Definition 4

Let $X$ be a topological space and $A$ any subset of $X$. A neighbourhood of $A$ is any subset of $X$ which contains an open set containing $A$. The neighbourhoods of a subset $\{ x \}$ consisting of a single point are also called neighbourhoods of the point $x$.

It is clear that every neighbourhood of a subset $A$ of $X$ is also a neighbourhood of each subset $B \subset A$; in particular, it is a neighbourhood of each point of $A$. Conversely, suppose $A$ is a neighbourhood of each of the points of a set $B$, and let $U$ be the union of the open sets contained in $A$; then $U \subset A$, and since each point of $B$ belongs to an open set contained in $A$, we have $B \subset U$; but $U$ is open by virtue of $(O_1)$, hence $A$ is a neighbourhood of $B$. In particular: