On Extensions of Topologies

Abstract

If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.