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 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:
 * $B_n = \phi_n \sqbrk {U_n}$

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

From Closed Ball is Path-Connected:
 * $B_n$ is path-connected

From Inverse of Homeomorphism is Homeomorphism:
 * $\phi^{-1}$ is a homeomorphism

and
 * $U_n = \phi^{-1} \sqbrk B_n$

From Continuous Image of Path-Connected Set is Path-Connected: Metric Space:
 * $U_n$ is path-connected

Hence $m$ has a local basis of path-connected sets.

Since $m$ was arbitrary, it follows that $M$ is locally path-connected by definition.