This is joint work with Maria Luisa Colasante, Universidad de los Andes,

Let X be a topological space. The topological
closure of the diagonal \Delta = {(x,x): x\in X} is a
symmetric relation on X. Our starting point is the
well-known proposition that a topological space
is T_2 if and only if cl(\Delta) = \Delta. We
investigate the closure of the diagonal on T_1
spaces and characterise those equivalence relations
that arise as the closure of the diagonal of some T_1