CW complex

As mentioned above, every collection of discrete points is a CW complex (of dimension 0).

Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.: the cellular attaching maps have no role in these computations. This is a very special phenomenon and is not indicative of the general case.

There is a technique, developed by Whitehead, for replacing a CW complex with a homotopy-equivalent CW complex that has a simpler CW decomposition.

Another way of stating the above is that a connected CW complex can be replaced by a homotopy-equivalent CW complex whose 0-skeleton consists of a single point.