Quasi-open map

In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps. However, continuous maps and quasi-open maps are not related.

Definition

A function $f : X \to Y$ between topological spaces $X$ and $Y$ is quasi-open if, for any non-empty open set $U \subset X$, the interior of $f(U)$ in $Y$ is non-empty.

Properties

Let $f: X \to Y$ be a function such that X and Y are topological spaces.

• If $f$ is continuous, it need not be quasi-open. Conversely if $f$ is quasi-open, it need not be continuous.
• If $f$ is open, then $f$ is quasi-open.
• If $f$ is a local homeomorphism, then $f$ is quasi-open.
• If $f: X \to Y$ and $g: Y \to Z$ are both quasi-open (such that all spaces are topological), then the function composition $h = g \circ f: X \to Z$ is quasi-open.