infinite dimension cell in a CW complex

Question

Lee "Introduction to Topological Manifolds" p. 132 speaks of an infinite dimensional CW complex (p. 132 2nd edition), with a cell of infinite dimensions. I'd like an example of such a CW complex and such a cell

Answer 1

I think you misread it. I didn't mention an "infinite-dimensional cell" -- but I did define an "infinite-dimensional CW complex." This is just a complex that's not finite-dimensional, which means that there is no integer $n$ such that every cell has dimension at most $n$; in other words, the dimensions of the cells are unbounded.

Here's an example of an infinite-dimensional complex. Note that for each $n\ge 1$, the sphere $\mathbb S^n$ can be obtained by attaching two $n$-cells to $\mathbb S^{n-1}$ (see Example 5.8(d) in my book). Now take the union of all of these cell complexes, and give it the topology coherent with the closed cells. This is a Cw complex in which there are two cells of each dimension.