Timeline for Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2014 at 4:55 | comment | added | Nik Weaver | Actually, on thinking it over I feel your counterexample is more interesting than either of my answers ... | |
| Sep 29, 2014 at 1:30 | comment | added | Nik Weaver | You're welcome, and thank you for pointing out my mistake. | |
| Sep 29, 2014 at 1:28 | history | edited | Nik Weaver | CC BY-SA 3.0 |
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| Sep 29, 2014 at 1:28 | vote | accept | Tristan Bice | ||
| Sep 29, 2014 at 1:27 | comment | added | Tristan Bice | Yep, looks good to me. Thanks again for answering both of my questions perfectly, and sorry they weren't more of a challenge! | |
| Sep 29, 2014 at 0:21 | history | edited | Nik Weaver | CC BY-SA 3.0 |
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| Sep 27, 2014 at 20:00 | comment | added | Tristan Bice | I've now changed the question to reflect this slightly different, and hopefully less trivial, problem. | |
| Sep 27, 2014 at 17:37 | comment | added | Tristan Bice | I'm afraid not Nik. Your last cover could still have a finite subcover of $Y$ (but not its closure). In fact, there are Hausdorff spaces where this happens (exercise!), I just don't know of any that are also locally compact. | |
| Sep 27, 2014 at 15:45 | comment | added | Nik Weaver | If $\overline{Y}$ is compact then any net in $Y$ has a cluster point in $\overline{Y} \subseteq X$. If it is not compact, find a family of open subsets of $X$ which covers $\overline{Y}$ but has no finite subcover, include $X\setminus\overline{Y}$ to get an open cover of $X$, and contradict (2). | |
| Sep 27, 2014 at 10:23 | comment | added | Tristan Bice | Certainly the answer to my original question was a lot easier than I thought it would be. Sorry if I'm missing something obvious again, but the answer to my last question on locally compact Hausdorff spaces is still not clear to me. Would you care to illuminate on this 'exercise'? Actually this was the question I was really interested in - I was expecting an answer to my original question would also answer this. | |
| Sep 26, 2014 at 22:52 | comment | added | Nik Weaver | Tristan, these exercises are left to the reader! | |
| Sep 26, 2014 at 22:43 | comment | added | Tristan Bice | Just one last question - is 1./2. equivalent to relative compactness, i.e. $\overline{Y}$ is compact, in a locally compact Hausdorff space? | |
| Sep 26, 2014 at 21:21 | comment | added | Tristan Bice | Incidentally, the same argument shows that you don't even need $T_1$ to show the equivalence of "every sequence in Y has a cluster point in X" and "every countable open cover of X has a finite subcover of Y" right? Do you still need $T_1$ to show these are equivalent to "every infinite subset of Y has a limit point in X"? | |
| Sep 26, 2014 at 21:18 | vote | accept | Tristan Bice | ||
| Sep 27, 2014 at 20:00 | |||||
| Sep 26, 2014 at 21:17 | comment | added | Tristan Bice | Ah, so my guess was wrong. Great answer - thanks Nik! | |
| Sep 26, 2014 at 20:11 | history | answered | Nik Weaver | CC BY-SA 3.0 |