Definition 4.4.1 (Neighbourhood). Let $(X, \topo)$ be a topological space and $A \subset X$. A neighbourhood of $A$ is a set $V \supset X$ where there exists $U \in \topo$ such that $A \subset U \subset V \subset X$.

The set $\cn_{X, \topo}(A) = \cn_{X}(A) = \cn_{\topo}(A) = \cn(A)$ denotes the collection of all neighbourhoods of $A$, and $\cn^{o}(A)$ denotes the set of open neighbourhoods of $A$.

If $A = \bracs{x}$ is a single point, the above definition and notation applies to $x$ directly.