There are multiple (incompatible) definitions of "compactly generated" or "$k$-space" in the literature. For a sample, see the various references mentioned in this question. See also the discussion here and here for example.
One starts with a topological space $(X,T)$, where $T$ is the topology. Sometimes some separation condition (like weak-Hausdorff, etc) is added to the definition, sometimes not, but we'll ignore that part here. What all the definitions have in common is that one takes the final topology on $X$ with respect to some family of continuous maps into $X$. One can either take all the inclusion maps from certain subspaces, or all the continuous maps from arbitrary spaces of a certain kind. And on the other hand, one can take the domains of these maps to be compact, or compact Hausdorff. That gives a total of four combinations. I am trying to unravel things and understand the relationships between these four possible definitions, spelled out explicitly below.
In each case, one starts with an appropriate family $\mathcal F$ of continuous functions into $X$ and one forms the final topology on $X$ with respect to $\mathcal F$. That topology is called the k-ification of $(X,T)$, denoted $T_{\mathcal F}$. The space $X$ is then called a $k$-space or compactly generated if its topology coincides with its k-ification. So here are the four possibilities with separate names for the purpose of discussion:
(1) CGSub: $\mathcal F=$ the family of all inclusions from compact subspaces of $X$. The corresponding definition of compactly generated space is the one in wikipedia.
(2) CG: $\mathcal F=$ the family of all continuous maps from arbitrary compact spaces.
(3) HCGSub: $\mathcal F=$ the family of all inclusions from compact Hausdorff subspaces of $X$. The wikipedia article calls the corresponding spaces "Hausdorff-compactly generated".
(4) HCG: $\mathcal F=$ the family of all continuous maps from arbitrary compact Hausdorff spaces. This is the definition in nlab and is the one more commonly used in algebraic topology.
Note that for CG and HCG, the families of maps are not sets but proper classes. The final topology is still perfectly well-defined, as explained here.
To understand the relationships between the various k-ifications, the following is useful.
Lemma 1: Let $X$ be a set, $\mathcal F$ a family of functions from topological spaces to $X$, and $\mathcal G$ a subfamily of $\mathcal F$. Then the respective final topologies satisfy $T_{\mathcal F}\subseteq T_{\mathcal G}$.
This implies these relationships between the k-ifications:
- $T_{CG}\subseteq T_{CGSub}\subseteq T_{HCGSub}$
- $T_{CG}\subseteq T_{HCG}\subseteq T_{HCGSub}$
Now combine this with
Lemma 2: Suppose $(X,T)$ is a topological space and $\mathcal F$ and $\mathcal G$ are two families of continuous functions from topological spaces to $X$. If each $f:Y\to X$ in $\mathcal F$ factors through an element $g:Z\to X$ in $\mathcal G$ (via some continuous function $Y\to Z$), then $T_{\mathcal G}\subseteq T_{\mathcal F}$.
We can apply this to CG and CGSub. Each continuous function in CG, namely $f:C\to X$ from a compact space $C$, has compact image and factors through the inclusion $f(C)\to X$ in CGSub. So $T_{CGSub}\subseteq T_{CG}$. Combining this with the previous inclusions gives this relationship between a topology and its k-ifications:
$$T\subseteq T_{CGSub}=T_{CG}\subseteq T_{HCG}\subseteq T_{HCGSub}.$$
And for the corresponding definitions of spaces, $X$ is CGSub if $T=T_{CGSub}$, etc. So the relationships between the various definitions of space properties are:
$$HCGSub\implies HCG\implies CG \iff CGSub.$$
In particular, "nlab compactly generated" (HCG) implies "wikipedia compactly generated" (CGSub). And for the wikipedia definition, it does not matter if it is defined via subspaces (CGSub) or via maps from arbitrary spaces (CG).
The implication HCG $\implies$ CGSub cannot be reversed. For example, as explained in this question, the one-point compactification of a space which is not CGSub (like the Arens-Fort space) is CGSub but not HCG (because HCG is preserved by open subspaces, but that's not the case in the example).
Also note: if $X$ is Hausdorff, all notions above coincide.
Apart from validating the above, I would be interested if anyone could point to a reference that "nlab definition" implies "wikipedia definition".
Specific question: Can the implication HCGSub $\implies$ HCG be reversed? In general a continuous map from a compact Hausdorff space does not seem to factor through an inclusion of a compact Hausdorff subspace of the codomain. So my guess would be no. But what would be an example of a HCG space that is not HCGSub? A compact example would be even better.
Minor question: HCGSub is called "Hausdorff-compactly generated" in wikipedia (note the hyphen!). That name seems very dubious. Never heard of it anywhere. Maybe someone can comment on that. Maybe for categorical reasons HCGSub is not a very useful notion compared with HCG, for uses in algebraic topology in particular?
Best Answer
Here is an example of an $HCG$ space which is not $HCGSub$.
The Sierpinski space is the two point space $\mathbb{S}=\{0,1\}$ with topology $\{\emptyset,\{1\},\mathbb{S}\}$. It is a $T_0$ space which is not $T_1$ and in particular not discrete. Its Hausdorff subspaces are the two singletons and the final topology they induce on $\mathbb{S}$ is the discrete topology. Thus $\mathbb{S}$ is not $HCGSub$.
On the other hand, $\mathbb{S}\cong [0,1]/(0,1]$, so $\mathbb{S}$ can be written as a quotient of the unit interval. Clearly this is sufficent for $\mathbb{S}$ to be $HCG$.
In general, a finite topological space (or more generally an anticompact space, that is, where all compact subsets are finite) will be $HCGSub$ exactly when it has the discrete topology. The reason is that in an anticompact $HCGSub$ space every compact Hausdorff subspace $K$ is finite Hausdorff, hence discrete. Thus every subset $A\subseteq X$ meets $K$ in an open subset of $K$, hence $A$ is open in $X$.
Every $HCGSub$ space is $T_1$ (because a singleton intersects every compact Hausdorff subspace $K\subseteq X$ in the empty set or in the singleton, which is closed in $K$, hence the singleton is closed in $X$). This provides another way to see that finite $HCGSub$ spaces are discrete.
As for the term `Hausdorff-compactly generated', I have never seen it in the literature. Looking through the Wikipedia revision history for the page you linked, it seems that the term originates with user Kaba3, whose profile includes a link to their personal homepage. You might consider getting in touch with them for a reference, as you seem also to have an interest in maintaining and improving the wiki page.
There is, of course, some merit in the `Hausdorff-compactly generated' notion. For starters it is perhaps more intuitive to work with. For example, there simply are no size issues that need resolving.
More relevant is the fact that any weak Hausdorff space is $HCG$ iff it is $HCGSub$. Indeed, a space $X$ is weak Hausdorff if and only if for each compact Haudorff space $K$ and each map $\phi:K\rightarrow X$, the image $\phi(K)$ is a closed, compact, Hausdorff subspace of $X$. Given that many authors work exclusively with weak Hausdorff spaces, one can understand the origins of the definition.