CW complex

Let $X$ be a topological space. A CW complex structure on $X$ consists of a filtration by subspaces

and, for each $n\ge 0$, a set $I_n$ and continuous attaching maps

such that $X^n$ is obtained from $X^{n-1}$ by attaching $n$-disks:

with the quotient topology, where the disjoint union has the topology in which a set is open exactly when its intersection with each summand is open. Here $D^n$ is the closed $n$-disk, $S^{n-1}=\partial D^n$, and for $n=0$ one uses $S^{-1}=\varnothing$ and $D^0=\{\ast\}$, so $X^0$ is a discrete set of points attached to $X^{-1}=\varnothing$.

The image of the interior of $D^n_\alpha$ is an open $n$-cell, denoted $e^n_\alpha$, and its characteristic map $D^n_\alpha\to X^n$ restricts to a homeomorphism from $\operatorname{int}(D^n_\alpha)$ onto $e^n_\alpha$. The cells are pairwise disjoint and $X^n$ is the union of the cells of dimensions at most $n$.

The structure satisfies two conditions:

1. Closure-finite: the closure of every cell meets only finitely many cells.
2. Weak topology: a subset $U\subseteq X$ is open if and only if $U\cap X^n$ is open in $X^n$ for every $n\ge0$.

The space $X$, together with this cell decomposition, is called a CW complex. Its $n$-skeleton is $X^n$, and the dimension of a cell $e^n_\alpha$ is $n$. A finite CW complex has finitely many cells; its dimension is the supremum of its cell dimensions (the empty complex has dimension $-1$).