Remark

Proposition 10 implies the following: if R is closed and if φ denotes the canonical mapping X → X/R, then for each x ∈ X and each neighbourhood U of the equivalence class of x in X, φ(U) is a neighbourhood of φ(x) in X/R. It should be carefully noticed that this statement by no means implies that for every neighbourhood V of x, φ(V) is a neighbourhood of φ(x); in other words (no. 3, Proposition 5) a closed equivalence relation is not necessarily open (Exercise 2). Conversely, an open equivalence relation is not necessarily closed (no. 1, Example 4);

for if $U$ is a neighbourhood in $X$ of an equivalence class $M$, then for each $x \in M$ and each neighbourhood $V \subset U$ of $x$, the saturation of $V$ is certainly a neighbourhood of $M$ in $X$, but this neighbourhood is not necessarily contained in $U$.

Finally, there are equivalence relations other than equality which are both open and closed (Exercise 3) and equivalence relations which are neither open nor closed ($§ 8$, Exercise 10).