Homotopy as Path of Paths
due by Wednesday, Aug 23, 2023
The Compactopen topology is a topology defined on the space $C(X, Y)$ of continuous functions between two topological spaces $X$ and $Y$. The compactopen topology is the topology with subbasis consisting of sets of the form $$V(K,U) = \{f: X \to Y: \textrm{$f$ continuous, $f(K)\subset V$}\},$$ where $K$ is a compact subset of $X$ and $U$ is an open subset of $Y$.
In particular, the compactopen topology gives a topology on the space of paths $\Omega(X) = C([0,1], X)$ in a topological space $X$.

Show that if $H: [0, 1] \times [0, 1] \to X$ is continuous (i.e., a homotopy), then the map $h: [0, 1] \to C([0, 1], X)$ defined by $h(t)(s) = H(s, t)$ is continuous.

Show that if $h: [0, 1] \to C([0, 1], X)$ is continuous, then the map $H: [0, 1] \times [0, 1] \to X$ defined by $H(s, t) = h(t)(s)$ is continuous.
Observe that functions $h: [0, 1]\to C([0, 1], X)$ are functions $h: [0, 1]\to \Omega(X)$ with $h(t)$ the path $h_t: s \mapsto H(s, t)$.