Connected and Locally Path-Connected Implies Path Connected

Theorem
Let $T$ be a connected and locally path-connected topological space.

Then $T$ is path-connected.

Proof
By:
 * Path Component is Closed
 * Path Component of Locally Path-Connected Space is Open

the path components of $T$ are clopen.

Because $T$ is connected, every path component equals $T$.

That is, $T$ is path-connected.