Problems of the theory of bitopological spaces.

*(Russian. English summary)*Zbl 0685.54019Summary: The bitopological literature is devoted mostly (\(\approx\) 95 %) to a generalization of the theory of topological space T on the category of bitopological spaces in the sense of Kelly KT and partly on the category of bitopological space in (the author’s) general sense BT. In the paper the following topics are considered: separation axioms, connectivity (local connectivity), compactness (local compactness), other special properties of coverings (their localizations), mappings, extensions, dimensions, hyperspaces, connections with other kind structures.

Generalizations from T (from KT) on BT are very rare - new ideas are necessary. All the more new ideas are necessary for a true bitopological theory, having no origin in the T-theory and even in KT-theory. In the paper some initial notions and results of a corresponding theory are mentioned. It concerns also some applications of bitopological theory: bitopological representations of some classes of continuous mappings and bitopologizations of different mathematical objects.

Let \(M\) be a class of T-mappings. A class \(S\) of bitopological structures is a bitopological representation of \(M\) if \(f:(X,\rho)\to (X',\rho ')\), \(f\in M\Leftrightarrow \exists \beta,\beta '\in S\) such that \(f:(X,\beta)\to (X',\beta ')\)-BT-mapping. In other words the class \(M\) of continuous mappings has a bitopological description. The set of all piecewise linear mappings \(f:R^ p\to R^ q\) has a bitopological representation: For each \(n\geq 0\) we can define such bitopological structure \(\beta^ n\) on \(R^ n\) that the piecewise linear mapping \(f\) from \(R^ p\) in \(R^ q\) is a BT-mapping from \((R^ p,\beta^ p)\) to \((R^ q,\beta^ q)\). It allows us to construct a bitopological version of the theory of piecewise linear manifolds. A bitopological manifold is a topological manifold with a bitopological structure which satisfies special conditions (homogeneous, co-ordinated, sequential). The paper presents some approach to a general theory of bitopological manifolds and also variants of notion of bitopological group.

Generalizations from T (from KT) on BT are very rare - new ideas are necessary. All the more new ideas are necessary for a true bitopological theory, having no origin in the T-theory and even in KT-theory. In the paper some initial notions and results of a corresponding theory are mentioned. It concerns also some applications of bitopological theory: bitopological representations of some classes of continuous mappings and bitopologizations of different mathematical objects.

Let \(M\) be a class of T-mappings. A class \(S\) of bitopological structures is a bitopological representation of \(M\) if \(f:(X,\rho)\to (X',\rho ')\), \(f\in M\Leftrightarrow \exists \beta,\beta '\in S\) such that \(f:(X,\beta)\to (X',\beta ')\)-BT-mapping. In other words the class \(M\) of continuous mappings has a bitopological description. The set of all piecewise linear mappings \(f:R^ p\to R^ q\) has a bitopological representation: For each \(n\geq 0\) we can define such bitopological structure \(\beta^ n\) on \(R^ n\) that the piecewise linear mapping \(f\) from \(R^ p\) in \(R^ q\) is a BT-mapping from \((R^ p,\beta^ p)\) to \((R^ q,\beta^ q)\). It allows us to construct a bitopological version of the theory of piecewise linear manifolds. A bitopological manifold is a topological manifold with a bitopological structure which satisfies special conditions (homogeneous, co-ordinated, sequential). The paper presents some approach to a general theory of bitopological manifolds and also variants of notion of bitopological group.