The Service Road to Geometry

$\def\S{\mathcal{S}} \def\dom{\operatorname{dom}} \def\CS{\mathrm{C}\mathcal{S}} \def\B{\mathcal{B}} \def\im{\operatorname{im}} \def\F{\mathcal{F}} \def\et{\operatorname{\acute Et}}$When one defines sheaves over a topological space $X$ as étale spaces over $X$, usually one first constructs a set $E$ along with a projection $p:E\to X$, and then topologizes $E$ in terms of a certain family of sections of $p$, verifying after that $p$ becomes a local homeomorphism. This is usually explained to the mathematical reader in an informal way, leaving the details of the construction to them. This post aims to fill up those missing details through the development of the mini-theory of "section bases" (Definition 2), which allow to characterize local homeomorphisms (Proposition 4). The last result, Corollary 9, explains then how to realize in general the construction referred on the first paragraph. Definition 1. Let $E$, $B$ be spaces and let $p:E\to B$ be a function (not necessaril