Equivalence of Definitions of Interior (Topology)

Theorem
Let $\left({T, \tau}\right)$ be a topological space.

Let $H \subseteq T$.

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

Then $H^\circ$ can alternatively be defined as the largest open set contained in $H$, in the sense that:


 * If $K$ is an open set such that $K \subseteq H$, then $K \subseteq H^\circ$.

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

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

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

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 $H$.

Let $U$ be the largest open set of $T$ contained in $H$.

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

Then by definition $K \subseteq U$.

It follows from Union is Smallest Superset: General Result that $U$ is the union of all open sets of $T$ contained in $H$.

That is:


 * $\displaystyle U = \bigcup_{K \mathop \in \mathbb K} K$

where $\mathbb K := \left\{{K \in \tau: K \subseteq H}\right\}$.

From the definition of the interior of $H$ it follows that $U = H^\circ$.

The implication has been demonstrated to hold in both directions, so the interior of $H$ can be defined as the largest open set of $T$ contained in $H$.