Interior point (topology)

Definition

Metric space

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

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

Relation to Neighbourhood

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

Claim: [ilmath]x[/ilmath] is interior to [ilmath]U[/ilmath] [ilmath]\implies[/ilmath] [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath]

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Proof

As [ilmath]x[/ilmath] is interior to [ilmath]U[/ilmath] we know immediately that:

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

So we have found an open set entirely contained in [ilmath]U[/ilmath] which also contains [ilmath]x[/ilmath], that is:

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

Thus [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath]

This completes the proof.

This implication can only go one way as in an arbitrary topological space (which may not have a metric that induces it) there is no notion of open balls (as there's no metric!) thus there can be no notion of interior point.

Topological space

In a topological space [ilmath](X,\mathcal{J})[/ilmath] and given an arbitrary subset of [ilmath]X[/ilmath], [ilmath]U\subseteq X[/ilmath] we can say that a point, [ilmath]x\in X[/ilmath], is an interior point of [ilmath]U[/ilmath] if:

• [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath]
  • Recall that if [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] we require [ilmath]\exists\mathcal{O}\in\mathcal{J}[x\in\mathcal{O}\subseteq U][/ilmath]

Relation to Neighbourhood

We can see that in a topological space that neighbourhood to and interior point of are equivalent. This site (like) defines neighbourhood to as containing an open set with the point in it. However some authors (notably Munkres) do not use this definition and use neighbourhood as a synonym for open set. In this case interior point of and neighbourhood to are not equivalent.

I don't like the term interior point as it suggests some notion of being inside, but the point being in the set is not enough for it to be interior! So I am happy with this and stand by the comments on the neighbourhood page

See also

• Interior
• Neighbourhood
• Open set
• Metric space