Connected and Locally Path-Connected Implies Path Connected

Theorem

Let $T = \struct {S, \tau}$ be a connected and locally path-connected topological space.

Then $T$ is path-connected.

Proof

By:

Path Component of Locally Path-Connected Space 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 $S$.

That is, $T$ is path-connected.

$\blacksquare$