Zero-Bidimension and Various Classes of Bitopological Spaces

TL;DR

This paper explores advanced properties of bitopological spaces, introducing new notions like relative normality and (i,j)-submaximality, and investigates their implications for dimensions, Baire spaces, and preservation under images.

Contribution

It introduces new concepts such as relative normality and (i,j)-submaximal spaces, and studies their properties and relationships within bitopological spaces, extending existing theories.

Findings

Sum theorem for zero-dimensional p-closed sets proved

Conditions for bitopological spaces to be (1,2)-Baire established

Preservation properties of (i,j)-submaximal and D-spaces analyzed

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.