bitopological

Bitopological refers to the study of bitopological spaces, a framework in topology where a single set is equipped with two distinct topologies. A bitopological space is typically described as a set X together with two topologies, often denoted tau1 and tau2. This setup enables the analysis of how two different notions of openness, convergence, and continuity interact on the same underlying set, and it supports the development of concepts that are not visible under a single topology.

Key ideas in bitology include pairwise concepts. A map between bitopological spaces is called pairwise continuous if it is continuous with respect to each topology on the domain and codomain. Similarly, pairwise convergence and pairwise compactness generalize their single-topology analogues to account for both tau1 and tau2. Researchers also study pairwise separation axioms, such as pairwise T1 or pairwise Hausdorff properties, as well as pairwise normality, which describe how separation conditions operate when both topologies influence the space. The field explores how properties in one topology relate to or contrast with properties in the other, and what new phenomena arise from their interaction.

Historical context and scope: Bitopology was introduced in the 1960s by James C. Kelly, who laid the groundwork for examining spaces with dual topologies. Since then, the study has produced a body of results detailing the interplay between the two topologies, with applications across analysis, dynamical systems, and theoretical computer science. While niche, bitopology provides a disciplined approach to convergence, continuity, and compactness in contexts where two topologies coexist and interact.