Open sets in compactly generated spaces

Question

There are multiple (incompatible) definitions of "compactly generated" or "$k$-space" in the literature. See the discussion here and here for example. Two commonly used ones are:

Definition 1: $X$ is compactly generated (call this CG1) if it has the final topology with respect to the inclusions from all its compact subspaces. This is the definition in wikipedia and in Willard.

Definition 2: $X$ is compactly generated (call this CG2) if it has the final topology with respect to all continuous maps from arbitrary compact Hausdorff spaces. This is the definition in nlab and is the one more commonly used in algebraic topology.

I have two questions about open sets in such spaces.

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For the first definition, Wikipedia claims that open sets in a CG1 space are also CG1. I don't think that's the case. It is shown in Eric Wofsey's anwer to this question that, in a CG1 space, open sets $U$ satisfying a certain regularity condition ((*) each $x\in U$ has a closed nbhood in $X$ contained in $U$) are also CG1. That seems to imply that it is not the case for arbitrary open sets.

What are examples of CG1 spaces with open sets that are not CG1?

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For the second definition, some sources say that all open sets in a CG2 space are CG2. And other sources say it is not the case. I was wondering who is right. (One difficulty is that different sources may use different terminology for CG2, so one has to double check that in each case.)

Sources that claim all open sets in a CG2 are CG2:

1.1 David Carchedi's answer to this question shows that all open sets in a CG2 space are CG2. The proof seems correct to me.

1.2 In [S] a CG2 space is called "compactly generated". Lemma 2.26 says that open sets in a CG2+WH space are CG2+WH (where WH = Weak Hausdorff). I have not checked, but I would imagine that the same proof would work without the WH condition.

Other sources claim or imply that not all open sets in a CG2 space are CG2:

2.1 In [M] a CG2 space is called a "$k$-space", and CG2+WH is called "compactly generated". But rephrasing in our explicit terminology, Problem 1(c) on p. 41 says that an open set in a CG2+WH space will be CG2+WH if the open sets satisfies the same regularity condition (*) above. This seems to imply it is not the case for all open sets.

2.2 In [R] a CG2 space is called a "$k$-space" and he uses "compactly generated" for CG2 + k-Hausdorff (a slightly different notion than WH). The paragraph after Remark 3.6 on p. 3 reads:

Observe that a general subspace of a k-space need not be a k-space. In particular, open subsets of a k-space can fail to be k-spaces. ...

In the terminology here, he is saying not all open in a CG2 are CG2.

So bottom line: are May and Rezk wrong about this?

References:

• [M] J.P. May, A Concise Course in Algebraic Topology (pdf here)
• [R] C. Rezk, Compactly generated spaces, 2018 (pdf here)
• [S] N. Strickland, The category of CGWH spaces, 2009 (pdf here)

Answer 1

To produce an example for your first question, take any space $Y$ which fails to be CG1. For instance, the uncountable product $Y=\mathbb{R}^\mathfrak{c}$ will work. Now let $X$ be the one-point compactification of $Y$. Thus $X$ is the disjoint union of $Y$ with an external point $\infty$, topologised so that points of $Y$ have their usual neighbourhoods and $\infty$ has a system of neighbourhoods given by the complements of closed compact subsets of $Y$.

Then $X$ is a compact space and in particular is CG1. It will be $T_1$ whenever $Y$ is, but will never be Hausdorff (it will always contain a non-closed compact subset). In any case, the point at $\infty$ is closed in $X$, so $Y$ is an open subset of $X$ which is not CG1.

As for your second question, it is difficult to tell for certain whether I have missed some assumption or definition one of the authors has made. However, as far as I can tell Rezk's statement is incorrect and May's, while not wrong, is incomplete. Carchedi's proof would seem to be correct:

Any open subset of a CG2 space is a CG2 space.

Here is another proof. From the definition it follows that a space $X$ is CG2 if and only if it is a quotient of a sum of compact Hausdorff spaces if and only if it is a quotient of a locally compact Hausdorff space. Thus let $X$ be CG2 and $\pi:L\rightarrow X$ a quotient map, where $L$ is locally compact Hausdorff. If $U\subset X$ is open, then the restriction $\pi|:\pi^{-1}(U)\rightarrow U$ is a quotient map. But $\pi^{-1}(U)$ is locally compact Hausdorff. Hence $U$ is CG2.

Finally, let us remark that a Hausdorff space is CG2 if and only if it is CG1. Hence the $T_1$ counterexamples produced for your first question are (close to) best possible.