Interior point (topology)
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Definition
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][1] 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]
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.
