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{{DISPLAYTITLE:{{M|n}}-cell}}{{Stub page|grade=A|msg=Looking alright so far...}}
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==Definitions==
===Closed {{M|n}}-cell===
A [[topological space]], {{Top.|X|J}}, is a "''closed {{N|cell}}''" if it is [[homeomorphic]] to the [[closed unit ball]]<ref group="Note">The closed unit ball, often denoted: {{M|\overline{\mathbb{B}^n} }} or {{M|\overline{\mathbb{B} }^n}}, is a [[closed ball]] of radius {{M|1}} based at the origin.
* {{M|\overline{\mathbb{B}^n}:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert\le 1\} }}, where the [[norm]] used is the standard [[Euclidean norm]]: {{M|\Vert x\Vert:\eq\sqrt{\sum^n_{i\eq 1} x_i^2} }}</ref> in [[R^n|{{M|\mathbb{R}^n}}]]{{rITTMJML}}.
* Often {{M|X}} will be a [[subset of]] another topological space and the [[topology]] will be the [[subspace topology]] {{M|X}} inherits from it.
===Open {{M|n}}-cell===
A [[topological space]], {{Top.|X|J}}, is a "''open {{N|cell}}''" if it is [[homeomorphic]] to the [[open unit ball]]<ref group="Note">The open unit ball, denoted: {{M|\mathbb{B}^n }}, is an [[open ball]] of radius {{M|1}} based at the origin.
* {{M|\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \vert\ \Vert x\Vert < 1\} }}, where the [[norm]] is the same as it is in the note for the closed unit ball</ref> in [[R^n|{{M|\mathbb{R}^n}}]]{{rITTMJML}}.
* Often {{M|X}} will be a [[subset of]] another topological space and the [[topology]] will be the [[subspace topology]] {{M|X}} inherits from it.
==Characterisations==
{{Caveat|There are (probably) other characterisations;}} like I should suspect any closed compact connected set with non-empty interior is a closed {{n|cell}}
===[[A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell|If {{M|X\in\mathcal{P}(\mathbb{R}^n)}} is compact and convex in {{M|\mathbb{R}^n}} with non-empty interior then it is a closed {{M|n}}-cell and its interior is an open {{N|cell}}]]===
{{:A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell/Statement}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Algebraic Topology}}