Cell complex

Definition

A cell complex is a topological space along with a cell structure, which is given as follows. We start off with the empty set, and at each stage, we attach a $n$-cell (for arbitrary $n$) using an attaching map from $S^{n-1}$ to the space constructed so far. The dimensions of the cells could be in any order, and the sequence of attachments could be transfinite.

A special case is a CW-complex where the cells are attached in increasing order of dimension.

Facts

• Any space which can be given a cell complex structure, is a homotopy-CW-space; in other words, it has the homotopy type of a CW-complex. Moreover, the number of cells of each dimension in the associated CW-complex is the same.