Characterization of Neighborhood by Basis

Theorem

Let $\struct {S, \tau}$ be a topological space.

Let $\BB$ be an analytic basis for $\tau$.

Let $N \subseteq S$.

Let $x \in N$.

Then $N$ is a neighborhood at $x$ if and only if:

$\exists B \in \BB : x \in B : B \subseteq N$

Proof

From Basis induces Local Basis:

$\BB_x = \set {B \in \BB: x \in B}$ is a local basis at $x$

By definition of local basis, $N$ is a neighborhood at $x$ if and only if:

$\exists B \in \BB_x : B \subseteq N$

By definition of $\BB_x$, $N$ is a neighborhood at $x$ if and only if:

$\exists B \in \BB : x \in B : B \subseteq N$

$\blacksquare$