Sheafification of a presheaf
Definition : Let \( \mathbf{F}\) and \( X \) be topological space, and \( \pi : \mathbf{F} \rightarrow X \) be a surjective map, then the triple \( ( \mathbf{F}, \pi, X ) \) is called an ètale space if \( \pi \) is a local homeomorphism , i.e. for all point \( p \in \mathbf{F} \), there exists an open neighborhood \( W \) of \( p \) such that \( \pi(W) \) is open in \( X \) and \( \pi|_W : W \rightarrow \pi(W) \) is a homeomorphism. \( \Omega \) will be used to denote the topology of \( X \) and \( \Omega_x \) for all the open subsets of \( X \) containing the point \( x \in X \). For all \( x \in X \), we say that \( \mathbf{F}_x := \pi^{-1}(x) \) is a stalk of \( \mathbf{F} \) at \( x \). If \( U \in \Omega \), \( \Gamma(U, \mathbf{F}) \) will denote the set of continuous maps \( \alpha: U \rightarrow \mathbf{F} \) such that \( \pi \circ \alpha = id \). The elements of \( \Gamma(U, \mathbf{F}) \) will be called a section of \( \mathbf{F} \) over \( U \). Let \( \mathbf{F}(U) =