In this pull request to the pi-Base database, we encountered this situation.
A space $X$ is said to be weakly Hausdorff provided for every compact Hausdorff space $K$ and every continuous $f:K\to X$, $f[K]$ is closed in $X$.
A space is $X$ said to be k-Hausdorff provided its diagonal $\Delta=\{(x,x):x\in x\}\subseteq X\times X$ is "k-closed".
In Quotients of k-semigroups a set is said to be k-closed if its complement is k-open, that is, its intersection with every compact set is open in the subspace. Call this $k_1$-closed, and its corresponding version of k-Hausdorff $k_1H$. There it is proven that every $k_1H$ space has the property that compact sets are closed, which in turn implies the space is weakly Hausdorff.
On the other hand, in Compactly Generated Spaces, a set is said to be k-closed if given any continuous map of a compact Hausdorff $K$ into the space, the set's inverse image is closed in $K$. Call this $k_2$-closed, and its corresponding version of k-Hausdorff $k_2H$. There it's shown that every weakly Hausdorff space is $k_2H$.
However, it seems weakly Hausdorff and k-Hausdorff aren't actually equivalent. So what gives?
(Hints: of course we have two different definitions of k-closed here. Additionally, some care should be given to checking when "compact" means "compact and Hausdorff".)
Best Answer
The bottom line is that the two notions of k-Hausdorff are not equivalent, and as usual with k-spaces/compactly generated spaces, different authors use the same terminology to mean different things, which can be confusing.
In general, the notions of k-closed set, k-open set, k-ification of a topological space, k-space (or compactly generated space, depending on the author), k-Hausdorff property, etc all depend on which family of "test" maps is used to define the corresponding final topology. The wikipedia article on compactly generated space has a good explanation of the general setup. See also Unraveling the various definitions of $k$-space or compactly generated space.
So given a topological space $(X,\tau)$, common choices are:
(Definition 1 = CG1 in Wikipedia, this in pi-base]) Take $\mathcal F_1$ to be the family of all inclusions from compact subspaces of $X$, or take $\mathcal F'_1$ to be the family of all continuous maps from arbitrary compact spaces to $X$. The final topology $\tau_1$ on $X$ with respect to $\mathcal F_1$ or with respect to $\mathcal F'_1$ is the same. Let's write $k_1X$ for that k-ification of $X$. This is the setup in [LB]. A k-closed-1 set is then a subset of $X$ that is closed in $k_1X$, or a $k_1$-closed set for short
(Definition 2 = CG2 in Wikipedia, this in pi-base]) Take $\mathcal F_2$ to be the family of all continuous maps from arbitrary compact Hausdorff spaces to $X$. Let's write $k_2X$ for the corresponding k-ification of $X$, namely $(X,\tau_2)$ with $\tau_2$ the final topology on $X$ with respect to $\mathcal F_2$. This is the setup in [R] and [S], more commonly used in algebraic topology. A k-closed-2 set is then a subset of $X$ that is closed in $k_2X$, or a $k_2$-closed set for short.
(Definition 3 = CG3 in Wikipedia, this in pi-base]) Take $\mathcal F_3$ to be the family of all inclusions from compact Hausdorff subspaces of $X$. Let's write $k_3X$ for the corresponding k-ification of $X$, namely $(X,\tau_3)$ with $\tau_3$ the final topology on $X$ with respect to $\mathcal F_3$.
Because $\mathcal F_3\subseteq\mathcal F_2\subseteq\mathcal F'_1$, we have inclusions of topologies $$\tau\subseteq\tau_1\subseteq\tau_2\subseteq\tau_3$$ and continuous identity mappings between the various k-ifications $$k_3X\to k_2X\to k_1X\to X.$$
So closed subsets of $X$ are $k_1$-closed, $k_1$-closed sets are $k_2$-closed, etc. And if a space $X$ is CG3 (i.e., $k_3X=X$), it is CG2 (i.e., $k_2X=X$); and so on.
Now for the definitions of k-Hausdorff. A space $X$ is k-Hausdorff is the diagonal $\Delta$ is k-closed in $X\times X$ (usual product space), i.e., if $\Delta$ is closed in the k-ification $k(X\times X)$. Of course that depends on the specific k-ification we are talking about. So there is the notions of $k_1$-Hausdorff (used in [LB]) and $k_2$-Hausdorff (used in [R]). We could even have $k_3$-Hausdorff. And because of the relationship between closed, $k_1$-closed, $k_2$-closed sets above, we have these implications between the properties for the space $X$:
$$\text{Hausdorff}\implies k_1\text{-Hausdorff}\implies k_2\text{-Hausdorff}.$$
More can be said. [LB] proves (Theorem 2.1) that $X$ being $k_1$-closed is equivalent to all compact subsets of $X$ being Hausdorff, and that implies that $X$ is KC (= all compact subsets are closed). On the other hand, [R] proves (Prop. 11.2) that weak Hausdorff implies $k_2$-Hausdorff. So we can insert those in the chain of implications already recorded in pi-base (namely $T_2$ implies KC implies weak Hausdorff implies US implies $T_1$, with US = "unique sequential limits"), and we get:
$$T_2\implies k_1\text{-Haus.}\implies KC\implies\text{weak Haus.}\implies k_2\text{-Haus.}.$$
I haven't given a thought about whether $k_2$-Hausdorff implies US, and the position of $k_3$-Hausdorff in this.
For an example of a space that is $k_2$-Haudorff and not $k_1$-Hausdorff, one can choose a space that is weak Hausdorff and not KC. A search in pi-base gives the square of the one-point compactification of the rationals: $X=\mathbb Q^*\times\mathbb Q^*$ with this reference.
Another example is the one-point compactification of the rationals itself: $X=\mathbb Q^*$. It is KC, hence weak Hausdorff and $k_2$-Hausdorff. But it is not $k_1$-Hausdorff, because it is compact but not Hausdorff (see [LB] Theorem 2.1). On the other hand, it is Frechet-Urysohn, hence sequential, hence CG2. So it is an example of CGWH space that is not $k_1$-Hausdorff.
One can also ask if any of the implication above become equivalences if we make an additional k-space assumption. An important case is the combination CGWH (= CG2 + weak Hausdorff). This is equivalent to CG3 + weak Hausdorff (Lemma 1.4(c) in [S]). Also, CG2 + $k_2$-Hausdorff imply weak Hausdorff ([R] Proposition 11.4 or [S] Proposition 2.14) and CG2 + weak Hausdorff imply KC (see here). So under the assumption of CG2, the three properties KC, weak Hausdorff, and $k_2$-Hausdorff are equivalent, and they are also equivalent to the combination of CG3 with any of them.
Does CGWH imply $k_1$-Hausdorff? No. See the example above of the one-point compactification of the rationals.
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