View source for Sphere (topological manifold)

{{Provisional page|grade=A*|msg=Would be good example}}
__TOC__
==Definition==
Let {{M|\mathbb{S}^n\subseteq\mathbb{R}^{n+1} }} denote the usual [[sphere|{{n|sphere}}]], of course defined by:
* {{M|\mathbb{S}^n:\eq\{x\in\mathbb{R}^{n+1}\ \big\vert\ \Vert x\Vert\eq 1 \} }} {{M|:\eq\left\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}\ \left\vert\ \sqrt{\sum^{n+1}_{i\eq 1}(x_i^2)\eq 1}\right.\right\} }} - where {{M|\Vert\cdot\Vert}} is the [[Euclidean norm]] on [[Euclidean n-space|Euclidean {{M|(n+1)}}-space]]


Note that in this article {{M|\mathbb{B}^n\subseteq\mathbb{R}^n}} with {{M|\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \big\vert\ \Vert x\Vert < 1\} }} is the [[open unit ball]] (with centre at the origin, as is implied by the name)<ref group="Note">This (may) be sometimes used to denote the [[closed unit ball|''closed'' unit ball]] instead.</ref>


We claim that {{M|\mathbb{S}^n}} is a [[topological manifold]] with the following standard {{M|2n+2}} [[charts]]{{rITSMJML}}:
* For {{M|i\in\{1,\ldots,n+1\}\subseteq\mathbb{N} }}:
** Define {{M|U^+_i:\eq\{(x_1,\ldots,x_{n+1})\in\mathbb{S}^n\ \big\vert\ x_i>0\} }}
*** {{M|\varphi^+_i:U^+_i\rightarrow\mathbb{B}^n}} given by {{M|\varphi^+_i:(x_1,\ldots,x_{n+1})\mapsto(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})}}
**** So that {{M|(U^+_i,\varphi^+_i)}} is a [[chart]]
** Define {{M|U^-_i:\eq\{(x_1,\ldots,x_{n+1})\in\mathbb{S}^n\ \big\vert\ x_i<0\} }}
*** {{M|\varphi^-_i:U^-_i\rightarrow\mathbb{B}^n}} given by {{M|\varphi^-_i:(x_1,\ldots,x_{n+1})\mapsto(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})}}
**** So that {{M|(U^-_i,\varphi^-_i)}} is a [[chart]]
==Proof of claims==
* Note that Hausdorff is inherited from {{M|\mathbb{R}^{n+1} }}
** As is {{M|\mathbb{S}^{n} }} being a [[second countable topological space]]
* The crux lies in this locally euclidean part.
** Define: {{M|f:\mathbb{B}^n\rightarrow\mathbb{R} }} given by {{M|f:x\mapsto\sqrt{1-\Vert x\Vert ^2} }}
*** We note that:
***# {{M|U^+_i}} is the {{link|graph|topological manifold}} of {{M|x_i\eq f(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})}}, and
***# {{M|U^-_i}} is the {{link|graph|topological manifold}} of {{M|x_i\eq-f(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})}}
{{Requires proof|grade=A|msg=Would be good to do [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 05:05, 12 March 2017 (UTC)}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Manifolds|Topological Manifolds|Topology}}