etale space

Last updated January 27, 2022

Definition

Let $B\in\text{Top}$, and $x\in B$.

• An etale space over $B$ is an object $E\in\text{Top}$ equipped with a map $$p:E\to B$$ such that $p$ is a local homeomorphism.
  • $E$ may be called the total space.
  • $p$ may be called the projection.
• $E_x=p^{-1}(x)$ is called the stalk of $p$ over $x$.

Relation to sheaves

Consider an arbitrary map $p:E\to B$. Recall (from sheaf) the sheaf of sections of $p$, denoted $$\Gamma_p(E/B).$$

1. If $p$ is etale, then the images of local sections form a base for the topology of the total space $E$.
2. If $p$ is etale, then the sheaf of sections