Timeline for Paracompact Hausdorff but not compactly generated?
Current License: CC BY-SA 3.0
9 events
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| Sep 13, 2015 at 20:02 | comment | added | Taras Banakh | It suffices to find a paracompact Hausdorff space $X$ whose $k$-coreflexion $k(X)$ is not regular. For sequential coreflexions such a compact Hausdorff space was constructed by Franklin and Rajagopalan in 1970 (see repository.cmu.edu/cgi/…). | |
| May 29, 2011 at 14:22 | history | bounty ended | David Carchedi | ||
| May 23, 2011 at 1:31 | comment | added | David Carchedi | Thanks, I'm aware of this result, but I'm not sure how to use it. In fact, this is and if and only if, i.e. it characterizes compactly generated spaces. Moreover, for compactly generated Hausdorff spaces, they are the obvious quotient of the disjoint union of all their compact subsets, and if $X$ is not compactly generated, this quotient is $k\left(X\right).$ This means when $X$ is paracompact Hausdorff, $k\left(X\right)$ is a quotient of a space which is is both locally compact and paracompact Hausdorff. I'm not sure where to go from here. | |
| May 22, 2011 at 19:13 | comment | added | David White | Every compactly generated space is a quotient of a locally compact Hausdorff space. That may help, but not in the naive way. You definitely can't conclude $k(X)$ is paracompact just because it's a quotient of a paracompact space. | |
| May 22, 2011 at 14:01 | history | bounty started | David Carchedi | ||
| May 19, 2011 at 5:40 | history | edited | David Carchedi | CC BY-SA 3.0 |
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| May 18, 2011 at 23:47 | comment | added | David Carchedi | Yes, it is indeed Hausdorff; I know that $k$ is a functor $$k:Haus \to CGH,$$ the question is whether or not it is paracompact. | |
| May 18, 2011 at 22:33 | comment | added | wildildildlife | It seems that it's certainly Hausdorff, as the topology of $k(X)$ is finer (if $U$ is open in $X$ then $U\cap K$ is open in $K$ for all compacta $K$, by definition of the subspace topology.) So the two separating sets that worked for $X$ still work for $k(X)$. | |
| May 18, 2011 at 15:07 | history | asked | David Carchedi | CC BY-SA 3.0 |