Proposition 10

Let R be an equivalence relation on a topological space X. Then R is closed if and only if every equivalence class M mod R possesses a fundamental system of neighbourhoods which are saturated with respect to R.

Suppose R is closed, and let U be an arbitrary open neighbourhood of M; since F = ĈU is closed in X, the saturation S of F with respect to R is closed in X. Since M is saturated with respect to R, we have M ∩ S = ∅, and thus V = ĈS is an open neighbourhood of M, saturated with respect to R and contained in U.

To prove the converse, let F be any closed subset of X. Let T be the saturation of F with respect to R, let x be a point of ĈT, and let M be the equivalence class of x; then M ∩ T = ∅ and a fortiori M ∩ F = ∅, so that U = ĈF is a neighbourhood of M. Hence there is a neighbourhood V ⊂ U of M such that V is saturated with respect to R; V does not meet F, hence does not meet T, so that ĈT is a neighbourhood of M and therefore of x. This shows that ĈT is open (§ 1, no. 2, Proposition 1), i.e. T is closed.