Remark

Let $F$ be a subset of $X$ such that for each point $x$ of $X$ there is a neighbourhood $V$ of $x$ such that $V \cap F$ is closed in the subspace $V$; then it follows from Proposition 3 of no. 1 that $F$ is closed in $X$. On the other hand, Proposition 5 below shows that in general there are locally closed sets which are not closed in $X$.