Homogeneous space: Difference between revisions

Latest revision as of 15:00, 24 October 2009

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

Symbol-free definition

A topological space is said to be homogeneous if given any two points in it, there is a homeomorphism from the topological space to itself that maps the first point to the second. In other words, the self-homeomorphism group is transitive on the points of the topological space.

Definition with symbols

A topological space $X$ is said to be homogeneous if it satisfies the following equivalent conditions:

For any points $x,y\in X$ , there is a homeomorphism $\varphi :X\to X$ such that $\varphi (x)=y$ .
The self-homeomorphism group of $X$ is transitive on $X$ .

Examples

Extreme examples

The empty space is homogeneous for trivial reasons.
The one-point space is homogeneous for trivial reasons.
The discrete topology and trivial topology both give homogeneous spaces.

Mainstream examples

The real line, Euclidean space, connected manifolds, and the underlying spaces of topological groups are all homogeneous.

Non-examples

The closed interval $[0,1]$ is not homogeneous, because there is no homeomorphism sending $0$ to any point in the open interval $(0,1)$ .
A pair of intersecting lines is not homogeneous because every homeomorphism fixes the point of intersection.

Relation with other properties

Stronger properties

property	quick description	proof of implication	intermediate notions
Underlying space of topological group	underlying space for a topological group	underlying space of topological group implies homogeneous
Underlying space of T0 topological group	underlying space for a T0 topological group
Connected manifold		connected manifold implies homogeneous	\|FULL LIST, MORE INFO
Compactly homogeneous space

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-A topological space is said to be '''homogeneous''' if given any two points in it, there is a homeomorphism from the topological space to itself that maps the first point to the second.
+A topological space is said to be '''homogeneous''' if given any two points in it, there is a homeomorphism from the topological space to itself that maps the first point to the second. In other words, the [[self-homeomorphism group]] is transitive on the points of the topological space.
+===Definition with symbols===
+A topological space <math>X</math> is said to be '''homogeneous''' if it satisfies the following equivalent conditions:
+# For any points <math>x,y \in X</math>, there is a homeomorphism <math>\varphi:X \to X</math> such that <math>\varphi(x) = y</math>.
+# The [[self-homeomorphism group]] of <math>X</math> is transitive on <math>X</math>.
+==Examples==
+===Extreme examples===
+* The [[empty space]] is homogeneous for trivial reasons.
+* The [[one-point space]] is homogeneous for trivial reasons.
+* The discrete topology and trivial topology both give homogeneous spaces.
+===Mainstream examples===
+* The real line, [[Euclidean space]], [[connected manifold]]s, and the underlying spaces of [[topological group]]s are all homogeneous.
+===Non-examples===
+* The closed interval <math>[0,1]</math> is not homogeneous, because there is no homeomorphism sending <math>0</math> to any point in the open interval <math>(0,1)</math>.
+* A pair of intersecting lines is not homogeneous because every homeomorphism fixes the point of intersection.
 ==Relation with other properties==
@@ Line 12: / Line 35: @@
 {| class="wikitable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions !! comparison
+! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 |-
 | [[Weaker than::Underlying space of topological group]] || underlying space for a [[topological group]] || [[underlying space of topological group implies homogeneous]] || ||
 |-
-| [[Weaker than::Underlying space of T0 topological group]] || underlying space for a [[T0 topological group]] ||
+| [[Weaker than::Underlying space of T0 topological group]] || underlying space for a [[T0 topological group]] || || ||
 |-
 | [[Weaker than::Connected manifold]] || || [[connected manifold implies homogeneous]] || || {{intermediate notions short|homogeneous space|connected manifold}}