Equivalence of Definitions of Interior (Topology)

Theorem
Let $\struct {T, \tau}$ be a topological space.

Let $H \subseteq T$.

Proof
Let $\mathbb K$ be defined as:
 * $\mathbb K := \set {K \in \tau: K \subseteq H}$

That is, let $\mathbb K$ be the set of all open sets of $T$ contained in $H$.

Then from definition 1 of the interior of $H$, we have:
 * $\ds H^\circ = \bigcup_{K \mathop \in \mathbb K} K$

That is, $H^\circ$ is the union of all the open sets of $T$ contained in $H$.

Let $K \subseteq T$ such that $K$ is open in $T$ and $K \subseteq H$.

That is, let $K \in \mathbb K$.

Then from Subset of Union it follows directly that $K \subseteq H^\circ$.

So any open set in $T$ contained in $H$ is a subset of $H^\circ$, and so $H^\circ$ is the largest open set of $T$ contained in $H$.

That is, $H^\circ$ is also the interior of $H$ by definition 2.

Hence both definitions are equivalent.