Zero-Bidimension and Various Classes of Bitopological Spaces

Abstract

The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is also new. The notion of almost $n$-dimensionality is considered from the bitopological point of view. Bitopological spaces in which every subset is i-open in its $j$-closure (i.e.,(i,j)-submaximal spaces) are introduced and their properties are studied. Based on the investigations begun in [5] and [14], sufficient conditions are found for bitopological spaces to be(1,2)-Baire in the class of p-normal spaces. Furthermore, (i,j)-I-spaces are introduced and both the relations between(i,j)-submaximal, (i,j)-nodec and (i,j)-I-spaces, and their properties are studied when two topologies on a set are either independent of each other or interconnected by the inclusion, S-, C- and N-relations or by their combinations. The final part of the paper deals with the questions of preservation of $(i,j)$-submaximal and $(2,1)\dd I$-spaces to an image, of $D$-spaces to an image and an inverse image for both the topological and the bitopological cases. Two theorems are formulated containing, on the one hand, topological conditions and, on the other hand, bitopological ones, under which a topological space is a $D$-space.