Timeline for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
Current License: CC BY-SA 4.0
8 events
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| May 26, 2023 at 4:12 | comment | added | Paul Fabel | Thanks for pointing out the potential different meanings of CG and the dependence on context. In the example at hand, the starting space is a sequential space (closed sets are precisely those closed under convergent sequences) and convergent sequences have unique limits. | |
| May 26, 2023 at 0:42 | comment | added | PatrickR | You are right that my example does not work. | |
| May 24, 2023 at 22:05 | comment | added | PatrickR | There are different notions of "compactly generated". In particular, CG-1 = Definition 1 and CG-2 = Definition 2 from en.wikipedia.org/wiki/Compactly_generated_space. (see also topology.pi-base.org/properties?filter=CG). Now CGWH means CG-2 + weak Hausdorff. $Y$ is compact, hence CG-1, but not every compact set is CG-2. That would need to be shown, either directly from the definition, or as a quotient of a locally compact Hausdorff space for example. (see also math.stackexchange.com/questions/4646084) | |
| May 24, 2023 at 11:18 | comment | added | Paul Fabel | $Y$ is compact and hence CG. $X$ is $T_2$ and hence compact subsets of $X$ are closed in $X$. Thus $Y$ is the Alexandroff compactification of $X$. In particular $Y$ is compact and infinite and contains $y$. | |
| May 24, 2023 at 5:10 | comment | added | PatrickR | What is the reason $Y$ is CG? It seems any compact subset of $Y$ containing $y$ is finite, and compact subsets not containing $y$ are either finite or sequences converging to $x$ (together with the limit point $x$). Then for fixed $n$ the subset $A_n=\{y\}\cup\{1/n+1/(m+n)\}$ is k-open (meets every compact in $Y$ is a relatively open set), but not open because its complement is not compact in $X$. What am I missing? | |
| Mar 24, 2023 at 8:17 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added a Wikipedia link
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| Mar 23, 2023 at 17:04 | history | edited | Tyrone | CC BY-SA 4.0 |
Mathjax together with a ruthless distaste for unnecessary spacebar usage.
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| Dec 11, 2015 at 4:45 | history | answered | Paul Fabel | CC BY-SA 3.0 |