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. In fact, we can construct a homotopy-equivalent CW-complex with the same number of cells in each dimension, as in the original cell complex.
• A product of cell complexes is a cell complex. More generally, given any fiber bundle with cell complex structures on the base and the fiber, we can combine those to give a cell complex structure to the total space. This is not possible for CW-complex structures since the order of attaching cells gets changed drastically.