CW-Complex

A CW-complex is a homotopy-theoretic generalization of the notion of a Simplicial Complex. A CW-complex is any Space $X$ which can be built by starting off with a discrete collection of points called $X^0$, then attaching 1-D Disks $D^1$ to $X^0$ along their boundaries $S^0$, writing $X^1$ for the object obtained by attaching the $D^1$s to $X^0$, then attaching 2-D Disks $D^2$ to $X^1$ along their boundaries $S^1$, writing $X^2$ for the new Space, and so on, giving spaces $X^n$ for every $n$. A CW-complex is any Space that has this sort of decomposition into Subspaces $X^n$ built up in such a hierarchical fashion (so the $X^n$s must exhaust all of $X$). In particular, $X^n$ may be built from $X^{n-1}$ by attaching infinitely many $n$-Disks, and the attaching Maps $S^{n-1}\to X^{n-1}$ may be any continuous Maps.

The main importance of CW-complexes is that, for the sake of Homotopy, Homology, and Cohomology groups, every Space is a CW-complex. This is called the CW-Approximation Theorem. Another is Whitehead's Theorem, which says that Maps between CW-complexes that induce Isomorphisms on all Homotopy Groups are actually Homotopy equivalences.

See also Cohomology, CW-Approximation Theorem, Homology Group, Homotopy Group, Simplicial Complex, Space, Subspace, Whitehead's Theorem