View source for CW-complex

{{Stub page|grade=A|msg=Leads to [[Delta-complex]] - very important for algebraic topology}}
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==Definition==
A ''CW-complex'' is a [[topological space]] {{Top.|X|J}} and a collection of ''[[disjoint]]'' [[open cells]] ([[open n-cell|open {{n|cells}}]] of various dimensions), {{M|\{e_\alpha\}_{\alpha\in I} }} where {{M|X\eq\bigcup_{\alpha\in I}e_\alpha}}, such that{{rEOATJRM}}:
# {{Top.|X|J}} is [[Hausdorff]]
# For each [[open n-cell|open {{M|m}}-cell]], {{M|e\in\{e_\alpha\}_{\alpha\in I} }}, there exists a [[continuous map]], {{M|f_\alpha:\overline{\mathbb{B}^n}\rightarrow X}} such that:
#* {{M|f_\alpha}} maps [[Interior (topology)|{{M|\text{Int}(\overline{\mathbb{B}^n})}}]] is mapped [[homeomorphism|homeomorphically]] onto {{M|e}}
#* {{M|f_\alpha}} maps [[Boundary (topology)|{{M|\partial\overline{\mathbb{B}^n} }}]] is mapped into a finite [[union]] of [[open n-cell|open {{N|cells}}]] of dimension strictly less than that of {{M|e}}
# A set {{M|A\in\mathcal{P}(X)}} is [[closed set|closed]] {{iff}} {{M|A\cap\overline{e_\alpha} }} is closed for each {{M|\alpha\in I}}

==References==
<references/>
{{Definition|Algebraic Topology|Homology|Homotopy Theory|Topology}}