The notion of exactness of bitopological extensions (Q1576018)

In the theory of extensions of topological spaces there is a natural notion of exactness of extensions. An extension \((X^\prime,\tau')\) of a topological space \((X,\tau)\) is called exact (in the sense of Ivanov) if for every two distinct points \(x\) and \(y\) in \(X'\), at least one of which does not belong to \(X\), there is a set \(F\) in \(X\) whose \(\tau^\prime\)-closure contains exactly one of these points (the points \(x\) and \(y\) are topologically distinguishable from within \(X\)). Under passage from the theory of extensions of topological spaces to the theory of extensions of bitopological spaces, the situation connected with this notion becomes somewhat more complicated. The fact is that in this situation we have bitopological spaces \((X,\beta)\) and \((X^\prime,\beta')\) and have to distinguish points \(x\) and \(y\) from within \(X\) while \(\beta^\prime\) is a topology on \(X^\prime\times X^\prime\). The author considers some variants of notion of exactness of extensions of bitopological spaces.