Topological Closure of Subset is Subset of Topological Closure/Proof 1

Proof
From Topological Closure is Closed, $\map \cl K$ is closed.

From Set is Subset of its Topological Closure:


 * $K \subseteq \map \cl K$

By Subset Relation is Transitive, it follows that:


 * $H \subseteq \map \cl K$

Hence, by definition of closure as the smallest closed set that contains $H$:


 * $\map \cl H \subseteq \map \cl K$