Birelator Spaces Are Natural Generalizations of Not Only Bitopological Spaces, But Also Ideal Topological Spaces

In 1962, W. J. Pervin proved that every topology T $$\mathscr {T}$$ on a set X can be derived from the quasi-uniformity U $$\mathscr {U}$$ on X generated by the preorder relations with A ∈ T $$A\in \mathscr {T}$$ . Thus, a quasi-uniform space X ( U ) $$X\hskip 0.2 mm(\hskip 0.2 mm\mathscr {U}\hskip 0.2 mm)$$ is a generalization of a topological space X ( T ) $$X\hskip 0.2 mm(\hskip 0.2 mm\mathscr {T}\hskip 0.2 mm)$$ , and a bi-quasi-uniform space X ( U , V ) $$X\,(\hskip 0.2 mm\mathscr {U}\hskip 0.2 mm, \,\mathscr {V}\hskip 0.2 mm)$$ is a generalization of a bitopological space X ( P , Q ) $$X\,(\hskip 0.2 mm\mathscr {P}\hskip 0.2 mm, \,\mathscr {Q}\hskip 0.2 mm)$$ , studied first by J. C. Kelly in 1963. Now, we shall show that a bi-quasi-uniform space X ( U , V ) $$X\,(\hskip 0.2 mm\mathscr {U}\hskip 0.2 mm, \,\mathscr {V}\hskip 0.2 mm)$$ is also a certain generalization of an ideal topological space X ( T , I ) $$X\,(\hskip 0.2 mm\mathscr {T}\hskip 0.2 mm, \,\mathscr {I}\hskip 0.2 mm)$$ studied first by K. Kuratowski in 1933. Actually, instead of a bi-quasi-uniform space X ( U , V ) $$X\,(\hskip 0.2 mm\mathscr {U}\hskip 0.2 mm, \,V\hskip 0.2 mm)$$ , we shall use a birelator space ( X , Y ) ( R , S ) $$({\hskip 0.2 mm}X\hskip 0.2 mm, \,Y\hskip 0.2 mm)\hskip 0.2 mm(\hskip 0.2 mm\mathscr {R}\hskip 0.2 mm, \,\mathscr {S}\hskip 0.2 mm)$$ , where X and Y are sets and R $$\mathscr {R}$$ and S $$\mathscr {S}$$ are relators (families of relations) on X to Y . Much more general results could be achieved by using corelations (functions of P ( X ) $$\mathscr {P}\hskip 0.2 mm({\hskip 0.2 mm}X\hskip 0.2 mm)$$ to P ( Y ) ) $$\mathscr {P}\hskip 0.2 mm({\hskip 0.2 mm}Y\hskip 0.2 mm)\hskip 0.2 mm\big )$$ instead relations on X to Y . However, a detailed theory of corelators has not been worked out yet.