Remark 4

Let $f: X\rightarrow Y$ be a continuous and separated mapping, $y$ a point

of Y and A a finite subset of $\overset{-1}{f}(y)$. Let us show that there exists a family $(V_a)_{a\in A}$ of pairwise disjoint sets such that for each $a\in A$, the set $V_a$ is a neighbourhood of $a$ in X. For this, for each subset $\{a, b\}$ of A with two elements, let us choose a neighbourhood $V_{(a,b)}$ of $a$ and a neighbourhood $V_{(b,a)}$ of $b$ in X such that $V_{(a,b)}\cap V_{(b,a)}=\emptyset$ (prop. 1, (iii)). Let $V_a$ denote the intersection of the family consisting of X and the sets $V_{(a,b)}$ for $b\in A,b=\not a$. The set $V_a$ is a neighbourhood of $a$ in X, and if $a,b$ are two distinct elements of A, the set $V_a\cap V_b$ is contained in $V_{(a,b)}\cap V_{(b,a)}$, hence is empty.