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.^[1]

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.^[1][2]

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.^[1]
• If ${\displaystyle f}$ is open, then ${\displaystyle f}$ is quasi-open.^[1]
• If ${\displaystyle f}$ is a local homeomorphism, then ${\displaystyle f}$ is quasi-open.^[1]
• The composition of two quasi-open maps is again quasi-open.^{[note 1][1]}

• Almost open map – Map that satisfies a condition similar to that of being an open map.
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