Local homeomorphism: Difference between revisions

Latest revision as of 02:39, 25 December 2010

This article defines a property of continuous maps between topological spaces

Definition

Let $X$ and $Y$ be topological spaces. A continuous map $f:X\to Y$ is termed a local homeomorphism if the following are true:

It is an open map
Every $x\in X$ has an open neighbourhood $U$ such that $f|_{U}$ is a homeomorphism to its image and $f(U)$ is itself an open subset of $Y$ .

Some variants of the definition of local homeomorphism also require the map to be surjective.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
covering map				\|FULL LIST, MORE INFO
etale map	(also sometimes called a sheaf map, though that term has other meanings)			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
open map	image of any open subset of the domain is open	local homeomorphism implies open map		\|FULL LIST, MORE INFO
quotient map (if surjective)

@@ Line 6: / Line 6: @@
 * It is an [[open map]]
-* Every <math>x \in X</math> has an open neighbourhood <math>U</math> such that <math>f|_U</math> is a homeomorphism to its image
+* Every <math>x \in X</math> has an open neighbourhood <math>U</math> such that <math>f|_U</math> is a homeomorphism to its image and <math>f(U)</math> is itself an open subset of <math>Y</math>.
+Some variants of the definition of local homeomorphism also require the map to be surjective.
 ==Relation with other properties==
 ===Stronger properties===
-* [[Covering map]]
+{| class="sortable" border="1"
-* [[Etale map]] (also sometimes called a sheaf map, though that term has other meanings)
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::covering map]] || || || || {{intermediate notions short|local homeomorphism|covering map}}
+|-
+| [[Weaker than::etale map]] || (also sometimes called a sheaf map, though that term has other meanings) || || || {{intermediate notions short|local homeomorphism|etale map}}
+|}
 ===Weaker properties===
-* [[Open map]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::open map]] || image of any open subset of the domain is open || [[local homeomorphism implies open map]] || || {{intermediate notions short|open map|local homeomorphism}}
+|-
+| [[quotient map]] (if surjective)|| || || ||
+|}