Local homeomorphisms: terminology

Question

Let $(E,\pi,X)$ be a local homeomorphism, so for any $x\in E$ there is an open set $U\ni x$ such that $\pi|U$ maps $U$ homeomorphically onto $\pi[U]\subseteq X$, and $\pi[U]$ is open in $X$.

Is there a standard terminology for such open sets $U$? "Even" $U$? "Flat" $U$? "Simple" $U$? "Schlicht" $U$?

I'm particularly interested in the context where $(E,\pi,X)$ is the etale space of a sheaf.

Answer 1

I would provocatively call it an étale neighbourhood of $x$ in $X$ over $E$. ;-) As it is the standard terminology in the analogue case in algebraic geometry, or differential/complex analytic geometry...