Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open

Let $(X, \mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) and let $x \in X$.

A set $N \subseteq X$ is a neighbourhood of $x$ if there is an open $U \in \mathcal{T}$ with $x \in U \subseteq N$. The family of all neighbourhoods of $x$ is written $\mathcal{N}(x)$ and called the neighbourhood filter at $x$. A neighbourhood that is itself open is an open neighbourhood.

Convention, and it is a live fork: in this library a neighbourhood need not be open. The competing convention, used by Munkres among others, defines a neighbourhood of $x$ to be an open set containing $x$. Both are in current use; this library follows the one above and writes "open neighbourhood" in full whenever openness is wanted, so that no statement here depends on which convention a reader brings.

A family $\mathcal{B}_x \subseteq \mathcal{N}(x)$ is a neighbourhood base at $x$ if every neighbourhood of $x$ contains a member of $\mathcal{B}_x$: for every $N \in \mathcal{N}(x)$ there is $B \in \mathcal{B}_x$ with $B \subseteq N$.

Four immediate consequences, established here because they are used constantly.

1. $X$ is a neighbourhood of each of its points, since $x \in X \subseteq X$ and $X$ is open by (T1); so $\mathcal{N}(x) \ne \varnothing$ and every point has at least one neighbourhood base, namely $\mathcal{N}(x)$ itself.
2. A superset of a neighbourhood of $x$ is a neighbourhood of $x$: if $x \in U \subseteq N \subseteq M$ with $U$ open then $x \in U \subseteq M$.
3. The intersection of two neighbourhoods of $x$ is a neighbourhood of $x$: if $x \in U \subseteq N$ and $x \in V \subseteq M$ with $U, V$ open, then $U \cap V$ is open by (T3) and $x \in U \cap V \subseteq N \cap M$. By iteration the same holds for any intersection of finitely many neighbourhoods of $x$.
4. A set is open exactly when it is a neighbourhood of each of its points. If $U$ is open and $x \in U$ then $x \in U \subseteq U$. Conversely, if $N$ is a neighbourhood of each of its points, choose for each $x \in N$ an open $U_x$ with $x \in U_x \subseteq N$; then $N = \bigcup_{x \in N} U_x$ is open by (T2). No choice principle is involved: $U_x$ may be taken to be the union of all open subsets of $N$ containing $x$, which is determined by $x$ and $N$, and is open by (T2).

Basic sets give neighbourhood bases. If $\mathcal{B}$ is a basis for $\mathcal{T}$ (Basis and subbasis for a topology, and the topology generated by a family of sets) then $\mathcal{B}_x := \{\, B \in \mathcal{B} : x \in B \,\}$ is a neighbourhood base at $x$ consisting of open sets. Each such $B$ is open and contains $x$, hence is a neighbourhood of $x$; and if $N \in \mathcal{N}(x)$, fix open $U$ with $x \in U \subseteq N$ and then $B \in \mathcal{B}$ with $x \in B \subseteq U$, which gives $B \in \mathcal{B}_x$ with $B \subseteq N$. A member of $\mathcal{B}_x$ is called a basic neighbourhood of $x$.

• The convention costs nothing and buys one thing. Every statement of the form "for every neighbourhood $N$ of $x$ ... " whose predicate is preserved when $N$ is enlarged is equivalent to the statement with $N$ restricted to open neighbourhoods: every neighbourhood contains an open one, and the predicate then passes to the larger set. Eventual-membership and the usual local-existence tests have this form; an arbitrary predicate need not. What the wider notion buys is that $\mathcal{N}(x)$ is a filter on $X$ in the sense of Filter on a set: consequence 1 is (F1), consequence 3 is (F3), consequence 2 is (F4), and (F2) holds because $x \in U \subseteq \varnothing$ is impossible, so $\varnothing$ is a neighbourhood of no point. Under the narrower convention the family of open sets containing $x$ fails (F4) as soon as some non-open set contains an open set around $x$, so the name "neighbourhood filter" would not be available.

• A neighbourhood base is not required to be closed under intersection, and the bases used below usually are not; what is required is only that its members be cofinal downwards among neighbourhoods.