CW complex= cell complex?; compact metrizable spaces

Question

I have a question about finite cell complexes and compact metrizable spaces.

In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\limits_{i\in \mathbb{N}} X_i$ of finite cell complexes.

My questions:

1)What is meant by cell complexes exactly? Are a cell complex and a CW complex something different? Wikipedia tells me that this cell complex = CW complex I would say.

2)What does "finite" mean exactly? I know that $X$ is a union of finite number of cells, but do we also have a "finiteness"-condition on all cells $X_i$?

3) Are the $X_i$ necessarily compact?

I appreciate your help.

Answer 1

In topological spaces, a cell complex is usually the same thing as a CW-complex. The name "cell complex" comes from the fact that there exists generalizations to other categories, but if you're interested in topological spaces then for all intents and purposes "cell complex" = "CW-complex".

A finite cell complex is a cell complex that has a finite number of cells. Each cell itself is compact. If $X$ is a finite cell complex, it is a quotient space of a disjoint union of a finite number of cells, and such a disjoint union is compact, hence $X$ is compact. It can in fact be shown that a cell complex is compact iff it is finite, cf. e.g. the book of Hatcher. So in particular in your theorem, yes, the $X_i$ are compact.

Note however some authors take a different approach, for example in this question there is a different definition: in Lee's definition, as far as I can tell, roughly speaking in a cell complex you don't have to glue cells in the order of their dimension, whereas in a CW-complex you do. The name "CW-complex" is never ambiguous though, and a finite cell complex (with Lee's definition) is always a CW-complex.