Quasi-open map

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 ${\displaystyle f:X\to Y}$ between topological spaces is called quasi-open if, for any nonempty open set ${\displaystyle U\subseteq X}$, the interior of ${\displaystyle f(U)}$ in ${\displaystyle Y}$ is nonempty.^[1][2] Such a function has also been called a quasi-interior map.^[3]

Properties

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

• If ${\displaystyle f}$ is continuous, it need not be quasi-open. For example, the constant map ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle f(x)=0}$ is continuous but not quasi-open.
• Conversely, if ${\displaystyle f}$ is quasi-open, it need not be continuous. For example, the map ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ defined by ${\displaystyle f(x)=x}$ if ${\displaystyle x<0}$ and ${\displaystyle f(x)=x+1}$ if ${\displaystyle x\geq 0}$ is quasi-open but not continuous.
• If ${\displaystyle f}$ is open, then ${\displaystyle f}$ is quasi-open.^[2] The converse is not true in general. For example, the continuous function ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,x\mapsto \sin(x)}$ is quasi-open but not open.
• If ${\displaystyle f}$ is a local homeomorphism, then ${\displaystyle f}$ is quasi-open.^[4]
• The composition of two quasi-open maps is quasi-open.^{[note 1][2]}

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 – 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
• Quotient map (topology) – Topological space constructionPages displaying short descriptions of redirect targets

Notes

1. This means that if ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ are both quasi-open, then the function composition ${\displaystyle g\circ f:X\to Z}$ is quasi-open.