Difference between revisions of "Interior point (topology)"
(Created page with "==Definition== Given a metric space {{M|(X,d)}} and an arbitrary subset {{M|U\subseteq X}}, a point {{M|x\in X}} is ''interior'' to {{M|U}}{{rITTGG}} if: * {{M|\exists\del...") |
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==Definition== | ==Definition== | ||
| + | ===Metric space=== | ||
Given a [[metric space]] {{M|(X,d)}} and an arbitrary subset {{M|U\subseteq X}}, a point {{M|x\in X}} is ''interior'' to {{M|U}}{{rITTGG}} if: | Given a [[metric space]] {{M|(X,d)}} and an arbitrary subset {{M|U\subseteq X}}, a point {{M|x\in X}} is ''interior'' to {{M|U}}{{rITTGG}} if: | ||
* {{M|\exists\delta>0[B_\delta(x)\subseteq U]}} | * {{M|\exists\delta>0[B_\delta(x)\subseteq U]}} | ||
| − | ==Relation to Neighbourhood== | + | ====Relation to Neighbourhood==== |
This definition is VERY similar to that of a [[neighbourhood]]. In fact that I believe "{{M|U}} is a neighbourhood of {{M|x}}" is simply a generalisation of interior point to [[topological space|topological spaces]]. | This definition is VERY similar to that of a [[neighbourhood]]. In fact that I believe "{{M|U}} is a neighbourhood of {{M|x}}" is simply a generalisation of interior point to [[topological space|topological spaces]]. | ||
Note that: | Note that: | ||
| Line 19: | Line 20: | ||
This completes the proof. | This completes the proof. | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
| + | This implication can only go one way as in an arbitrary topological space (which may not have a [[metric space|metric]] that [[topology induced by a metric|induces it]]) there is no notion of [[open ball|open balls]] (as there's no metric!) thus there can be no notion of interior point.<ref group="Note">At least not with this definition of ''interior point'', this probably motivates the topological definition of interior point</ref> | ||
| + | ===Topological space=== | ||
| + | In a [[topological space]] {{M|(X,\mathcal{J})}} and given an arbitrary subset of {{M|X}}, {{M|U\subseteq X}} we can say that a point, {{M|x\in X}}, is an ''interior point'' of {{M|U}}{{rITTBM}} if: | ||
| + | * {{M|U}} is a [[neighbourhood]] of {{M|x}} | ||
| + | ** Recall that if {{M|U}} is a neighbourhood of {{M|x}} we require {{M|\exists\mathcal{O}\in\mathcal{J}[x\in\mathcal{O}\subseteq U]}} | ||
| + | ====Relation to Neighbourhood==== | ||
| + | We can see that in a [[topological space]] that [[neighbourhood]] to and interior point of are equivalent. This site (like<ref name="ITTBM"/>) defines neighbourhood to as containing an open set with the point in it. However some authors (notably Munkres) ''do not'' use this definition and use neighbourhood as a synonym for [[open set]]. In this case ''interior point of'' and ''neighbourhood to'' are not equivalent. | ||
| + | |||
| + | I don't like the term ''interior'' point as it suggests some notion of being inside, but the point being in the set is not enough for it to be interior! So I am happy with this and stand by the comments on the [[neighbourhood]] page | ||
==See also== | ==See also== | ||
* [[Interior]] | * [[Interior]] | ||
| Line 24: | Line 34: | ||
* [[Open set]] | * [[Open set]] | ||
* [[Metric space]] | * [[Metric space]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Metric Space|Functional Analysis|Topology}} | {{Definition|Metric Space|Functional Analysis|Topology}} | ||
Revision as of 03:02, 29 November 2015
Contents
Definition
Metric space
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.
This implication can only go one way as in an arbitrary topological space (which may not have a metric that induces it) there is no notion of open balls (as there's no metric!) thus there can be no notion of interior point.[Note 1]
Topological space
In a topological space [ilmath](X,\mathcal{J})[/ilmath] and given an arbitrary subset of [ilmath]X[/ilmath], [ilmath]U\subseteq X[/ilmath] we can say that a point, [ilmath]x\in X[/ilmath], is an interior point of [ilmath]U[/ilmath][2] if:
- [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath]
- Recall that if [ilmath]U[/ilmath] is a neighbourhood of [ilmath]x[/ilmath] we require [ilmath]\exists\mathcal{O}\in\mathcal{J}[x\in\mathcal{O}\subseteq U][/ilmath]
Relation to Neighbourhood
We can see that in a topological space that neighbourhood to and interior point of are equivalent. This site (like[2]) defines neighbourhood to as containing an open set with the point in it. However some authors (notably Munkres) do not use this definition and use neighbourhood as a synonym for open set. In this case interior point of and neighbourhood to are not equivalent.
I don't like the term interior point as it suggests some notion of being inside, but the point being in the set is not enough for it to be interior! So I am happy with this and stand by the comments on the neighbourhood page
See also
Notes
- ↑ At least not with this definition of interior point, this probably motivates the topological definition of interior point
