Introduction to anti-bitopological spaces

Abstract

The main aim of this study is to introduce the notion of anti-bitopological space and to reintroduce
some basic concepts of topology in this novel framework. We point out that there are at least three possible
and not necessarily equivalent methods of defining openness (and thus, the notion of open set) with respect to
two anti-topologies simultaneously. We choose one of these approaches and concentrate on it. This allows us to
define anti-bitopological interior and closure in some specific manner. Moreover, we prove some initial lemmas
on anti-bitopological boundary. Finally, we study the problem of subspaces.