Homogeneous space

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:


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

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.