Complete Radial interval topology

[yhx-12243]

This is a great counterexample about CW complexes.

S135|¬P118 is a little bit long, maybe we can transfer it to mse.

P109, P177 seems unknown, but both two are property that all CW complexes share.

[prabau]

P240 (CW complex): it's pretty obvious that the 0-skeleton is discrete. And it's also easy to check from looking at it that all the 1-cells together are "attached to the 0-skeleton in one step" (i.e., according to the definition we have), i.e., as a quotient of a direct sum, etc. No need to write the details.
But I am curious, is there some basic result about CW complexes that allows to see this more directly maybe?

[prabau]

P118 (Has a σ-locally finite network = False): The proof is a little too long for pi-base.

If you want I can write a mathse question tomorrow and you (and others) can answer there?
Or do you prefer to do something else?

For now, if you want, you can remove P118 from this PR and we can merge the rest. And then do P118 in a separate PR. Or not.

[yhx-12243]

Yes we can put it into mathse.

[prabau]

OK, I'll ask a question tomorrow.

[felixpernegger]

P109, P177 seems unknown, but both two are property that all CW complexes share.

Do you have some reference for this? In any case, you could also edit my post in #1769 so we dont forget about this

[yhx-12243]

P109, P177 seems unknown, but both two are property that all CW complexes share.

Do you have some reference for this? In any case, you could also edit my post in #1769 so we dont forget about this

CW complex are stratifiable. (H of zb:0175.19802)
Stratifiable spaces are monotonically normal. (Theorem 10.5.16 of zb:0546.00022)
Stratifiable spaces are σ-space. (Theorem 10.5.9 of zb:0546.00022)

However pi-base does not have “Stratifiable” property now. So it can be postponed away.

[prabau]

https://math.stackexchange.com/questions/5136509

[prabau]

@yhx-12243 Can you expand the proof of Theorem 1 in the mathse post based on your comment?

[prabau]

@yhx-12243 thanks for the clarification. We should also add P117 (Has a σ-locally finite network).

For Theorem 2, sorry I did not have the time today to look at it in detail. Will do it tomorrow.