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 $f : X \to Y$ Script 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 $U \subseteq X$ Script 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 $f : X \to Y$ 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.^[1]
If $f$ is open, then $f$ is quasi-open.^[1]
If $f$ is a local homeomorphism, then $f$ is quasi-open.^[1]
The composition of two quasi-open maps is again quasi-open.^{[note 1]}^[1]

Notes

↑ This means that if $f : X \to Y$ and $g : Y \to Z$ are both quasi-open (such that all spaces are topological), then the function composition $g \circ f : X \to Z$ is quasi-open.

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References

↑ ^a ^b ^c ^d ^e ^f Script error: No such module "Citation/CS1".
↑ Script error: No such module "Citation/CS1".

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[3] This means that if $f : X \to Y$ and $g : Y \to Z$ are both quasi-open (such that all spaces are topological), then the function composition $g \circ f : X \to Z$ is quasi-open.

[Kim-1] ↑ ^a ^b ^c ^d ^e ^f Script error: No such module "Citation/CS1".

[2] Script error: No such module "Citation/CS1".

[1]

[2]

[note 1]

Quasi-open map

Contents

Definition

Properties

See also

Notes

References

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Quasi-open map

Definition

Properties

See also

Notes

References

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