Set is Subset of its Topological Closure

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

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

Then:

$H \subseteq H^-$

Proof

From the definition of closure, we have:

$H^-$ is the union of $H$ and its limit points.

From Subset of Union it follows directly that:

$H \subseteq H^-$

$\blacksquare$