Quasi-open map

Template:Short description 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.

Definition

A function Template:Math between topological spaces Template:Mvar and Template:Mvar is quasi-open if, for any non-empty open set Template:Math, the interior of Template:Math in Template:Mvar is non-empty.

Properties

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

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

See also

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References

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