In topology, a branch of mathematics, a quasi-open map (also called quasi-interior map) is a function that generalizes the notion of open map.
Definition
A function between topological spaces is called quasi-open if, for any nonempty open set , the interior of in is nonempty.Definition 3 on page 7 Such a function has also been called a quasi-interior map.
Properties
Let be a map between topological spaces. If is continuous, it need not be quasi-open. For example, the constant map defined by is continuous but not quasi-open. Conversely, if is quasi-open, it need not be continuous. For example, the map defined by if and if is quasi-open but not continuous. If is open, then is quasi-open. The converse is not true in general. For example, the continuous function is quasi-open but not open. If is a local homeomorphism, then is quasi-open. The composition of two quasi-open maps is quasi-open. This means that if and are both quasi-open, then the function composition is quasi-open.
See also
Notes
References