Interpreting a homotopy as a function from $[0,1]$ to a function space? Let $X,Y$ be topological spaces. A homotopy is a continuous $H:X\times[0,1]\to Y$. Currying, we get $H:[0,1]\to X\to Y$. Is there a way to interpret $H$ as a continuous function of type $H:[0,1]\to (X\to Y)$, where $(X\to Y)$ is the space of continuous function from $X$ to $Y$?
There are a few questions.

*

*Given topological spaces $X,Y$, is there natural space of continuous functions from $X$ to $Y$?

*Given a function out of a product space $f:X\times Y\to Z$, is there a natural way to curry?

*If the above doesn't work for general spaces, is it possible for $Y=[0,1]$?

 A: Q: Given two topological spaces, $X$ and $Y$, is there a natural space of continuous functions from one space to the other?A: Yes.  It is the collection of all continuous functions from $X$ to $Y$.  (From another vantage point, one sometimes encounters an arbitrary set $X$, a topological space $Y$, and a set ${\cal F}$ of functions from $X$ to $Y$.  One can then ask, What minimal topology will make all the functions in ${\cal F}$ continuous?)
Since you call this collection of functions a ``space'', I suspect you might be looking to find out whether there is some natural (topological?) structure on this collection.  If so, yes, there is: the compact-open topology.  I think it leads to a general answer to your question about regarding a homotopy as a continuous (in some sense) mapping from $[0, 1]$ to the set of continuous mappings from $X$ to $Y$.  An efficient source to grasp this from is the book "Homotopic topology" by Fomenko, Fuchs, and Gutenmaher.
A: The answer is in [1].
$\newcommand{\Top}{\mathbf{Top}} % topological spaces$

*

*Yes, the compact open topology.

*No. A topological space $X$ is exponentiable if the functor $-\times X: \Top\to\Top$ has a right adjoint $(-)^X$. [1] shows that $X$ is exponentiable iff it is core compact. If $X$ is Hausdorff, $X$ is core compact iff it is locally compact.

*Again, no. What matters is $X$, not $Y$.

[1]: Escardó, Martín; Heckmann, Reinhold, Topologies on spaces of continuous functions., Topol. Proc. 26(2001-2002), No. 2, 545-564 (2002). ZBL1083.54009.
