Equivalence of Definitions of Interior (Topology)

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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

Definition $1$ is equivalent to Definition $2$

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.

$\Box$


Definition $1$ is equivalent to Definition $3$

\(\ds x\) \(\in\) \(\ds \bigcup_{K \in \mathbb K} K\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists K \in \mathbb K: \, \) \(\ds x\) \(\in\) \(\ds K\) Definition of Union of Set of Sets
\(\ds \leadstoandfrom \ \ \) \(\ds \exists K \subseteq H: \, \) \(\ds K\) \(\in\) \(\ds \tau \wedge x \in K\) Definition of $\mathbb K$
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds x \text{ is an interior point of } H\) Definition of Interior Point (Topology)

Therefore:

$\ds \bigcup_{K \mathop \in \mathbb K} K = \set {x \mid \text {$x$ is an interior point of $H$} }$

$\blacksquare$


This article is complete as far as it goes, but it could do with expansion.
In particular: to include 2-3 equivalence
Why? Once you've proved $1 \iff 2$ and $1 \iff 3$, $2 \iff 3$ follows trivially
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