Open map: Difference between revisions

Latest revision as of 02:40, 25 December 2010

This article defines a property of continuous maps between topological spaces

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.

Property	Meaning	Proof of implication	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

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

@@ Line 24: / Line 24: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::quotient map]] || || [[open map implies quotient map]] || || {{intermediate notions short|quotient map|open map}}
+| [[Stronger than::quotient map]] (if surjective) || || [[surjective open map implies quotient map]] || || {{intermediate notions short|quotient map|open map}}
 |-
 | [[Stronger than::inductively open map]] || || || || {{intermediate notions short|inductively open map|open map}}
 |}