View source for Locally Euclidean topological space

{{Stub page|grade=A**|msg=This is a '''notes quality page''' it is actively being worked upon and should not be taken to be as reliable as a normal page
* I am currently doing the proofs for equivalent definitions [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 16:55, 19 February 2017 (UTC)}}
: {{Caveat|Need to do locally euclidean '''of dimension {{n}}'''!}}
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==Definition==
Let {{Top.|X|J}} be a [[topological space]], we say it is ''locally Euclidean'' if:
* {{M|\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\underbrace{\exists V\in\mathcal{O}(\mathbb{R}^n)\exists\varphi\in\mathcal{F}(U,V)[U\cong_\varphi V]}_{\exists\text{homeomorphism }\varphi:U\rightarrow\text{some open subset of }\mathbb{R}^n } }}
** In words: for all points, {{M|p\in X}}, there is an {{M|n}} such that there is an [[open neighbourhood]] of {{M|p}} which is [[homeomorphic]] to some [[open set]] in {{M|\mathbb{R}^n}} for the given {{M|n}].
==Equivalent definitions==
===Some open ball at the origin===
'''Claim: '''
* {{M|\Big(\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\exists V\in\mathcal{O}(\mathbb{R}^n)\exists\varphi\in\mathcal{F}(U,V)[U\cong_\varphi V]\Big)}} {{M|\iff}} {{M|\Big(\forall p\in X\exists n\in\mathbb{N}_0\exists U\in\mathcal{O}(p;X)\exists\epsilon\in\mathbb{R}_{>0}\exists\varphi\in\mathcal{F}(U,B_\epsilon(0;\mathbb{R}^n)[U\cong_\varphi B_\epsilon(0;\mathbb{R}^n)]\Big)}}
'''Proof: '''
===={{M|\implies}}====
* Let {{M|p\in X}} be given
** Choose {{M|n:\eq n'}} where {{M|n'}} is the {{M|n\in\mathbb{N}_0}} posited to exist by the LHS of the {{M|\iff}}
*** We now obtain:
**** {{M|U'\in\mathcal{O}(p;X)}} posited to exist by the LHS,
***** {{M|V'\in\mathcal{O}(\mathbb{R}^n)}} posited to exist by the LHS, such that:
****** {{M|\varphi':U'\rightarrow V'}} posited to exist by the LHS is a homeomorphism.
*** Recall that "''[[an open set in a metric space contains an open ball about all of its points]]''", this means:
**** {{M|\exists\delta\in\mathbb{R}_{>0}[B_\delta(\varphi'(p);\mathbb{R}^n)\subseteq V']}}
*** As [[open balls]] are [[open sets]] and "''[[the pre-image of an open set under a homeomorphism is open]]''" we see:
**** {{M|\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)}} is open in {{M|X}}
*** We must now show that {{M|p\in\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)}} (so we can say {{M|\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)\in\mathcal{O}(p,X)}} shortly)
**** Note that {{M|f(p)\in B_\delta(\varphi'(p);\mathbb{R}^n)}} as {{M|d(\varphi'(p),\varphi'(p)):\eq 0}} regardless of the [[metric]] used
***** As {{M|0<\delta}} we see {{M|\varphi'(p)\in B_\delta(\varphi'(p);\mathbb{R}^n)}}
****** Thus {{M|p\in\varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)}} - by the definition of {{link|pre-image|function}}
*** Choose: {{M|U:\eq \varphi'^{-1}\big(B_\delta(\varphi'(p);\mathbb{R}^n)\big)}}, by the discussion above we see {{M|U\in\mathcal{O}(p,X)}}
**** Choose: {{M|\epsilon\eq\delta}}, we found {{M|\delta}} above, recall {{M|\delta\in\mathbb{R}_{>0} }}, so obviously {{M|\epsilon\in\mathbb{R}_{>0} }}
{{Requires work|grade=A|msg=The bulk of the proof is done here, the "not trivial part" is that {{M|\varphi'}} restricts to a homeomorphism onto {{M|B_\delta(\varphi'(p);\mathbb{R}^n)}}, then compose that with a translation, we use [[translations are homeomorphisms]] and we're basically done. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 16:55, 19 February 2017 (UTC)
* "''[[Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself]]''" will be needed and is ready}}

=References=
<references/>
{{Definition|Topology|Manifolds|Topological Manifolds|Smooth Manifolds}}