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
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 is said to be homogeneous if it satisfies the following equivalent conditions:
- For any points , there is a homeomorphism such that .
- The self-homeomorphism group of is transitive on .
- 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.
- The real line, Euclidean space, connected manifolds, and the underlying spaces of topological groups are all homogeneous.
- The closed interval is not homogeneous, because there is no homeomorphism sending to any point in the open interval .
- A pair of intersecting lines is not homogeneous because every homeomorphism fixes the point of intersection.
Relation with other properties
|property||quick description||proof of implication||proof of strictness (reverse implication failure)||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|