Separation properties at \(p\) for the topological category of Cauchy spaces (Q452798)

Cauchy spaces, designed to provide a general setting for studying classical Cauchy convergence, are pairs \(\langle X,\mathcal C\rangle\), where \(\mathcal C \) is a family of proper filters on a set \(X\) -- the Cauchy filters on \(X\) -- satisfying: (1) the filter of supersets of a point is Cauchy; (2) a filter containing a Cauchy filter is also Cauchy; and (3) if \(F\) and \(G\) are Cauchy and each member of \(F\) intersects each member of \(G\), then \(F\cap G\) is Cauchy. A function from a Cauchy space \(\langle X,\mathcal C\rangle\) to a Cauchy space \(\langle Y,\mathcal D\rangle\) is Cauchy continuous if the image of each filter in \(\mathcal C\) generates a filter in \(\mathcal D\). These ingredients constitute the topological category \textbf{CHY}. In a series of papers \textit{M. Baran} [Math.\ Balk., New Ser. 10, No. ~1, 39--48 (1996; Zbl 1036.54502)] defined separation properties at a point, and generalized this to point-free definitions via the generic element method of topos theory for an arbitrary topological category over sets. In the present paper, it is the author's stated goal to: (1) give a characterization for each of the generalized versions of T\({}_i\) \((i = 0, 1, 2)\) and Pre T\({}_2\) at a point in \textbf{CHY}; (2) to examine how these generalizations are related; and (3) to show that each of the generalized (point-free) separation properties implies its corresponding version at a point in \textbf{CHY}.