On strict and simple type extensions

Abstract

Let (Y, τ) be an extension of a space (X, τ′) · p ∈ Y, let . For U ∈ τ′, let . In 1964, Banaschweski introduced the strict extension Y#, and the simple extension Y+ of X (induced by (Y, τ)) having base {o(U) : U ∈ τ′} and , respectively. The extensions Y# and Y+ have been extensively used since then. In this paper, the open filters ℒp = {W ∈ τ′ : W⫆intxclx(U) for some , and = {W ∈ τ′ : intxclx(W) ∈ ℒp} = ∩{𝒰 : 𝒰 is an open ultrafilter on on X are used to define some new topologies on Y. Some of these topologies produce nice extensions of (X, τ′). We study some interrelationships of these extensions with Y#, and Y+ respectively.