CW-complex

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Leads to Delta-complex - very important for algebraic topology

Definition

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

1. $(X,\mathcal{ J })$ is Hausdorff
2. For each open $m$-cell, $e\in\{e_\alpha\}_{\alpha\in I}$, there exists a continuous map, $f_\alpha:\overline{\mathbb{B}^n}\rightarrow X$ such that:
   • $f_\alpha$ maps $\text{Int}(\overline{\mathbb{B}^n})$ is mapped homeomorphically onto $e$
   • $f_\alpha$ maps $\partial\overline{\mathbb{B}^n}$ is mapped into a finite union of open $n$-cells of dimension strictly less than that of $e$
3. A set $A\in\mathcal{P}(X)$ is closed if and only if $A\cap\overline{e_\alpha}$ is closed for each $\alpha\in I$