Paracompact Hausdorff but not compactly generated? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:58:06Z http://mathoverflow.net/feeds/question/65340 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65340/paracompact-hausdorff-but-not-compactly-generated Paracompact Hausdorff but not compactly generated? David Carchedi 2011-05-18T15:07:47Z 2011-07-17T22:10:12Z <p>I'm sorry to be asking a (possibly) elementary question, but I've run into a problem in point-set topology; I've just read that there exists paracompact Hausdoff spaces which are not compactly generated. I ask the following:</p> <p><strong>Question:</strong> If $X$ is paracompact Hausdorff, is its compactly generated replacement, $k\left(X\right),$ paracompact Hausdorff?</p> <p>Recall: The inclusion $i:CGH \to Haus$ of compactly generated Hausdorff spaces into Hausdorff spaces has a right adjoint $k,$ which replaces the topology of $X$ with the following topology:</p> <p>$U \subset X$ is open in $k\left(X\right)$ if and only if for all compact subsets $K \subset X,$ $U \cap K$ is open in $K$.</p> <p>Another way of describing this topology is that it is the final topology with respect to all maps into $X$ with compact Hausdorff domain. (For the experts, $CGH$ is the mono-coreflective Hull of the category of compact Hausdorff spaces in the category of Hausdorff spaces)</p> http://mathoverflow.net/questions/65340/paracompact-hausdorff-but-not-compactly-generated/65380#65380 Answer by wildildildlife for Paracompact Hausdorff but not compactly generated? wildildildlife 2011-05-18T22:33:31Z 2011-05-18T22:33:31Z <p>(This should be a comment, but my rep is too low.)</p> <p>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 $k(X)$. </p>