Proposition 11

The following properties of a topological space $X$ are equivalent:

(O_{III}) The set of closed neighbourhoods of any point of $X$ is a fundamental system of neighbourhoods of the point.

(O'_{III}) Given any closed subset $F$ of $X$ and any point $x \notin F$ there is a neighbourhood of $x$ and a neighbourhood of $F$ which do not intersect.

(O_{III}) \Longrightarrow (O'_{III}): If $F$ is closed and $x \notin F$, then there is a closed neighbourhood $V$ of $x$ contained in the neighbourhood $C_F$ of $x$; $V$ and $C_V$ are neighbourhoods of $x$ and $F$ respectively, and have no point in common.

(O'_{III}) \Longrightarrow (O_{III}): If $W$ is an open neighbourhood of $x \in X$, then there is a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $C_W$ which are disjoint, and therefore $\overline{U} \subset W$.