Finite Product Topologies Modulo An Ideals

Abstract

Given a topological space (X,$\tau $) and an ideal$\Im $ in X, a finer topology $\tau $* in X can be associated with $\tau $ and$\Im $. Given two topological spaces (X,$\tau $${}_{1}$), (Y,$\tau $${}_{2}$) and ideals $\Im $, $\vartheta$ in X, Y respectively, an ideal $\Im $x $\vartheta$ in X x Y, called the product ideal of $\Im $and $\vartheta$, in X x Y. We investigate inclusion relations between $\tau $${}_{1}$* x $\tau $${}_{2}$* and ($\tau $${}_{1}$ x $\tau $${}_{2 }$)* and the conditions under which $\tau $${}_{1}$* x $\tau $${}_{2}$* = ($\tau $${}_{1}$ x $\tau $${}_{2 }$)* and we extend the theorem for finite case.