Topology/Manifolds

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Definition (Topological Manifold) A topological space M is called an n-dimensional topological manifold if,

1. Every point has an open neighbourhood , that is homeomorphic to an n-dimensional open Euclidean ball Bⁿ.
2. M is Hausdorff.

Note: As a convention, the ball B⁰ is a single point. Any space with the discrete topology is a 0-dimensional manifold.

Note that all topological manifolds are clearly locally connected.

[edit] Theorem

A topological manifold is connected if an only if it is pathwise connected.

[edit] Proof

Since all topological manifolds are clearly clearly locally connected, the theorem immediately follows.

[edit] Whitney Embedding Theorem