Connected manifold implies homogeneous

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., connected manifold) must also satisfy the second topological space property (i.e., homogeneous space)
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Statement

Any connected manifold is homogeneous, viz given a connected manifold and two points in it, there is a self-homeomorphism of the manifold that takes the first point to the second.

Proof

Proof outline

The proof involves three steps:

We show that given any two points inside a closed disk in Euclidean space, there is a homeomorphism of the disc that takes one point to the other and is identity on the boundary.
We use the gluing lemma for closed subsets to show that if two points lie inside a Euclidean open subset of the manifold, then there is a homeomorphism of the manifold that takes one point to the other. To apply the gluing lemma for closed subsets, we use the fact that the disc is compact, and that the manifold is Hausdorff, and hence the disc is a closed subset of the manifold.
We finally use the fact that the manifold is locally Euclidean to show that the orbit of any point under the action of the self-homeomorphisms is both open and closed, and then use connectedness of the manifold to show that it is the whole manifold.

A further abstraction

The proof outline can be abstracted by defining the notion of a compactly homogeneous space -- a space in which given any two points, there is a homeomorphism between them that fixes the complement of a compact set. The above proof then generalizes to the fact that a connected homogeneous space in which every point is contained in a compactly homogeneous open set, is itself compact homogeneous.