The étalé space associated to the sheaf of continuous functions over $\mathbb{R} $?

Question

We consider the sheaf $C$ of continuous functions over the reals, that's it, for any open subset $U$ of the reals we put $C(U) $ as the set of all continuous functions from $U$ to $\mathbb {R} $. There is an equivalence between the category of sheaves over a topological space $X$ and the category of étalé space over $X$. So for this sheaf, there must be an étalé space $p:E\longrightarrow \mathbb {R} $ such that the sheaf of sections is isomorphic to this sheaf. My question is there a description of the space $E$? What properties does $E$ have? $E$ is Haussdorf, compact, separable, connected,...?