I'm reading the book Topology and Groupoids.
Given a topological space $X$, a subset $A$ is called compactly-closed, if for any continuous map (called test maps) $t:C\rightarrow X$ from a compact Hausdorff space $C$ to $X$, the preimage $t^{-1}(A)$ is closed in $C$. $X$ is a $k$-space if all compactly-closed subset is closed.
One can see that if $X$ is a $k$-space, then a map $f:X\rightarrow Y$ is continuous if and only if its composition with any test map $t:C\rightarrow X$ is continuous. But all the test maps don't form a set, which is hard to deal with. (We'd like to make some categorical constructions like colimits, which requires the index category to be small)
There's a lemma that, in order to test for a $k$-space we need look only at a set of test maps.
The set $C_X$ is constructed as follows. Since $X$ is a $k$-space,
for each non-closed subset $B$ of $X$ there is a compact Hausdorff space $C_B$
and map $t : C_B \rightarrow X$ 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 $C_X$ be the set of all these
$t$. That this set has the required property is clear.
My question is about the choice step. For each $B$, the class of all maps $t$ available is not a set too. Then the choice step need to made a choice among a set-indexed family of proper classes. I don't know how to justify this step, with my poor understanding towards axiomatic set theory (just knows a little about ZFC system).
Is the Axiom of Choice available for this situation?