Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?).
What strategies could one follow in order to compute the number of path components of $Map(X,Y)$? I believe this number should also be the number of $[X,Y]$ homotopy classes of maps $X\rightarrow Y$, is it not? (If there is disparity between the two, I am interested in the second).
I am reading some literature about the topology of $Map(X,Y)$ and there are some interesting results about the topology of the connected components, but what about the number of the connected components themselves?
For example, intuitively, I can say
$[X_1\vee X_2, Y]\approx [X_1,Y]\times[X_2,Y]$
Is there a result for the a generic union? E.g., how far is the following from the truth when $X=X_1\cup X_2$?
$[X,Y]\approx[X_1/(X_1\cap X_2),Y]\times [X_2/(X_1\cap X_2),Y]\times [X_1\cap X_2,Y]$
Finally, what if I am interested in smooth maps instead of continuous? If the manifolds I am working with are 'well behaved', can I forget about the difference?