Interior point (topology)

Definition

Given a metric space $(X,d)$ and an arbitrary subset $U\subseteq X$, a point $x\in X$ is interior to $U$ if:

• $\exists\delta>0[B_\delta(x)\subseteq U]$

Relation to Neighbourhood

This definition is VERY similar to that of a neighbourhood. In fact that I believe "$U$ is a neighbourhood of $x$" is simply a generalisation of interior point to topological spaces. Note that:

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Claim: $x$ is interior to $U$ $\implies$ $U$ is a neighbourhood of $x$

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Proof

As $x$ is interior to $U$ we know immediately that:

• $\exists\delta>0[B_\delta(x)\subseteq U]$
• We also see that $x\in B_\delta(x)$ (as $d(x,x)=0$, so for any $\delta>0$ we still have $x\in B_\delta(x)$)
• But open balls are open sets

So we have found an open set entirely contained in $U$ which also contains $x$, that is:

$\exists\delta>0[x\in B_\delta(x)\subseteq U]$

Thus $U$ is a neighbourhood of $x$

This completes the proof.

See also

• Interior
• Neighbourhood
• Open set
• Metric space