Left and right homotopies have the same equivalence relations Let $f, g \colon A \longrightarrow X$ be continuous functions of topological spaces. We say that $f,g$ are left-homotopic if there exists a continuous map $H \colon A \times I \longrightarrow X$ such that $H(a, 0) = f(a) $ and $H(a,1)  = g(a) $. We say that $f,g$ are right-homotopic if there exists a continuous map $K \colon A \longrightarrow X^I$ where $X^I = \mathsf{Top}(I, X)$ is the space of continuous maps with the compact-open topology, such that $K(a)(0) = f(a)$ and $K(a)(1) = g(a)$. In $X^I$ a subbase for the topology is given by the collection of subsets $V(K, U) \subseteq X^I$ where $K \subseteq I$ is compact, $U \subseteq X$ is open and
$$
V(K, U) = \left\{ \lambda \in X^I \vert  \ \lambda( K) \subseteq U\right\}.
$$
I am trying to show that the equivalence classes for left and right homotopy are equivalent. I have been able to show left $\implies$ right: Given a left homotopy $H \colon A \times I \longrightarrow X$ define $K \colon A \longrightarrow X^I$ by $K(a)(t) = H(a,t)$. Choose $\tilde K \subseteq I$ compact and $U \subseteq X$ open. Then
$$
K^{-1}  ( V (\tilde K, U ) ) = \displaystyle\bigcap_{t \in \tilde K} (H i_t )^{-1} (U)
$$
where $\iota_t \colon A \longrightarrow A \times I$ is the map $a \mapsto (a,t)$. It is sufficient to check inverse images of the subbase.
Now given right homotopy $K \colon A \longrightarrow X^I$ define $H \colon A \times I \longrightarrow X$ by $H(a,t) = K(a)(t)$. I presume this is the correct function. How do I show it is continuous?
 A: The main result you need is that if $A,Y,X$ are topological spaces and the space $X^Y$ of continuous functions $Y \to X$ is given the compact open topology then the exponential map 
$$e: X^{A \times Y}  \to (X^Y)^A$$
which sends $f: A \times Y \to X $ to the function which on $a \in A$ has value the function $y \mapsto f(a,y)$, this map $e$ is well defined and is a bijection if $Y$ is locally compact, i.e. each point of $Y$ has a base of compact neighbourhoods. The case you are interested in is when  $Y=I$, the unit interval, which is Hausdorff and locally compact. 
For more information on this sort of question see the wiki entry on convenient categories and also, to plug my account, Section 5.9 of Topology and Groupoids. In category theory, this discussion is about the notion of cartesian closed categories. 
Lawvere has given an intuitive description of these ideas. One sees $X^Y$ as a kind of phase space of positions of $y \in Y$ in terms of $X$. Then $(X^Y)^I$ is the space of paths in the phase space. On the other hand, $(X^I)^Y$ is a phase space of paths.  
Later: Just to answer Paul's question on the continuity of $e$; this question is easier to answer if one knows exactly what is $(X^Y)^A$. I published  the answer in my first two  papers, available from my publication list.  There is a natural topology on the set product $A \times Y$ giving a space $A \times_S Y$: it takes the usual product topology and the takes the final topology with respect to all subsets of the form $\{a\} \times Y, A \times C$ where $ a \in A$ and $C$ is compact in $Y$. (This all works in the Hausdorff case. There is a modification for the non Hausdorff case.) This product is associative. Then one finds that the natural map 
$$e_S: X^{A \times_S Y}  \to (X^Y)^A$$
is a bijection! One can also use the associativity to prove this is a homeomorphism. In modern terminology, we can make the category $Top$ of spaces and continuous maps into a monoidal closed category. Of course this monoidal structure is not symmetric. Too bad! The identity  map $ A \times_S Y \to A \times Y$ is continuous, so we get an embedding $X^{A \times Y} \to X^{A \times_S Y}$, which proves the continuity of $e$ without getting ones hands dirty!
With regard to the non Hausdorff case, this is dealt with in a paper on Convenient Categories by Booth and Tillotson in the Pacific J. Math. 1980. One replaces compact subsets of a space $Z$ by test maps from a compact Hausdorff space to $Z$, as in Section 5.9 of T&G. 
The second paper on "Function spaces and product topologies" (1964) gives stronger results than were published in 1968 by someone else. 
