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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 $\mathfrak{M}_X$?

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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

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  • $\begingroup$ 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. $\endgroup$ – Rick Sternbach Nov 11 '18 at 12:29

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