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

Venezuela.

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

space.