Equivalence of Definitions of Connected Manifold

Theorem
Let $M$ be a topological manifold.

Definition 1 implies Definition 2
Let $M$ be connected.

From Topological Manifold is Locally Path-Connected:
 * $M$ is locally path-connected.

From Connected and Locally Path-Connected Implies Path Connected:
 * $M$ is path-connected.

Definition 2 implies Definition 1
Follows immediately from Path-Connected Space is Connected.