Definition 5

In a topological space $X$, a fundamental system of neighbourhoods of a point $x$ (resp. of a subset $A$ of $X$) is any set $\mathfrak{G}$ of neighbourhoods of $x$ (resp. $A$) such that for each neighbourhood $V$ of $x$ (resp. $A$) there is a neighbourhood $W \in \mathfrak{G}$ such that $W \subset V$.

If $\mathfrak{G}$ is a fundamental system of neighbourhoods of a subset $A$ of $X$, then every finite intersection of sets of $\mathfrak{G}$ contains a set of $\mathfrak{G}$.

Examples. 1) In a discrete space (no. 1) the set $\{ x \}$ alone constitutes a fundamental system of neighbourhoods of the point $x$. 2) On the rational line $\mathbf{Q}$ the set of all open intervals containing a point $x$ is a fundamental system of neighbourhoods of this point. So is the set of open intervals $]x - 1/n, x + 1/n[$, and the set of closed intervals $[x - 1/n, x + 1/n]$, where $n$ runs through all the integers $> 0$, or through any infinite strictly increasing sequence of integers $> 0$. \* There are analogous results for the real line. \*