Theorems around CW complexes

[felixpernegger]

Continuation of #1764 (which I closed so this is more organised):

The following paper https://epub.ub.uni-muenchen.de/4524/1/4524.pdf contains several useful theorems.

First some definitions: A CW complex is finite/countable if there only exist finitely/countably many cells. It is finite dimensional if #1768. It is locally finite/countable if each closed cells (i.e. image of characteristic map) meets only finitely/countably many other closed cells.

Note:

• Finite <=> Compact, Locally finite <=> Locally compact, Locally countable <=> Locally σ -compact, see this comment.
• I believe Countable <=> Separable <=> Hereditarily Separable (*) should hold (basically use that disks are separable).

Now the paper says (here all is a CW complex):

Theorem A:

• Countable + Finite dimensional + Locally compact => Embedabble in euclidean space (see Theorem Suggestion: A compact CW complex embeds in Euclidean space #1764 for a simplification)

Theorem B:

• Locally compact <=> Metrizable <=> First countable

In pibase we only need: Locally compact => Metrizable, First countable => Locally compact

(Theorem C is redundant if indeed (*) holds)

Theorem D (simplified under (*), previous theorems and general theorems):

• Embedabble in euclidean space => Finite dimensional

(There is also Lemma 3.2 but that follows from the main theorems)

[GeoffreySangston]

I couldn't find a reference discussing Separable => Countable but here's an argument I found using the power of pi-base: Choose an interior point from every cell in $X$ to form a subset $S \subset X$. Since the intersection of $S$ with each cell is finite (i.e., a singleton), Lemma 5.13 of Lee's Topological Manifolds (referenced on MSE) implies $S$ is discrete, and $S$ is closed by the characterization of closed sets in a CW complex (dual to the one given in the definition for open sets). The singletons in a closed discrete subspace form a discrete family (see P88). Since CW Complexes are collectionwise normal (π-Base, Search for CW complex + ~Collectionwise normal), it follows that there exists an open cover $\{U_s : s \in S \}$ of $S$ such that each pair is disjoint.

Suppose $X$ is separable. Since separable is an open hereditary property, there exists a countable dense subset $D$ of $\bigcup_{s \in S} U_s$. The map $D \to S$ sending each point of $D$ to the unique element of $S$ such that $U_s$ contains it is a surjection. Hence $S$ is countable.

[GeoffreySangston]

What if we just add Finite CW Complex / Countable CW Complex / Locally Finite CW Complex / Locally Countable CW Complex to pi-base, since we can fully characterize them, and their property pages can mostly reference the property page for CW Complex?

[felixpernegger]

What if we just add Finite CW Complex / Countable CW Complex / Locally Finite CW Complex / Locally Countable CW Complex to pi-base, since we can fully characterize them, and their property pages can mostly reference the property page for CW Complex?

Yes I think this is a good idea.

[prabau]

In the same way that a space can have different finite CW structures for example, is it possible that a space has one finite CW structure and one infinite one? (If I understand, it's not possible because such a space would be compact, and hence the number of cells must be finite.)

How about locally finite vs. not locally finite?

Locally countable vs. not?

By the way, does "countable CW complex" use countable in the sense of either finite or countably infinite number of cells?

[yhx-12243]

In the same way that a space can have different finite CW structures for example, is it possible that a space has one finite CW structure and one infinite one?

If I remind correctly, it is impossible. Finiteness and Countablyness is a CW complex invariant across all CW realizations, based on homeomorphs.

By the way, does "countable CW complex" use countable in the sense of either finite or countably infinite number of cells?

Mainly countable (≤ ℵ₀) is more useful.

[prabau]

@GeoffreySangston A maybe slightly easier argument for the end of #1769 (comment), once you have the pairwise disjoint family of open sets $U_s$ for $s\in S$.

Separable CW complexes are hereditarily Lindelof
(π-Base, Search for CW complex + Separable + ~Hereditarily lindelof)
hence $S$ is countable.

[GeoffreySangston]

@prabau I'm happy with either one. I think getting to the pairwise disjoint family is kind of black boxy so I'd prefer we actually found some reference doing this more directly in the context of CW complexes. (I can look more.)

[felixpernegger]

@prabau are you okay with adding finite, countable, locally countably, locally finite and finite dimensional cw complex as its own property? (we can also only do a subset thereof, note we can characterise all properties with cw complex except finite dimensional)
If yes Ill make PR

[prabau]

I'll answer a little later.

[felixpernegger]

In any case, as def I would write something like (i.e. for finite CW, for countable etc we may need some stackexchange post):

A compact CW complex.

Equivlently:

• every CW strcutre has finitely many cells
• some CW strcutre has finite many cells.

For proof of equivalence see Lemmas XY in {{zb:AB}} (Hatcher)

[prabau]

@GeoffreySangston @yhx-12243 @StevenClontz etc
I would like one of you to concur before we move forward on this.

Writing down my thinking here.

In general we don't add a property that is a combinations of two other pi-base properties A + B (example: "compact Hausdorff") unless there is a standard separate name for it. But we are not in this situation here.
Something like "finite CW complex" does not mean ("CW complex" + finite) as a combination of two pi-base properties. It rather means "homeomorphic to a CW complex with a finite number of cells". So there should not be an objection on that ground.

One slight confusion that may arise is when we strings together multiple links when describing spaces or in theorems: ... is {Pxxx} {Pyyy} {Pzzz}. That could expand to "... is locally countable finite CW complex ..." which would be hard to parse. But in practice I don't think that will be much of a problem. Hope it would be ok for the other names as well. I assume these are standard names?

As for why adding this property? If we don't add it, we could still make use of the theorem that finite CW complexes are the compact ones, which we could use when justifying traits. But having it as a separate property seems convenient. It is easy to verify for specific spaces, and by asserting it, we would immediately derive other properties. So that seems like it would be helpful.

So a priori I don't have an objection. I would still like someone else to confirm that it's worth adding before you go ahead.

---

Assuming this moves ahead, we should change the basic layout of the definitions. The main property really should be " ... a finite number of cells" and not "compact CW complex". The equivalence with (CW complex + compact) should be in some theorems. Although it could be mentioned also in the definition.

Also IMPORTANT, please write separate PRs for each definition. As you know, you have a tendency to underestimate the number of comments that something will generate. And you are fond of "big PRs", that end up being bogged down because people will not take the time to go through those. Splitting things into separate PRs allows them to move in parallel and the easy ones get quickly accepted and everything nicely moves forward. Thanks in advance.

[felixpernegger]

Assuming this moves ahead, we should change the basic layout of the definitions. The main property really should be " ... a finite number of cells" and not "compact CW complex". The equivalence with (CW complex + compact) should be in some theorems. Although it could be mentioned also in the definition.

I disagree with this and prefer my phrasing, because we defined a CW complex as having some CW structure (usually one doesnt do it this way), so its a bit awkward if this is supposed to hold for some CW structure of the space or for all (though its equivalent).
(But this isnt super important to me, other way would work as well)

[StevenClontz]

Agreed on one new property definition per PR.

As for the canonical name, what's the literature say? Pg 520 of https://pi.math.cornell.edu/~hatcher/AT/AT.pdf seems to indicate "finite CW complex" as the name, defined as having finitely-many cells, with compactness as a proposition.