Local Homeomorphism

A function $p:E\to B$ between Topological Spaces is a local homeomorphism if for every $e:E$, there is an Open Neighborhood $U\in\mathcal{O}({E})$ such that

• The Direct Image $p_{*}(U)$ of $U$ is open in $B$
• The restriction of $p\mid_{U}:U\to p_{*}U$ is a Homeomorphism.

In other words, $E$ "looks locally" like $B$.

As Etale Spaces

When dealing with local homeomorphisms, we want to think about $E$ as a space that is parameterized by $B$. This seems like a situation that is asking for display.