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.

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

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

• If ${\displaystyle f}$is continuous, it need not be quasi-open. Conversely if ${\displaystyle f}$is quasi-open, it need not be continuous.
• If ${\displaystyle f}$is open, then ${\displaystyle f}$is quasi-open.
• If ${\displaystyle f}$is a local homeomorphism, then ${\displaystyle f}$is quasi-open.
• The composition of two quasi-open maps is again quasi-open.

• 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 spacePages 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