A new pointfree form of subspaces

Abstract

This paper concerns sub-topolocales as a generalization of subspaces which are determined by their lattice of open sets. We first study topolocales (dual of topoframes), and sub-topolocales with the connectivity properties of them.Then we show that every sub-topolocale of a regular (resp., completely regular) topolocale is regular (resp., completely regular). Finally, we show that sub-topolocales of a normal topolocale are not necessary normal unless we rewrite this for the special cases.