Quasi-open map

Short description: Generalization of open map in topology

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 $f:X\to Y$ between topological spaces is called quasi-open if, for any nonempty open set $U\subseteq X$, the interior of $f(U)$ in $Y$ is nonempty. Such a function has also been called a quasi-interior map.

Properties

Let $f:X\to Y$ be a map between topological spaces.

• If $f$ is continuous, it need not be quasi-open. For example, the constant map $f:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $f(x)=0$ is continuous but not quasi-open.
• Conversely, if $f$ is quasi-open, it need not be continuous. For example, the map $f:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $f(x)=x$ if $x<0$ and $f(x)=x+1$ if $x\ge 0$ is quasi-open but not continuous.
• If $f$ is open, then $f$ is quasi-open. The converse is not true in general. For example, the continuous function $f:\mathbb{ℝ}\to \mathbb{ℝ},x\mapsto \sin (x)$ is quasi-open but not open.
• If $f$ is a local homeomorphism, then $f$ is quasi-open.
• The composition of two quasi-open maps is quasi-open.

See also

• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Closed graph – Graph of a map closed in the product space
• Closed linear operator – Linear operator whose graph is closed
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Proper map – Map between topological spaces with the property that the preimage of every compact is compact