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.[1]
Definition
A function f : X → YScript error: No such module "Check for unknown parameters". between topological spaces Template:Mvar and Template:Mvar is quasi-open if, for any non-empty open set U ⊆ XScript error: No such module "Check for unknown parameters"., the interior of f ('U)Script error: No such module "Check for unknown parameters". in Template:Mvar is non-empty.[1][2]
Properties
Let be a map between topological spaces.
- If is continuous, it need not be quasi-open. Conversely if is quasi-open, it need not be continuous.[1]
- If is open, then is quasi-open.[1]
- If is a local homeomorphism, then is quasi-open.[1]
- The composition of two quasi-open maps is again quasi-open.[note 1][1]
See also
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Notes
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- ↑ This means that if and are both quasi-open (such that all spaces are topological), then the function composition is quasi-open.
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References
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