Timeline for Making CW-complexes metrizable
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2022 at 15:22 | vote | accept | Jeremy Brazas | ||
| Dec 14, 2022 at 13:40 | answer | added | Tyrone | timeline score: 5 | |
| Dec 13, 2022 at 18:19 | comment | added | Jeremy Brazas | @Tyrone Thanks! Please submit this as an answer. I'd like to accept it. | |
| Dec 13, 2022 at 13:39 | comment | added | Tyrone | Sure. It's Theorem 2.1 in R. Cauty's Rétractions dans les espaces stratifiables. | |
| Dec 13, 2022 at 13:03 | comment | added | Jeremy Brazas | @Tyrone Do you know of a reference? I'm a little skeptical of the final claim if $K$ is say a Sierpinski carpet that fills up a 2-cell. | |
| Dec 13, 2022 at 11:31 | comment | added | Tyrone | This is tue. In fact, $X$ (no restriction on its number of cells) admits a continuous metric $d$ so that 1. $X_{Met}=(X,d)$ is an ANR 2. $X\rightarrow X_{Met}$ is a homotopy equivalence 3. $d$ the homotopies witnessing the previous equivalence can be chosen to fix pointwise any given compact subset $K\subseteq X$. | |
| Sep 23, 2021 at 14:38 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| Sep 23, 2021 at 14:21 | history | edited | Jeremy Brazas | CC BY-SA 4.0 |
Although I didn't forget about this question, I did neglect to make my question more detailed and explicit after the discussion in the comments. This has finally been done.
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| Sep 23, 2021 at 5:31 | answer | added | Ethan Dlugie | timeline score: 9 | |
| Mar 9, 2013 at 2:30 | answer | added | Sergey Melikhov | timeline score: 8 | |
| Jan 15, 2013 at 14:30 | comment | added | Jeremy Brazas | I did not mean to stress the "non-explicit homotopy" as much as "preferred CW-structure" but I will see if I can make the construction work for me. Regardless, I still hope someone might know the answer to my question. Thanks again! | |
| Jan 15, 2013 at 0:09 | comment | added | Misha | Jeremy: The explicit construction Igor refers to is on the last page of math.cornell.edu/~hatcher/AT/AT-exercises.pdf | |
| Jan 14, 2013 at 21:48 | comment | added | Igor Belegradek | Jeremy, actually the homotopy equivalence can be described explicitly (increasing dimension by one), i.e. the locally finite complex can be obtained as a mapping telescope of an exhausion of your given countable complex by finite subcomplexes (as explained somewhere in Hatcher's "Algebraic topology" text). Maybe this is be enough for your purposes. | |
| Jan 14, 2013 at 15:21 | comment | added | Jeremy Brazas | Thank you for these comments @Igor and @Misha. I am aware of Whitehead's result, however, I don't want to lose my preferred CW-structure with a non-explicit homotopy equivalence. What I am asking seems plausible when you consider the 1-dimensional case. For instance, a countably infinite wedge of circles is not first countable but you can weaken the topology at the basepoint so that it embeds in $\mathbb{R}^2$. The homotopy inverse of the continuous (but non-open) identity map comes from collapsing a small closed ball about the basepoint. | |
| Jan 14, 2013 at 4:53 | comment | added | Misha | Jeremy: See discussion and references at mathoverflow.net/questions/90570/… | |
| Jan 13, 2013 at 19:49 | comment | added | Igor Belegradek | A classical result of Whitehead says: every countable finite dimensional CW complex is homotopy equivalent to a locally finite CW complex of the same dimension, which is of course metrizable. | |
| Jan 13, 2013 at 17:24 | history | asked | Jeremy Brazas | CC BY-SA 3.0 |