Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls

Theorem
Let $M$ be a locally Euclidean space of some dimension $d$.

Let $m \in M$.

Then:
 * there exists a countable local basis $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$.

Proof
By definition of a locally Euclidean space:
 * there exists an open neighbourhood $U$ of $m$ which is homeomorphic to an open subset $V$ of Euclidean space $\R^d$.

By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the induced by the metric:

By definition of the induced topology:
 * $\exists \epsilon > 0 : \map {B_\epsilon} {\map \phi m} \subseteq V$

Consider the set of open balls:
 * $\ds \BB_m = \set{\map {B_{\dfrac \epsilon n}} {\map \phi m} : n \in \N_{>0}}$

lemma 1

 * $\BB_m$ is a countable local basis of $\map \phi m$

Now if $\phi$ is a homeomorphism $U \to \R^d$, then by definition $\phi^{-1}$ is continuous.

Therefore by Continuous Image of Compact Space is Compact, $\phi^{-1} \sqbrk C \subseteq M$ is compact.

Furthermore $m \in \phi^{-1} \sqbrk C$ because $\map \phi m \in C$.

Thus every point of $M$ has a compact neighborhood.