Let $E$ be the open interval $(0, 1)$ with the usual metric topology and $\Bbb R$ be given the usual topology.
Let $\Bbb R^J$ be given the topology of pointwise convergence $T_p$, where the index set $J$ comprises all continuous functions $f : E \to \Bbb R$.
Define a function $T : E \to \Bbb R^J$ by $T(x) = (f(x))_{f∈J}$ . Suppose that the image set $T(E)$ is given the subspace topology as a subspace of $(\Bbb R^J, T_p)$.
I want to show that $T$ is a homeomorphism from $E$ onto $T(E)$ but am not sure what method to employ even hmm.
Is there an explicit map to find? or find continuous maps $f$ and $g$ s.t. $f\circ g=g\circ f= i$ where $i$ is the identity map?