Local Basis Test for Isolated Point

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

Let $H \subseteq S$.

Let $x \in H$.

Let $\BB_x$ be a local basis of $x$.

Then $x$ is an isolated point of $H$ :
 * $\exists U \in \BB_x : U \cap H = \set x$

Necessary Condition
Let $x \in H$ be an isolated point of $H$.

By definition of an isolated point:
 * $\exists U \in \tau: U \cap H = \set x$

By definition of a local basis of $T$:
 * $\exists V \in \BB_x : x \in V \subseteq U$

From Set Intersection Preserves Subsets:
 * $V \cap H \subseteq U \cap H = \set x$

From Singleton of Element is Subset:
 * $\set x \subseteq V \cap H$

From set equality:
 * $V \cap H = \set x$

Sufficient Condition
Let $U \in \BB_x : U \cap H = \set x$.

By definition of local basis of $T$:
 * $U \in \tau$

Then $x$ is an isolated point of $H$ by definition.