Interior (topology)

See Task:Merge interior page into interior (topology) page - this hasn't been done yet Alec (talk) 19:27, 16 February 2017 (UTC)

Definition

Let [ilmath](X,\mathcal{J})[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath], the interior of [ilmath]A[/ilmath], with respect to [ilmath]X[/ilmath], is denoted and defined as follows:

• [math]\text{Int}(A):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} } U[/math] - the interior of [ilmath]A[/ilmath] is the union of all open sets contained inside [ilmath]A[/ilmath].
  • We use [ilmath]\text{Int}(A,X)[/ilmath] to emphasise that we are considering the interior of [ilmath]A[/ilmath] with respect to the open sets of [ilmath]X[/ilmath].

Equivalent definitions

• [math]\text{Int}(A)\eq\bigcup_{x\in\{y\in X\ \vert\ y\text{ is an interior point of }A\} } \{x\} [/math] (see interior point (topology) as needed for definition)
  • Claim 1: this is indeed an equality

Caveat:Unproved, suspected from current version of interior page - Alec (talk) 19:27, 16 February 2017 (UTC)

See also

• List of topological properties
  • Boundary - denoted [ilmath]\partial A[/ilmath]
  • Closure - denoted [ilmath]\overline{A} [/ilmath]

Proof of claims

Grade: B

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Would be good to do