Topological manifolds are locally compact and locally path connected

Let $O$ be any open neighbourhood of $p$. By [F1] choose a chart [F1, choose] $(U,\phi )$ at $p$ and an open Euclidean ball $B(c,r)$ such that $p\in {\phi }^{-1}[B(c,r)]\subseteq O$ and the closure of ${\phi }^{-1}[B(c,r)]$ in $M$ is compact. This already gives a compact neighbourhood basis at $p$, so $M$ is locally compact.

If the manifold dimension is $n=0$, then every point is open, so $M$ is [L1, A1, step 1.1] locally path connected trivially. If $n\ge 1$, then [L1] says the Euclidean ball $B(c,r)$ is path connected. Since $\phi$ is a homeomorphism on $U$, [A1] makes ${\phi }^{-1}[B(c,r)]$ path connected. Thus every neighbourhood of $p$ contains an open path-connected neighbourhood of $p$.

Step 1.1 proves local compactness and step 2.1 proves local path [step 1.1, step 2.1, L2] connectedness. The role of [L2] is only to justify that the compact-neighbourhood conclusion in step 1.1 matches the Euclidean local model used to produce the coordinate balls.