Timeline for "All retracts are closed" and "all compacts are closed"
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2022 at 18:55 | comment | added | Steven Clontz | I've contributed the RC property to pi-Base in case it helps future investigators. | |
| Nov 28, 2022 at 18:44 | comment | added | Christian Remling | @StevenClontz: In other words, I tried to get a counterexample as a one-point compactification of a space with no nontrivial retracts. If your bet is correct, it would refute this strategy in general (since the first space is metric). | |
| Nov 28, 2022 at 18:39 | comment | added | Christian Remling | @StevenClontz: I fooled around with (perhaps) somewhat similar ideas in several versions of my deleted answer below, which you probably can't see: Start out with a space with few or no retracts ("Cook continuum"), add a point $\infty$, and give the extra point only big neighborhoods (an extreme case would be to make the whole space the only neighborhood). Then there should be few retracts, but many compact sets (because just having $\infty$ in the set really helps), so maybe we can get RC, but not KC. But one has to carefully fine tune things at $\infty$, and I gave up after too many mistakes. | |
| Nov 28, 2022 at 17:41 | comment | added | Steven Clontz | My bet was yes, but with equally sparse evidence. (-: | |
| Nov 28, 2022 at 17:00 | comment | added | Paul Fabel | Nice question! If forced to bet, I would bet no, despite having no counterexample in mind, | |
| Nov 28, 2022 at 16:09 | comment | added | Steven Clontz | I think the next point of attack is this: are RC and KC equivalent for one-point compactifications of T2 spaces? | |
| Nov 28, 2022 at 15:21 | comment | added | Paul Fabel | As noted, thanks to Steven's inquiry in the comments of the original question, I noticed the mistake in my original answer and corrected it yesterday, and, serendipitously, also constructed a non-example similar to the one at hand. It's nice to revisit the question, thanks to Steven and everyone else for exposing the mistake, and helping to set things right. | |
| Nov 28, 2022 at 14:58 | history | edited | Steven Clontz | CC BY-SA 4.0 |
added 113 characters in body
|
| Nov 27, 2022 at 18:55 | comment | added | Jianing Song | Thanks for your work! What a pity for the "example", of course :) | |
| Nov 25, 2022 at 2:17 | history | edited | Steven Clontz | CC BY-SA 4.0 |
added 972 characters in body
|
| Nov 25, 2022 at 1:01 | comment | added | Steven Clontz | Something is fishy here: upon closer inspection, Fabel's answer proves RC by asserting it is compact and KC., which seems to contradict the result I cite from Between T1 and T2. | |
| Nov 25, 2022 at 0:18 | history | answered | Steven Clontz | CC BY-SA 4.0 |