Basis Test for Isolated Point

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

Let $\BB$ be a synthetic basis of $T$.

Let $H \subseteq S$.

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

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 synthetic basis of $T$:
 * $\exists V \in \BB: x \in V \subseteq U$

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 $U \in \BB : U \cap H = \set x$.

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

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