Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?

We recall the notion of a $$k$$-space (or compactly generated space) to fix our notations. For every topological space $$X$$, we can define a category $$\mathfrak{M}_X$$. The class of objects of $$\mathfrak{M}_X$$ is the class of continuous mappings $$u: K \to X$$ from a compact Hausdorff space $$K$$ to $$X$$. Let $$v : C \to X$$ be another object in $$\mathfrak{M}_X$$, the set of morphisms from $$u$$ to $$v$$ is the continuous mappings $$f: K \to C$$ such that $$u = v\circ f$$. Let $$\tau$$ be the collection of open sets in $$X$$. Define $$k(\tau)$$ to be the final topology on $$X$$ with respect to the class $$\mathrm{Obj}(\mathfrak{M}_X)$$. In other words, for any subset $$U$$ of $$X$$, $$U \in k(\tau)$$ if and only if $$u^{-1}(U)$$ is open in $$K$$ for any object $$u : K \to X$$ in $$\mathfrak{M}_X$$. Let $$k(X)$$ be the space $$X$$ equipped with the topology $$k(\tau)$$ which is finer than $$\tau$$ by definition. Then we call $$X$$ a $$k$$-space if $$k(X) = X$$, or more precisely, $$\tau = k(\tau)$$.

If we adopt the approach of the Grothendieck's axiom on universe to approach the above categorical setting. We can fix a universe $$\mathcal{U}$$ with $$X \in \mathcal{U}$$. The category $$\mathfrak{M}_X$$ is redefined as the category of $$\mathcal{U}$$-small continuous mappings $$u : K \to X$$ from a compact Hausdorff $$K$$ which is also $$\mathcal{U}$$-small. To emphasize this dependence on our universe $$\mathcal{U}$$, we denote this category by $$\mathfrak{M}_{X,\mathcal{U}}$$. Similarly, the final topology on $$X$$ with respect to $$\mathrm{Obj}(\mathfrak{M}_{X, \mathcal{U}})$$ is denote by $$k_{\mathcal{U}}(\tau)$$.

My question is, if we have another universe $$\mathcal{V}$$ with $$X \in \mathcal{V}$$, does $$k_{\mathcal{U}}(\tau) = k_{\mathcal{V}}(\tau)$$?

A sufficient condition would be there is a small (small is for any universe containing $$X$$) final subcategory $$\mathfrak{S}_X$$ of $$\mathfrak{M}_X$$, as is the case when $$X$$ is assumed to be weakly Hausdorff (we can take $$\mathfrak{S}_X$$ to be the full subcategory of $$\mathfrak{M}_X$$ generated by all embeddings $$K \hookrightarrow X$$ where $$K$$ is a closed compact Hausdorff subspace of $$X$$). But for general topological space $$X$$, can we find a small final subcategory $$\mathfrak{S}_X$$ of $$\mathfrak{M}_X$$?

The following part of 5.9.1 of Topology and Groupoids shows for a particular space $$X$$ how to reduce the role of a universe.

If $$X$$ is a k-space, there is a set $$\mathcal C_{X}$$ of maps $$t : C_{t} \to X$$ for compact Hausdorff spaces $$C_t$$ such that a set $$A$$ is closed in $$X$$ if and only if $$t^{-1}(A)$$ is closed in $$C_{t}$$ for all $$t \in \mathcal C_{X}$$.

The proof goes as follows: for each non-closed subset $$B$$ of $$X$$ there is a compact Hausdorff space $$C_{B}$$ and map $$t: C_{B} \to X$$ such that $$t^{-1}[B]$$ is not closed in $$C_B$$. Choose one such $$C_{B}$$ and one such $$t$$ for each non-closed $$B$$, and let $$\mathcal C_{X}$$ be the set of all these $$t$$. That this set has the required property is clear.

This leads to standard properties of such k-spaces.

The following paper uses various classes of compact Hausdorff spaces to construct and study convenient categories:

"Monoidal closed, Cartesian closed and convenient categories of topological spaces" Peter I. Booth and J. Tillotson, 88 (1980), No. 1, 35–53 DOI: 10.2140/pjm.1980.88.35

• This solves my problem so elegantly. I guess the trick is to consider how to detect the complement of $\tau_X$ in $2^X$ instead of $\tau$ itself. – Rick Sternbach Nov 11 '18 at 12:29