# Equivalence of Definitions of Interior (Topology)

## Theorem

The following definitions of the concept of **interior** in the context of **topology** are equivalent:

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

Let $H \subseteq T$.

### Definition 1

The **interior** of $H$ is the union of all subsets of $H$ which are open in $T$.

That is, the **interior** of $H$ is defined as:

- $\ds H^\circ := \bigcup_{K \mathop \in \mathbb K} K$

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

### Definition 2

The **interior** of $H$ is defined as the largest open set of $T$ which is contained in $H$.

### Definition 3

The **interior** of $H$ is the set of all interior points of $H$.

## 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:

- $\displaystyle H^\circ = \bigcup_{K \mathop \in \mathbb K} K$

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 $H$.

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

Hence both definitions are equivalent.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors