International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 2, Pages 239-247

On strict and simple type extensions

Mohan Tikoo

Department of Mathematics, Southeast Missouri State University, Cape Girardeau 63701, Missouri, USA

Received 14 August 1996; Revised 21 November 1996

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.


Let (Y,τ) be an extension of a space (X,τ)pY, let 𝒪yp={WX:Wτ,pW}. For Uτ, let o(U)={PY:U𝒪yp}. In 1964, Banaschweski introduced the strict extension Y#, and the simple extension Y+ of X (induced by (Y,τ)) having base {o(U):Uτ} and {U{p}:pY,andUOyp}, respectively. The extensions Y# and Y+ have been extensively used since then. In this paper, the open filters p={Wτ:Wintxclx(U) for some U𝒪yp}, and 𝒰p={Wτ:intxclx(W)𝒪yp}={Wτ:intxclx(W)p}={𝒰:𝒰 is an open ultrafilter on X,𝒪yp𝒰} 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.