Set Closure is Smallest Closed Set
Jump to navigation
Jump to search
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.