Definition 2

Let E and B be topological spaces, let $p: E\rightarrow B$ be a mapping and let $x$ be a point of E. One says that the mapping $p$ is topologically etale at $x$ if there exists a neighbourhood U of $x$ in E and a neighbourhood V of $p(x)$ in B such that $p$ induces a homeomorphism of U onto V.

One says that the mapping $p$ is topologically etale if it is topologically etale at every point $x$ of E.

When no confusion is possible, cf. example 3 below, one shall say etale instead of topologically etale. Instead of saying that $p: E\rightarrow B$ is an etale mapping, one also says that the B-space $(E, p)$ is an etale B-space, or simply that E is a space spread over B, when no doubt is possible as to the mapping $p$.