Theorem Suggestion: A compact CW complex embeds in Euclidean space

A finite CW complex is a complex with finitely many cells in total, this is equivalent to the CW complex being compact. Similar countable.

It is locally finite if every cell (image of characteristic map) meets only finite many other cells, this is equivalent to locally compact, see https://ncatlab.org/nlab/show/locally+finite+cell+complex

Now Theorem 4.1 in https://epub.ub.uni-muenchen.de/4524/1/4524.pdf is a stronger version of the assertion.

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I'm not sure if we can encode "finite dimensional CW complex" as of now (could posibly be considered its own property if not in the future)

Our CW complex sources also have:

Proposition A.1 of Hatcher implies Compact CW complex <=> Finite CW complex

Proposition 3.6 of Lundell-Weingram implies Locally finite <=> Locally compact. Also has Locally countable <=> Locally $\sigma$-compact.

I also don't think we can get finite dimensional other than by assuming compactness.

@GeoffreySangston

@GeoffreySangston I think countable CW complex is equivalent to separable CW complex?

I close this issue in favour of making a more comprehensive one

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