A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties:
For each n-dimensional open cell C in the partition of X, there exists a continuous map f from the n-dimensional closed ball to X such that
The restriction of f to the interior of the closed ball is a homeomorphism onto the cell C, and
- the image of the boundary of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than n.
A subset of X is closed if and only if it meets the closure of each cell in a closed set.
I am confused by this definition, because I don't see how for example a torus is a CW complex (I've been told it is), given that the cells have to be open.
Here's my problem: I'd consider a torus to consist of one open 2-cell (the image of an open square/disk onto the main shape of the torus), and two 1-cells, namely the left and right side of the square glued together and the top and bottom sides glued together.
The problem with this is, because the sides have to be open cells, we havent included the corners of the square. If we would include that, it wouldn't be an open cell anymore. So in this construction, either one or two points of the torus are not included in the partitioning into cells.
What mistake am I making?