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

Properties

Let f : X → Y {\displaystyle f:X\to Y} be a map between topological spaces.

  • If f {\displaystyle f} is continuous, it need not be quasi-open. For example, the constant map f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 0 {\displaystyle f(x)=0} is continuous but not quasi-open.
  • Conversely, if f {\displaystyle f} is quasi-open, it need not be continuous. For example, the map f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = x {\displaystyle f(x)=x} if x < 0 {\displaystyle x<0} and f ( x ) = x + 1 {\displaystyle f(x)=x+1} if x ≥ 0 {\displaystyle x\geq 0} is quasi-open but not continuous.
  • If f {\displaystyle f} is open, then f {\displaystyle f} is quasi-open. The converse is not true in general. For example, the continuous function f : R → R , x ↦ sin ⁡ ( x ) {\displaystyle f:\mathbb {R} \to \mathbb {R} ,x\mapsto \sin(x)} is quasi-open but not open.
  • If f {\displaystyle f} is a local homeomorphism, then f {\displaystyle 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 – Property of functions in topologyPages displaying short descriptions of redirect targets
  • Closed linear operator – Linear operator whose graph is closed
  • Open and closed maps – Functions that send open (resp. closed) subsets to open (resp. closed) subsets
  • Proper map – Mathematical map between topological spaces
  • Quotient map (topology) – Topological space constructionPages displaying short descriptions of redirect targets

Notes