Open map
This article defines a property of continuous maps between topological spaces
Definition
A continuous map of topological spaces is termed an open map if the image of any open subset of the domain space is an open subset of the range space.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| homeomorphism | a continuous bijective map with continuous inverse | homeomorphism implies open | Covering map|FULL LIST, MORE INFO | |
| covering map | a continuous surjective map that locally looks like a product with a discrete space | covering map implies open | Local homeomorphism|FULL LIST, MORE INFO | |
| local homeomorphism | a continuous map such that every point has an open neighborhood to which the restriction of the map is a homeomorphism with the image again being open | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| quotient map (if surjective) | surjective open map implies quotient map | |FULL LIST, MORE INFO | ||
| inductively open map | |FULL LIST, MORE INFO |