Bitopological spaces

A. A. Ivanov

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part IV, Tome 122 (1982), pp. 30-55

Résumé

A notion of bitopological space introduced by Y. C. Kelly as a triple $(X,\tau_1,\tau_2)$ where $\tau_1$ and $\tau_2$ are topological structures on a set $X$ is well known. The author considers here more general notion of bitopological space as a pair $(X,\beta)$ where $\beta$ is a topological structure on $X\times X$ called bitopological structure on $X$. Any triple $(X,\tau_1,\tau_2)$ naturally defines a topological space $(X\times X, \tau_1\times\tau_2)$ and consequently a bitopological space $(X, \tau_1\times\tau_2)$. Since $\tau_1\times\tau_2$ is a very special case of a topological structure on $X\times X$ the above defined notion is a true generalization of the previous notion of bitopological space. Any mapping $f\colon X_1\to X_2$ is called bicontinuous mapping $(X_1,\beta_1)$ into $(X_2, \beta_2)$ if $f\times f\colon (X_1\times X_1, \beta_1)\to (X_2\times X_2, \beta_2)$ is a continuous mapping. The paper contains some basic notions and initial results of general theory of bitopological spaces.