Difference between revisions of "Interior (topology)"
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* {{MM|\text{Int}(A):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} } U}} - the ''interior'' of {{M|A}} is the [[union]] of all [[open sets]] contained inside {{M|A}}. | * {{MM|\text{Int}(A):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} } U}} - the ''interior'' of {{M|A}} is the [[union]] of all [[open sets]] contained inside {{M|A}}. | ||
** We use {{M|\text{Int}(A,X)}} to emphasise that we are considering the interior of {{M|A}} with respect to the open sets of {{M|X}}. | ** We use {{M|\text{Int}(A,X)}} to emphasise that we are considering the interior of {{M|A}} with respect to the open sets of {{M|X}}. | ||
| + | ===Equivalent definitions=== | ||
| + | * {{MM|\text{Int}(A)\eq\bigcup_{x\in\{y\in X\ \vert\ y\text{ is an interior point of }A\} } \{x\} }}<ref group="Note">see ''[[interior point (topology)]]'' as needed for definition</ref> | ||
| + | ** See ''[[the interior of a set in a topological space is equal to the union of all interior points of that set]]'' for proof. | ||
==Immediate properties== | ==Immediate properties== | ||
| − | + | * {{M|\text{Int}(A)}} is [[open set|open]] | |
| − | * {{ | + | ** By definition of {{M|\mathcal{J} }} 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== | ==See also== | ||
* [[List of topological properties]] | * [[List of topological properties]] | ||
** {{link|Boundary|topology}} - denoted {{M|\partial A}} | ** {{link|Boundary|topology}} - denoted {{M|\partial A}} | ||
** {{link|Closure|topology}} - denoted {{M|\overline{A} }} | ** {{link|Closure|topology}} - denoted {{M|\overline{A} }} | ||
| − | == | + | ==Notes== |
| − | + | <references group="Note"/> | |
==References== | ==References== | ||
<references/> | <references/> | ||
| + | {{Requires references|grade=B|msg=Where did I get the interior point version from? Looking at the [[interior]] page (as of now, by ignoring the redirect [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 20:10, 16 February 2017 (UTC)) it seems: | ||
| + | * [[Books:Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene]] | ||
| + | * [[Books:Introduction to Topology - Bert Mendelson]] | ||
| + | have something to say}} | ||
{{Definition|Topology|Metric Space|Functional Analysis}} | {{Definition|Topology|Metric Space|Functional Analysis}} | ||
Latest revision as of 20:10, 16 February 2017
- 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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