International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 2, Pages 239-247
doi:10.1155/S0161171298000349

On strict and simple type extensions

Abstract

Let (Y,τ) be an extension of a space (X,τ′)⋅p∈Y, let 𝒪yp={W∩X:W∈τ,p∈W}. For U∈τ′, let o(U)={P∈Y:U∈𝒪yp}. 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 {U∪{p}:p∈Y,and U∈Oyp}, 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 U∈𝒪yp}, and 𝒰p={W∈τ′:intxclx(W)∈𝒪yp}={W∈τ′:intxclx(W)∈ℒp}=∩{𝒰:𝒰 is an open ultrafilter on X,𝒪yp⊂𝒰} 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.