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 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 $\R^d$.

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.