# Set Closure is Smallest Closed Set

## Theorem

### Topology

Let $T$ be a topological space.

Let $H \subseteq T$.

Let $H^-$ denote the closure of $H$ in $T$.

Then $H^-$ is the smallest superset of $H$ that is closed in $T$.

### Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $S$ be a subset of $X$:

$S \subseteq X$

Let $S^-$ be the closure of $S$.

Then $S^-$ is the smallest closed set which contains $S$.

### Closure Operator

Let $S$ be a set.

Let $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$.

Then $\map \cl T$ is the smallest closed set (with respect to $\cl$) containing $T$ as a subset.