The Étalé space (Espace Étalé) is a topological space associated to a presheaf $\sheaf F$ on a space $X$ The Étalé space is defined to be the disjoint union of stalks of the sheaf $\sheaf F$

$$\sheaf E_{\sheaf F} \equiv \coprod_{x\in X} \sheaf F_x$$

Over each open set $U\subset X$ there is a set of sections $\Gamma(U,\sheaf F)$ A basis for the topology on the Étalé space is formed by taking the open sets to be of the form $\sheaf U_s = \{s_x, x\in U\}$ for $s\in \Gamma(U,\sheaf F)$ and $s_x$ the germ of $s$ at $x$ There is a natural map $\pi\!:\!\sheaf E_{\sheaf F} \rightarrow X$ which takes germs $s_x$ in the stalk $\sheaf F_x$ over $x$ to $x$

Let $s\in \Gamma(U,\sheaf F)$ and $s^\prime \in \Gamma(U^\prime,\sheaf F)$ with $U\cap U^\prime \ne \emptyset$ At each point $x\in U \cap U^\prime$ where $s_x = s^\prime_x$ by the definition of germs there exists an open set $V\subset U\cap U^\prime$ containing $x$ such that $s$ and $s^\prime$ restrict to the same section on $V$ ($s|_V = s^\prime|_V$ . This verifies that $\{\sheaf U_s\}$ form a basis for $\sheaf E_{\sheaf F}$

Then there is another presheaf, $\widetilde{\sheaf F}$ whose sections are the continuous functions from $X$ to $\sheaf E_{\sheaf F}$ assigning an element $s(x)\in \sheaf F_x$ to each point $x \in X$ This presheaf forms a sheaf equivalent to the sheafification of the presheaf $\sheaf F$