For $X/{\sim}$ a quotient space, 
$$
Map(X/{\sim},Y)\subset Map(X,Y).
$$
But is this inclusion always a homeomorphism on its image? (Assuming compact-open topology on the mapping spaces.) If not what would be the most general setting to make it true? We can also assume that $X$ and $Y$ are compactly generated.


A related question: if $q\colon X\to X/{\sim}$ is a quotient map and $X/{\sim}$ is compact, does always exist a compact $Y\subset X$ such that $q(Y)=X/{\sim}$. 



Mapping space from a quotient space
The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.


It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexample to the first question for $Y=\mathbb R$.


