When does taking the product $- \times X$ preserve reflexive coequalizers in $\mathsf{Top}$?

Question

By $\mathsf{Top}$ I mean the category of topological spaces and continuous homomorphisms.

I am aware that if $(X,\tau_X)$ is a topological space, then the functor $- \times (X,\tau_X) : \mathsf{Top} \to \mathsf{Top}$ does not preserve pushouts in general. Therefore, I would not expect it to preserve all coequalizers.

As far as I can see, the main question is that if $f,g:(Y,\tau_Y) \to (Z,\tau_Z)$ are two continuous maps, then their coequalizer in $\mathsf{Top}$ is computed by performing the coequalizer $Q$ in $\mathsf{Set}$ and the endowing it with the quotient topology $\tau_Q$. Therefore, the coequalizer of $(f \times X,g \times X)$ is $(Q\times X,\tau_{Q \times X})$ with $\tau_{Q \times X}$ the quotient topology, while $(Q,\tau_Q)\times (X,\tau_X) = (Q \times X,\tau_Q \times \tau_X)$ where $\tau_Q \times \tau_X$ is the product topology. Since in this case the quotient topology $\tau_{Q \times X}$ is, in general, finer than the product topology $\tau_Q \times \tau_X$ we have that the identity $$(Q \times X,\tau_{Q \times X}) \to (Q \times X, \tau_Q \times \tau_X)$$ is a continuous bijective homomorphism which is not, in general, an homeomorphism.

However, there are particular instances (depending on $X$ or on the particular coequalizer) for which $- \times X$ does preserve coequalizers. For example:

1. If $X$ is locally compact Hausdorff, then $- \times X$ preserves all colimits,
2. If $X$ is a topological group and $f,g:A \to B$ are two open maps of topological $X$-sets, then the coequalizer map $q : B \to Q$ in $\mathsf{Set}$ becomes open in $\mathsf{Top}$ and one can show that $(Q \times X, q \times X) \cong \mathsf{coeq}(f \times X,g \times X)$ (the product topology becomes the quotient topology in this case).

So I am wondering, in (almost) reverse order of strength,

Question 1: Under which conditions on $X$, $ - \times X$ preserves all reflexive coequalizers? (i.e. equalizers of parallel pairs $f,g : A \to B$ admitting a common section $s : B \to A$, that is, $fs = id_B = gs$)

Question 2: If $G$ is a topological group, $X$ is a topological $G$-set, $f,g:A \to B$ are continuous morphisms of topological $G$-sets admitting a common section $s$, does $- \times X$ preserve the coequalizer of $f$ and $g$ in $\mathsf{Top}$?

Question 3: If, as in 2., $X$ is a topological group and $f$ and $g$ admits a common section, but they are not open, could I still conclude something?

Question 4: If $X$ is just a topological monoid instead of a group, does the conclusion of 2. still hold? Under which additional conditions do we get that $q: B \to Q$ is open?

Any comment would be greatly appreciated.

Answer 1

Answer to 1: since $Y \mapsto X \times Y$ always commutes with coproducts, $Y \mapsto X \times Y$ commutes with all colimits if and only if it commutes with reflexive coequalisers.

The spaces that satisfy this property are called core-compact.