Interior
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Definition
Given a set [ilmath]U\subseteq X[/ilmath] and an arbitrary metric space, [ilmath](X,d)[/ilmath] or topological space, [ilmath](X,\mathcal{J})[/ilmath] the interior of [ilmath]U[/ilmath], denoted [ilmath]\text{Int}(U)[/ilmath] is defined as[1][2]:
- [ilmath]\text{Int}(U):=\{x\in X\vert\ x\text{ is interior to }U\}[/ilmath] - (see interior point for the definition of what it means to be interior to)
Note that, unlike interior point which is basically a synonym for neighbourhood (taking the definition of neighbourhood as discussed on its page) the interior is a meaningful and distinct definition. In accordance with the topological definition of interior point (requiring that [ilmath]U[/ilmath] be a neighbourhood to some [ilmath]x\in X[/ilmath]) we see that:
- [ilmath]\text{Int}(U)[/ilmath] is the set of all points [ilmath]U[/ilmath] is a neighbourhood to.
Immediate properties
Let [ilmath]U\subseteq X[/ilmath] be an arbitrary subset of a topological space [ilmath](X,\mathcal{J})[/ilmath] (as all metric spaces are topological, they are included), then:
Claim 1: [ilmath]\text{Int}(U)[/ilmath] is open
Recall that for [ilmath]U[/ilmath] to be open we require that [ilmath]\text{Int}(U)=U[/ilmath], so for [ilmath]\text{Int}(U)[/ilmath] to be open we require that:
- [ilmath]\text{Int}(\text{Int}(U))=\text{Int}(U)[/ilmath]
TODO: Be bothered to fill in this proof
