Proposition 3

Let $(x_i)_{1 \leq i \leq n}$ be a finite family of distinct points of a Hausdorff space $X$; then each $x_i$ has a neighbourhood $V_i$ in $X$ such that the $V_i$ ($1 \leq i \leq n$) are mutually disjoint.

The proof is by induction on $n$: the case $n = 2$ is just the axiom (H). Let then $W_i$ ($1 \leq i \leq n - 1$) be a neighbourhood of $x_i$ such that the $W_i$ are mutually disjoint. On the other hand, for $1 \leq i \leq n - 1$ there is a neighbourhood $T_i$ of $x_i$ and a neighbourhood $U_i$ of $x_n$ which do not intersect. If we take $V_i$ to be $W_i \cap T_i$ for $1 \leq i \leq n - 1$ and

the conditions of the proposition are satisfied.