The definition of homotopy in algebraic topology In this post, let $I=[0,1]$.
Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the fundamental group often begin with the basic notion of a homotopy of curves (or more generally, continuous functions between topological spaces) and describe it intuitively as "a continuous deformation of one curve into another". They often supplement this statement with some nice picture, like this one in Wikipedia. When I was taught algebraic topology, I too had heard a motivating explanation as above and was shown a picture of this sort. From this I could already guess what a (supposedly) natural formal definition would be. I expected it to look something like this:

Let $X$ be a topological space and let
  $f,g : I \to X$ be two curves in $X$.
  Then a homotopy between $f$ and  $g$
  is a family of curves $h_t: I \to X$
  indexed by $t \in I$ (the "time"
  parameter) such that $h_0 = f$, $h_1 =
> g$ and the function $t \mapsto f_t$ is
  continuous from $I$ to $C(I,X)$ (the
  space of curves in $X$ with domain
  $I$, equipped with some suitable
  topology).

However, the definition given (which is used in every book on algebraic topology which I sampled) is similar, but not quite what I thought. It is defined as a continuous function $H: I \times I \to X$ such that $H(s,0)=f(s)$ and $H(s,1)=g(s)$ for all $s \in I$.
This actually quite surprised me, for several reasons. First, the intuitive definition of a homotopy as a "continuous deformation" contains no mention of points in the space $X$ - it gives the feeling that it is the paths that matter, not the points of the underlying space (though obviously one needs the space in question to define the space of paths $C(I,X)$). However, the above definition, while formally almost equivalent to the definition I thought of (up to a definition of a "good" topology on $C(I,X)$), makes the underlying space $X$ quite explicit, it appearing explicitly in the range of the homotopy.
Moreover, many of the properties related to homotopies, the fundamental group and covering spaces can be expressed using the vocabulary of category theory, using universal properties. Now, from a categorical-theoretic point of view, wouldn't one want to suppress the role of the underlying space as much as one can (in favor of its maps and morphisms)?
Additionally, the definition of homotopy (as used) seems notationally inconvenient to me, in that it is less clear which of the two variables is the time parameter (each mathematician has his own preference, it seems). Also, the definition of many specific homotopies looks needlessly complicated in this notation, IMO. For instance, if $f,g$ are two curves in $\mathbb{R}^n$ then they are homotopic, and one can write the obvious homotopy either as $H(s,t)=tf(s)+(1-t)g(s)$ or as $h_t = tf+(1-t)g$. Maybe that's just me, but the second notation seems much more natural and easier to understand than the first one. Formulae of this sort appear frequently in the study of the fundamental group of various spaces (and in the verification that the fundamental group is indeed a group), and using the $H(s,t)$ notation makes these formulae much more cumbersome, in my opinion.
So, to sum up, I have two questions:

1) For a topological space $X$, can
  $C(I,X)$ be (naturally) topologized so
  that "my" definition of homotopy (see above) and
  the usual definition coincide (by
  setting $h_t (x) = H(x,t)$)?
2) If so, why isn't such a definition
  preferred? See my arguments above.

 A: In regards to your question 2, I think there are at least two reasons people may like the standard, general definition of homotopy between maps.  First, it lets you get right to the point without defining an extra space and a somewhat complicated topology on that space.  Of course at some point, one doesn't mind these complications, but for a beginning student (e.g. the reader of a basic topology book) thinking of homotopies as maps $X\times I \to Y$ is probably simpler.
The second thing is that I think it's often easier to check continuity of a specific homotopy by viewing it as a map $X\times I \to Y$.  For example, there are lots of places in topology where one uses what are essentially piecewise linear homotopies $I\times I \to Y$.  These are formed by cutting up the square into closed sets in some nice way, such that on each piece the homotopy is a linear function in two variables.  Then by the "gluing lemma," to check continuity you need only check that the maps agree on the overlaps.  I think you'll find examples of this if you work through the details of basic arguments with the fundmental group, for instance.  Some wonderful examples show up in May's Geometry of Iterated Loop Spaces.
All that being said, mapping spaces with the compact-open topology are really important, and it's a shame that they're not emphasized in many topology text books.  Ioan James' books are a good source, though (he has two books about fiber-wise topology that say everything I've ever needed to know) and Munkres has a good discussion too.
A: *

*The answer to your first question is yes (at least if you are in a "convenient" category of topological spaces): If you imbue C(I,X) with the compact-open topology this works out.  This is an instance of the standard adjunction between products and exponential objects.


Peter May's concise intro to algebraic topology develops these ideas quite nicely, and Mac Lane's categories for the working mathematician has an entire section on compactly generated Hausdorff spaces which treats this material as well.
A: There is a natural topology on the function space called the 
compact-open topology.
In extreme levels of generality, your two definitions are different (they are the same for eg locally compact spaces).  Let me give a more general discussion of this.
Let $X$, $Y$, and $Z$ be spaces, and for spaces $A$ and $B$ let $A^B$ be the space of continuous maps $B \rightarrow A$ with the compact-open topology.  There is then a natural bijection $X^{Y \times Z} \rightarrow (X^Y)^Z$.  However, this is not in general a homeomorphism.  The fix is to work in the category of compactly generated spaces.  May's book "A Concise Course in Algebraic Topology" has a nice description of them.
A: The answer to (1) is no: the exponential law $C(X \times Y, Z) \cong C(X, C(Y, Z))$ is only valid for some nice topological spaces; for instance I think that Hausdorff and compactly generated is ok.
One of the reasons why the category of simplicial sets is nicer than that of topological spaces is exactly the fact that the exponential law is always valid.
So what happens for spaces which are not nice? Well, for these spaces the path space $C(I, X)$ can behave bad, so the usual definition which does not involve $C(I, X)$ works better.
A: The answer to (1) is yes:  it follows from the "exponential law"  $C(X\times I, Y) \cong 
C(I, C(X,Y))$.  
I'll hazard an answer to (2):  If you are dealing with spaces such as manifolds or finite complexes, the definition in terms of maps $X\times I \to Y$ allows you to study homotopy
using the same kinds of spaces.
