View source for Interior point (topology)

==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:
* {{M|\exists\delta>0[B_\delta(x)\subseteq U]}}

====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]].
Note that:
{{Begin Inline Theorem}}
Claim: {{M|x}} is interior to {{M|U}} {{M|\implies}} {{M|U}} is a neighbourhood of {{M|x}}
{{Begin Inline Proof}}
'''Proof'''
: As {{M|x}} is interior to {{M|U}} we know immediately that:
:* {{M|\exists\delta>0[B_\delta(x)\subseteq U]}}
:* We also see that {{M|x\in B_\delta(x)}} (as {{M|1=d(x,x)=0}}, so for any {{M|\delta>0}} we still have {{M|x\in B_\delta(x)}})
:* But [[open ball|open balls]] are [[open set|open sets]]
: So we have found an open set entirely contained in {{M|U}} which also contains {{M|x}}, that is:
:: {{M|\exists\delta>0[x\in B_\delta(x)\subseteq U]}}
: Thus {{M|U}} is a neighbourhood of {{M|x}}
This completes the proof.
{{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==
* [[Interior]]
* [[Neighbourhood]]
* [[Open set]]
* [[Metric space]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Metric Space|Functional Analysis|Topology}}