View source for Locally Euclidean topological space of dimension n

{{DISPLAYTITLE:Locally Euclidean topological space of dimension {{N}}}}
{{Stub page|grade=A**|msg=See [[Locally euclidean]], it's the same thing but with {{M|n}} fixed before the {{M|\forall p\in x}} part. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:07, 19 February 2017 (UTC)
* Fleshed out a little bit but more work is needed [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 12:47, 20 February 2017 (UTC)}}
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==Definition==
: {{Caveat|I think this {{M|n}} might have to be unique}} as later (see [[topological manifold]]) we'll talk about the "well-defined-ness" of {{M|n}}!<ref group="Note">As usual, [[well-defined-ness]] means we have an [[equivalence relation]] in play, and we're [[quotient by an equivalence relation|quotienting something]]. I'm not quite sure what yet though!
* I would have thought that a "locally euclidean of dimension n" space is really just something such that there exists an {{N}} for all points...
{{XXX|Solve this}}</ref>
Let {{Top.|X|J}} be a [[topological space]] and let {{M|n\in\mathbb{N}_0}} be given. We say that {{M|X}} is ''locally Euclidean of dimension {{M|n}}'' if:
* {{M|\forall p\in X\exists U\in\mathcal{O}(p;X)\exists\epsilon\in\mathbb{R}_{>0}\exists \varphi\in\mathcal{F}\big(U,B_\epsilon(0;\mathbb{R}^n)\big)\big[U\cong_\varphi B_\epsilon(0;\mathbb{R}^n)\big]}}
* {{Caveat|Or perhaps...}}
** {{M|\exists n\in\mathbb{N}_0\forall p\in X\exists U\in\mathcal{O}(p;X)\exists\epsilon\in\mathbb{R}_{>0}\exists \varphi\in\mathcal{F}\big(U,B_\epsilon(0;\mathbb{R}^n)\big)\big[U\cong_\varphi B_\epsilon(0;\mathbb{R}^n)\big]}}
*Where the dimension, {{N}}, is the {{M|n}} that must exist in the first quantifying clause.
{{XXX|Verdict needed after investigation}}
==Equivalent definitions==
We posit that there must be an open ball of radius {{M|\epsilon}} about {{M|0\in\mathbb{R}^n}}, it actually works if:
# We require there to be any open set containing {{M|p}} to be [[homeomorphic]] to any open set of {{M|\mathbb{R}^n}}
# We require there be an open set containing {{M|p}} homeomorphic to all of {{M|\mathbb{R}^n}}
# We require there be an open set containing {{M|p}} homeomorphic to the [[open unit ball]], {{M|\mathbb{B}^n}}
See the [[Locally euclidean]] page for more information.
==Notes==
<references group="Note"/>
==References==
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{{Definition|Topology|Manifolds|Smooth Manifolds|Topological Manifolds}}