Locally Euclidean Space is Locally Path-Connected

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

Then $M$ is locally path-connected.

Proof
Let $m \in M$ be arbitrary.

From User:Leigh.Samphier/Topology/Element of Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls:
 * there exists a local basis $\family{U_n}_{n \in \N}$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$.

For all $n \in \N$, let:
 * $B_n = \phi \sqbrk {U_n}$

where $B_n$ is an open ball of $\R^d$ and $\phi: U_n \to B_n$ is a homeomorphism