Equivalence of Definitions of Interior (Topology)

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Let $\left({T, \tau}\right)$ 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:

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

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

Definition 2

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


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 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