  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 $f\in C({\mathbf
I})$ 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 $U\cap
f^{-n}(V)\not=\emptyset.$ 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 $\ell_2$.
