Manifolds have a Countable Basis of Precompact Coordinate Balls

Last updated Nov 1, 2022

Let $M$ be a Topological Manifold of Manifold Dimension $n$. Then there exists a Topological Basis, $\mathcal{B}$, for $M$ s.t.

1. $\mathcal{B}$ is Countable
2. Each $B \in \mathcal{B}$ is a Coordinate Ball
3. Each $B \in \mathcal{B}$ is Precompact

TODO

• Precompact
• Continuous Open Maps transfer Bases
• Homeomorphisms transfer Bases
• A Topological Basis can be thinned out by a smaller one
• Euclidean Space is Second Countable
• Euclidean Space has a Countable Precompact Basis of Balls
• Euclidean Balls are Precompact
• Homeomorphisms preserve Precompactness
• Basis Gluing Lemma