Interior is Subset of Interior of Closure

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Let $H^\circ$ denote the interior of $H$.

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

Then:

$H^\circ \subseteq \left({H^-}\right)^\circ$

Proof

From Set is Subset of its Topological Closure, we have $H \subseteq H^-$.

The result follows directly from Interior of Subset.

$\blacksquare$