# CW-complex

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Leads to Delta-complex - very important for algebraic topology

## Contents

## Definition

A *CW-complex* is a topological space [ilmath](X,\mathcal{ J })[/ilmath] and a collection of *disjoint* open cells (open [ilmath]n[/ilmath]-cells of various dimensions), [ilmath]\{e_\alpha\}_{\alpha\in I} [/ilmath] where [ilmath]X\eq\bigcup_{\alpha\in I}e_\alpha[/ilmath], such that^{[1]}:

- [ilmath](X,\mathcal{ J })[/ilmath] is Hausdorff
- For each open [ilmath]m[/ilmath]-cell, [ilmath]e\in\{e_\alpha\}_{\alpha\in I} [/ilmath], there exists a continuous map, [ilmath]f_\alpha:\overline{\mathbb{B}^n}\rightarrow X[/ilmath] such that:
- [ilmath]f_\alpha[/ilmath] maps [ilmath]\text{Int}(\overline{\mathbb{B}^n})[/ilmath] is mapped homeomorphically onto [ilmath]e[/ilmath]
- [ilmath]f_\alpha[/ilmath] maps [ilmath]\partial\overline{\mathbb{B}^n} [/ilmath] is mapped into a finite union of open [ilmath]n[/ilmath]-cells of dimension strictly less than that of [ilmath]e[/ilmath]

- A set [ilmath]A\in\mathcal{P}(X)[/ilmath] is closed
*if and only if*[ilmath]A\cap\overline{e_\alpha} [/ilmath] is closed for each [ilmath]\alpha\in I[/ilmath]