However the rational numbers ℚ with euclidean topology are first countable (because they are metrizable), but not compactly generated.

So that example appears to be false.

Edit: Sorry, I was wrong. ℚ is compactly generated. Ignore this comment. ]]>

Remove a dead link to the “bimorphism” page since it was specifically decided to not have such a page.

]]>A reasonable point; thanks. The entry has been updated.

]]>Re #84/#85: any indiscrete space with more than one point is also a (non-T₁) counterexample; $\alpha \mathbf{Q}$ shows that even weak Hausdorffness (in particular *KC*-ness or T₁-ness) is not enough.

Added the example of the one-point compactification of $\mathbb{Q}$ as a compact space that is not compactly generated.

]]>Re #84: thanks. Some of that should be incorporated more explicitly. Let me get to that in a bit.

]]>There is an example of a compact space which is not compactly generated here https://math.stackexchange.com/questions/783082/is-every-compact-space-compactly-generated

bruno_h

]]>I think it is still true that Fréchet spaces (i.e. the tvs ones) are compactly generated, since they are metrisable, hence first countable, hence sequential. Just an unfortunate clash of naming there.

]]>Fixed reference to Fréchet spaces (under class of first-countable examples) to refer and link to Fréchet-Uryson spaces

]]>I have added the references on (aspects of) local cartesian closure

Peter I. Booth, Ronnie Brown,

*Spaces of partial maps, fibred mapping spaces and the compact-open topology*, General Topology and its Applications**8**2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$Peter I. Booth, Ronnie Brown,

*On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps*, General Topology and its Applications**8**2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$Peter May, Johann Sigurdsson, §1.3.7-§1.3.9 in:

*Parametrized Homotopy Theory*, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)

Then I replaced (in the paragraph here) the line which was vaguely referring to May&Sigurdsson with concrete pointers to page and verse in these references.

]]>I have expanded the statement of regularity (here) by saying that not only compactly generated Hausdorff spaces but also cg weakly Hausdorff spaces form a regular category.

(This is from the same page 3 in Cagliari, Matovani and Vitale 1995, in the Remark below the Theorem.)

]]>Added (here) pointer to

- Rainer M. Vogt,
*Convenient categories of topological spaces for homotopy theory*, Arch. Math 22, 545–555 (1971) (doi:10.1007/BF01222616)

and grouped this together with Escadro-Lawson & Simpson 2004 and with Gaucher 2009 under “discussion in the generality of subcategory-generated spaces, including $\Delta$-generated spaces”.

]]>added this pointer, for relation to $\Delta$-spaces:

- Philippe Gaucher, Section 2 of:
*Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (arXiv:0710.3553)

I have added boosted versions of the statements about quotients/colimits of LCH spaces here and here, following discussion in another thread (here)

]]>I have started a new Properties-subsection “Relation to locally compact Hausdorff spaces” (here)

Regarding the statement that a Hausdorff space is a k-space iff it is the quotient of a locally compact Hausdorff space:

Is that quotient then also the coequalizer in *weak* Hausdorff spaces (?) Need to think about this tomorrow when I am more awake…

$\,$

also, I have split the References-section into two subsections, and re-organized slightly, for more systematics

]]>I have tried to give more logical structure to the (old) section on the coreflection (now here) by adding numbered environments and cross-pointers, and disentangling the construction of the coreflection (which is still lacking a word on why it’s actually a coreflection) from the discussion of the equivalence $k Top \simeq Top_k$.

In the course of doing this I ended up doing similar edits to the Defintion section here (adding numbered environments and cross-pointers).

Finally, I have started a new Properties-section “Reflection into weak Hausdorff spaces” (here), showing the other adjunction.

In doing so, I have stuck – for the time being – with the notation “$k Top$” long used in the previous section, and so now $Top_{CGWH}$ is denoted “$h k Top$” here. I am not saying this is a good idea, necessarily, but it’s momentarily the least disruptive to the old material in the entry.

]]>Okay, I found the proof: This “MacTutor biography” clarifies it. It’s the only source that I have seen so far which admits that this David Gale authored “Compact sets of functions and function rings”.

Okay, great, have created a `category:people`

-entry *David Gale* now.

Seems very likely. The Wikipedia Gale worked on games, but was supervised by the topologist Tucker, who incidently also supervised Isbell (and John Nash and Marvin Minsky).

]]>Okay, thanks. It looks like, with your help, we saved this information last-minute from disappearing into forgottenness.

By the way, I used to assume that this David Gale is not the David Gale who wrote about k-spaces – but since you said above that our David Gale passed away in 2008, maybe that’s the same person after all? Do you know?

]]>Okay, I have further adjusted the footnote (see here).

But please modify as you see the need.

]]>Woops. Thanks for catching. As you noticed, I had this mixed up already in the lecture notes statement pointed. Thanks for fixing!

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