. We reproduce the following two examples from [1] for our purpose. Proof.…”

“…Let τ = {∅, R} ∪ {(x, ∞) : a ∈ R} be the topology on R. Then (R, τ ) is a normal space and therefore it is P s-normal space. From Example 4, [1], (R, τ ) is not a C-Tychonoff space. Then by Theorem 3.2 it is not a P s-Tychonoff space.…”

“…Let H be the set of all points of G with atmost contably many non-zero co-ordinates. Let M = G × H. By Example 4 in [1] M is a Tychonoff space and not C-normal. Therefore M is a P s-Tychonoff space.…”