On the Theory of Relative Bitopological and Topological Properties

TL;DR

This paper develops a comprehensive theory of relative properties in bitopological and topological spaces, including separation axioms, compactness, dimension, and Baire spaces, enhancing existing mathematical frameworks.

Contribution

It introduces and investigates new relative versions of separation, compactness, and dimension properties in bitopological spaces, strengthening foundational theories.

Findings

Extended the classification of relative separation axioms.

Introduced relative compactness and pseudocompactness concepts.

Analyzed relative dimension and Baire space properties.

Abstract

In the first part of the work (Sections 2-6) a special attention is given to relative separation axioms and relative connectedness, in particular, many relative versions of p-T_0, p-T_1, p-T_2, (i,j)- and p-regularities, (i,j)- and p-complete regularities, p-real normality and p-normality are discussed. Moreover, relative properties of (i,j)- and p-compactness types, including relative versions of (i,j)- and p-paracompactness, (i,j)- and p-Lindeofness, (i,j)- and p-pseudocompactness are also introduced and investigated. The second part (Sections 7-12) is devoted, on the one hand, to relative bitopological inductive and covering dimension functions and, on the other hand, to relative versions of Baire spaces for both the topological and the bitopological case. At the end, note that relative (bi)topological properties play a special role not only in the development of respective theories, but also in the strengthening of the previously known results.