A natural topology on space of continuous functions

Question

Let $X$ and $Y$ be two topological spaces. Let $C(X,Y)$ be set of all maps from $X$ to $Y$. Does there exists a natural map topology on $C(X,Y)$? By main motivation is to define two maps $f,g$ as homotopically equivalent if there exists a continuous path between these two functions in this topology of $C(X,Y)$.

Answer 1

You want your spaces to be good, say locally compact, probably Hausdorff. Then the compact open topology is perfect for this. $C(X,Y)$ is an exponential object in that continuous maps $Z \to C(X,Y)$ correspond in the obvious way to continuous maps $X \times Z \to Y$, and one is continuous if and only if the other is too. In particular, its path components correspond to homotopy classes of maps $X \to Y$, as you desire.

With the subspace topology, $\text{Homeo}(X) \subset C(X,X)$ becomes a topological group, and it acts continuously on both $C(X,Y)$ and $X$ itself; similarly with $\text{Homeo}(Y)$.

It should be noted that this space $C(X,Y)$ captures quite a lot of the topology of the individual spaces $X$ and $Y$ (as well as data about how they map into one another). For instance, suppose $Y$ is a $K(G,n)$. Then it is a theorem of Thom's that $C(X,Y)$ is homotopy equivalent to $$\prod_{q=1}^n K(H^q(X;\pi),n-q).$$ In particular, this captures the theorem for $n=1$ that continuous maps $X \to K(G,1)$ are in bijection with homomorphisms $\pi_1(X) \to G$.

Answer 2

Yes, there are various topologies one can consider on sets of continuous maps, which are useful to solve different problems. You are trying to find a topology which makes the category of topological spaces Cartesian closed; this means that for any triple of spaces $X,Y,Z$ you would like there to be a natural bijection $C(X \times Y, Z) \xrightarrow{\cong} C(X,\underline{C}(Y,Z))$ given by sending a map $f$ to $x \mapsto f(x,-)$. Here $\underline{C}(Y,Z)$ denotes the set $C(Y,Z)$ together with some appropriate topology. However for this to work you must restrict to some nice class of spaces. One such class is the class of compactly generated spaces, and then the topology on sets of continuous maps is the $k$-ification (a certain refinement) of the compact-open topology. For details see http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf (in particular Theorem 5.5). A more digestible account focusing on the smaller class of weakly Hausdorff compactly generated spaces is given in http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf.
In the case you are interested in the above bijection becomes $C(I \times X, Y) \xrightarrow{\cong} C(I,\underline{C}(X,Y))$, so homotopies between continuous maps $X \to Y$ are exactly paths in $\underline{C}(X,Y)$.
Remark: You also get the bijections $C(I \times X, Y) \xrightarrow{\cong} C(X \times I, Y) \xrightarrow{\cong} C(X,\underline{C}(I,Y))$. So you can also view any homotopy between maps $X \to Y$ as a continuously varying family of paths in $Y$ indexed by the points in $X$.