Considering $X$ as a tychonoff space and $Y$ any metric space, and taking the fine topology on function space $C(Y, \Bbb R)=C(Y)$ and $C(X, \Bbb R)=C(X)$. Then the continuous map $g \in C(X, Y)$ induces a function $g_* : C(Y) \rightarrow C(X)$, as $g_*(f) = \phi(f, g) = g \circ f, f \in C(Y)$.
How to prove the induced map; $g_* : C(Y) \rightarrow C(X)$, induced from the map $g \in C(X, Y)$, endowed with fine topology is an embedding. Taking $g$ as continuous, onto and open map. How to prove that $g_*$ is an embedding.
I can prove it continuous and bijection but i don't know how to prove it open map. Can anybody help?