Set is Subset of its Topological Closure

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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:

$H \subseteq H^-$

$\blacksquare$