Mohan Tikoo

Copyright © 1998 Mohan Tikoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.