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 $Y=[0,1]$?