Interior (topology)
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Revision as of 20:10, 16 February 2017 by Alec (Talk | contribs) (Added link to proof of equivalent definition - which I have now shown - added more references, removed some un-needed sections)
- See Task:Merge interior page into interior (topology) page - this hasn't been done yet Alec (talk) 19:27, 16 February 2017 (UTC)
Contents
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[1]:
- [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][Note 1]
Immediate properties
- [ilmath]\text{Int}(A)[/ilmath] is open
- By definition of [ilmath]\mathcal{J} [/ilmath] being a topology it is closed under arbitrary union. The interior is defined to be a union of certain open sets, thus their union is an open set.
See also
Notes
- ↑ see interior point (topology) as needed for definition
References
Grade: B
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