Hausdorff metric, the construction of a Hausdorff quotient space, and boundaries of symmetric spaces (Q1392318)

As is known, in many interesting cases, the quotient space of a metric space (topological space, algebraic variety) turns out to be non-Hausdorff. In algebraic geometry, a number of methods are known for constructing a Hausdorff quotient space from a non-Hausdorff one (the closure in the Chow scheme, blow-up of singularities, and the spectrum of an algebra of invariant functions; see [\textit{M. M. Kapranov}, Adv. Ser. Math. 16(2), 29-110 (1993; Zbl 0811.14043)]). The aim of the present note is to offer a very simple construction (essentially, of general topological nature) for a Hausdorff quotient space and to discuss some of its applications.