I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric 
$$d(f,g)=\sup\_x d(f(x),g(x)).$$
Can we say the space $C(X,X)$ is compact? 



Compactness of $C(X,X)$
Indeed this space is (almost, see the comment by YCor) never compact except for trivial cases (i.e. the compact set $X$ having only finitely many points).
For more information on the topology (which is colloquially known as the compact-open topology) of the space you are asking about, see e.g. the book Engelking: General topology. (Chapters 2.6 and 3.4 will have most of the information you could possibly need). 


Thanks for the clarification by @YCor, the never compact of the original answer was a bit hasty and comes from my use of these spaces where $X$ is always locally euclidean (i.e. the $X$ should be a manifold). Which is of course not necessary.


