Fpqc morphism

In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully [flat morphism]s of schemes.
Sometimes an fpqc morphism means one that is faithfully flat and quasi-compact. This is where the abbreviation fpqc comes from: fpqc stands for the French phrase "fidèlement plat et quasi-compacte", meaning "faithfully flat and quasi-compact".
However, it is more common to define an fpqc morphism of schemes to be a faithfully flat morphism that satisfies the following equivalent conditions:

Every quasi-compact open subset of Y is the image of a quasi-compact open subset of X.
There exists a covering of Y by open affine subschemes such that each is the image of a quasi-compact open subset of X.
Each point has a neighborhood such that is open and is quasi-compact.
Each point has a quasi-compact neighborhood such that is open affine.

Examples: An open faithfully flat morphism is fpqc.
An fpqc morphism satisfies the following properties:

The composite of fpqc morphisms is fpqc.
A base change of an fpqc morphism is fpqc.
If is a morphism of schemes and if there is an open covering of Y such that the is fpqc, then f is fpqc.
A faithfully flat morphism that is locally of finite presentation is fpqc.
If is an fpqc morphism, a subset of Y is open in Y if and only if its inverse image under f is open in X.