Timeline for Making CW-complexes metrizable
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 24, 2021 at 23:13 | comment | added | Allen Hatcher | @EthanDlugie: I was referring to your statement that a set is open if it is open in each cell. If this were true with the usual definition of cells then every subcomplex would be an open set. For a set to be open it needs to be open in the closure of each cell. | |
| Sep 23, 2021 at 21:31 | comment | added | Ethan Dlugie | @AllenHatcher are you referring to how I kept saying "the interior of a cell"? I think I just wrote that to avoid any of that sort of confusion. I think the argument works exactly the same regardless of how you interpret "cell", no? | |
| Sep 23, 2021 at 16:01 | comment | added | Allen Hatcher | The argument in this answer seems to be using a nonstandard definition of what a cell in a CW complex is. According to the standard definition, as in Whitehead's original paper and most recent sources, a CW complex is the disjoint union of its cells, so cells are "open cells". Unfortunately some authors in the intervening years took "cell" to mean the closure of what Whitehead called a cell. With this definition a "cell" need not be contractible. For example, every smooth closed connected manifold is a "cell" by this definition. | |
| Sep 23, 2021 at 13:51 | comment | added | Jeremy Brazas | Nice. This is exactly the example I note in the comments above. Even though $X$ is not regular you can let $Y=\bigcup_{n\in\mathbb{N}}\{(x,y)\in\mathbb{R}^2\mid (x-1-\frac{1}{n})^2+y^2=(1+\frac{1}{n})^2\}$. Now $Y$ is something of a metrizable version of $X$. There is a continuous bijection $X\to Y$. The homotopy inverse $Y\to X$ quotients by a small closed ball about the origin. | |
| Sep 23, 2021 at 6:24 | comment | added | David Roberts♦ | That's a nice construction! | |
| Sep 23, 2021 at 5:31 | history | answered | Ethan Dlugie | CC BY-SA 4.0 |