Neighbourhood

of a point $x$ (of a subset $A$) of a topological space

Any open subset of this space containing the point $x$ (the set $A$). Sometimes a neighbourhood of the point $x$ (the set $A$) is defined as any subset of this topological space containing the point $x$ (the set $A$) in its interior (cf. also Interior of a set): in this case the first definition is that of an open neighorhood. A set $N$ is a neighbourhood of the set $A$ if and only if it is a neighbourhood of each point $x \in A$.

In the first definition, the open neighbourhoods are precisely the open sets of the topology.

In the second definition, the system of neighbourhoods $\mathfrak{N}(x)$ of a point $x$ satisfy the following four properties:

1. $x \in N$ for every $N \in \mathfrak{N}(x)$;
2. If $M \supset N$ for $N \in \mathfrak{N}(x)$, then $M \in \mathfrak{N}(x)$;
3. If $N_1, N_2 \in \mathfrak{N}(x)$ then $N_1 \cap N_2 \in \mathfrak{N}(x)$;
4. For each $N \in \mathfrak{N}(x)$ there exists $M \in \mathfrak{N}(x)$ such that $N \in \mathfrak{N}(y)$ for each $y \in M$.

In the opposite direction, properties (1)--(4) may be taken as the definition of a topology; a set $N$ is open if it is in $\mathfrak{N}(x)$ for each of its elements $x$: these are the Hausdorff neighbourhood axioms.

References

• Franz, Wolfgang. General topology (Harrap, 1967).