A couple of days ago, thanks to Strom's excellent book Modern Classical Homotopy Theory, I started reading up on compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces (the best decision in my life so far). The classical article by Steenrod A Convenient Category of Topological Spaces and Strickland's openly available exposition The Category of CGWH Spaces have been extremely valuable resources.

Not only have these texts shown me there is a conceptually satisfying way to deal with the topology in algebraic topology without any funny business, but for the first time ever, I understand the importance of the categorical viewpoint (and Yoneda's lemma).

I have several times tried to grasp algebraic topology in the past, and have gone some way each time, however I was always held back by the topology, and have come to StackExchange on a couple occasions to ask questions. I learnt about classifying spaces for vector bundles and principal bundles and thus know of the usefulness of paracompact spaces, so I want to know wether my new paradise (the category of CGWH spaces) contains them.

1) Are paracompact (resp. regular, normal...) spaces compactly generated?

1') Are paracompact (resp. regular, normal...) Hausdorff spaces compactly generated?

2) If not, is the $k$-ification of a paracompact (resp. regular, normal...) space still paracompact (resp. regular, normal...)?

3) Are there free online resources that give a thorough account of paracompactness and other such separation properties (such as regularity, normality, metrizability) and their interplay?

The article on nLab linked me to a set of lecture notes I have thus far only saved on my pc (http://www.helsinki.fi/~hjkjunni/ the top.$1$ to top.$10$).


The answer to all versions of (1) and (1') is no.

Let $D$ be an uncountable set and $p$ a point not in $D$. Let $X=\{p\}\cup D$, and topologize $X$ by making each point of $D$ isolated and making $V\subseteq X$ a nbhd of $p$ iff $p\in V$ and $X\setminus V$ is countable. In other words, if $\tau$ is the topology on $X$, $$\tau=\wp(D)\cup\{X\setminus C:C\subseteq D\text{ and }|C|\le\omega\}\;.$$

($X$ has been called the one-point Lindelöfization of the uncountable discrete space $D$.)

$X$ is easily seen to be paracompact, Hausdorff, and hereditarily normal, but $X$ is not a $k$-space: the only compact subsets of $X$ are the finite subsets, and they don’t generate the topology.

The $k$-ification of a Hausdorff space is certainly Hausdorff, since the new topology is finer than the original one; I don’t know about the other properties off the top of my head.

  • $\begingroup$ Thank you for that counter-example! I think it shows us that the $k$-ification of a paracompact space need not be paracompact. Indeed, the only compact subspaces of $X$ are the finite ones, and a subset $O\subset X$ is $k$-open iff for any compact subset $K$ of $X$, $K\cap O$ is open in $K$: since $K$ is discrete, this places no restriction on $K\cap O$, so $k\tau$ is the discrete topology on $X$, which is not paracompact because $X$ is uncountable. $\endgroup$ – Olivier Bégassat Jun 12 '12 at 8:29
  • $\begingroup$ @Olivier: The discrete topology on any set is paracompact: every open cover even has a discrete open refinement, which is certainly locally finite. The cardinality doesn’t matter. $\endgroup$ – Brian M. Scott Jun 12 '12 at 8:32
  • $\begingroup$ ohhhhh! I was thinking of countable covers. You are right! $\endgroup$ – Olivier Bégassat Jun 12 '12 at 8:32

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