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 $\R^d$.

Let $\phi: U \to V$ be a homeomorphism.

By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the topology induced by the metric:
 * $\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$

where $x = \tuple {x_1, x_2, \ldots, x_d}, y = \tuple {y_1, y_2, \ldots, y_d} \in \R^d$.

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

where $\map {B_\epsilon} {\map \phi m}$ is the open ball of radius $\epsilon$ and center $\map \phi m$

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

From Null Sequence induces Local Basis in Metric Space:
 * $\BB_m$ is a countable local basis of $\map \phi m$ in $\R^d$

From Local Basis of Open Subspace iff Local Basis:
 * $\BB_m$ is a countable local basis of $\map \phi m$ in $V$

For each $n \in \N_{>0}$, let:
 * $U_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}} {\map \phi m}}$

By definition of continuity:
 * $\forall n \in \N_{>0}: U_n$ is an open neighbourhood of $m$

Consider the set of open neighbourhoods of $m$:
 * $\BB'_m = \set{U_n : n \in \N_{>0}}$

By definition of countable set:
 * $\BB'_m$ is a countable set

From Homeomorphic Image of Local Basis is Local Basis:
 * $\BB'_m$ is a local basis of $m$ in $U$

From Local Basis of Open Subspace iff Local Basis:
 * $\BB'_m$ is a local basis of $m$ in $M$

From Restriction of Homeomorphism is Homeomorphism:
 * $\forall n \in \N_{>0}: U_n = \phi^{-1} \sqbrk {\map {B_{\epsilon / n}} {\map \phi m}}$ is homeomorphic to the open ball $\map {B_{\epsilon / n}} {\map \phi m}$

The result follows.