currying of mapping spaces bijective for locally compact spaces
1. Proposition
Let 
, 
and 
be topological spaces, such that 
is locally compact
Then for the mapping spaces equipped with the compact open topology, Currying
is bijective
2. Proof
2.1. injective
since currying inside category set is an isomorphism, we conclude, that it is injective here.
2.2. surjective
Let 
.
Then $q is continuous and furthermore for
or
Here 
is continuous as composition of continuous maps (cf. evaluation map from a locally compact as continuous map)
Furthermore, by construction (since they agree on all values)
