The topological structure of function space of transitive maps

Let $C(mathbf I)$ be the set of all continuous self-maps from ${mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $fin C({mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $mathbf{I}$, there exists a positive integer $n$ such that $Ucap f^{-n}(V) ot=emptyset.$ We note $T(mathbf{I})$ and $overline{T(mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(mathbf I)$. In this paper, we show that $T(mathbf{I})$ and $overline{T(mathbf{I})}$ are homeomorphic to the separable Hilbert space $ell_2$.