I'm trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia)
For an equivalence relation $\sim$ on $X\times X$, we denote with $X/\!\sim$ the quotient space and with $q:X\rightarrow X/\!\sim$ the quotient map.
I'm trying to convince myself that if $\sim$ is induced by a continuous function $f:X\rightarrow Y$ (i.e., $x\sim x^\prime$ iff $f(x)=f(x^{\prime}$), then $X/\!\sim$ is a retraction of $X$.
QUESTION 1 How do one formalize within category theory that a quotient is induced by a map in the category, and not by an arbitrary set-theoretic relation (which is looks as an external concept)?
QUESTION 2 As I said I'm trying to prove that $X/\!\sim$ is a retract of $X$. For this I should basically construct a continuous function (a "section") $s: X/\!\sim\rightarrow X$ such that $q\circ s=id$. The natural candidate is $s([x]_{\sim})=x$ for some chosen $x$ "representing" the equivalence class. Of course I can construct a set-function with this property. But how to prove that it is continuous?
QUESTION 3 Under what hypothesis a functor $F$ preserve quotients? Both in general and in the particular case of $F$ an endofunctor in compact Polish spaces.
An interesting functor is $\textbf{V}$, the Vietoris functor mapping $X$ to the space of its compact subsets which in Compact Polish spaces are precisely the closed sets with the hit-miss topology. This is interesting because the points in $\mathbb{V}(X)$ are closed sets, hence, $V(X)$ somehow topologies the (complement of the) topology on $X$ !
QUESTION 4 As a particular case of question 3 above, does $\mathbb{V}(X)$ preserve quotients?
Thanks!!!