Equivalent Definitions of Locally Euclidean

Last updated Nov 1, 2022

Let $M$ be a Topological Space. Then The following are Equivalent:

1. $M$ is Locally Euclidean of dimension $n$
2. For each $p \in M$, there exists a Coordinate Chart containing $p$ to $\mathbb{R}^{n}$
3. For each $p \in M$, there exists a Coordinate Chart containing $p$ to $B(\mathbb{R}^{n})$

$(2) \Rightarrow (1)$: Since $\mathbb{R}^{n}$ is Open (by definition of a Topological Space), we have for each $p \in M$ an Open $U \ni p$ and Open $\hat{U} = \mathbb{R}^{n}$ that is homeomorphic to $U$. Thus $M$ is Locally Euclidean of dimension $n$. $\checkmark$

$(3) \Rightarrow (2)$: This follows because The Euclidean Open Unit Ball is Homeomorphic to the Euclidean Space. $\checkmark$

$(1) \Rightarrow (3)$: Let $p \in M$ and let $(U, \varphi)$ be a Coordinate Chart for $p$ to $\hat{U} \subset \mathbb{R}^{n}$. Such a Coordinate Chart exists because $M$ is Locally Euclidean. Let $\hat{p} := \varphi_{p}(p)$. Since $\hat{U} := \varphi_{p}(U_{p})$ is Open in $\mathbb{R}^{n}$, by definition of the Metric Topology, there exists some radius $\epsilon > 0$ so $B_{\epsilon}(\hat{p}) \subset \hat{U}$. Since $\varphi_{p}$ is a Continuous Function (by definition of Homeomorphism), $B := \varphi_{p}^{-1}(B_{\epsilon}(\hat{p})) \subset M$ is Open and contains $p$. Thus, after noting that The Restriction of Homeomorphism is a Homeomorphism in the Subspace Topology we can construct Coordinate Chart $(B, \varphi_{p} {\big|}_{B})$ containing $p \in M$. Every Open Ball is Homeomorphic to the Open Unit Ball in the Euclidean Space, so we can create a Coordinate Chart containing $p$ to the Open Unit Ball by Function Composition. Since $p \in M$ was arbitrary, we have established $(3)$. $\checkmark$ $\blacksquare$

• Euclidean Space
• Open Unit Ball