show that A $T_2$-space is pathwise connected iff it is arcwise connected.
This is Corollary. 31.6 in the book called general topology by Willard the author write
Proof. By the theorem 31.5 and Theorem 31.2, every path is arcwise connected. ■
Theorem 31.5 : - A Hausdorff space $X$ is a continuous image of the unit interval $I$ iff it is a Peano space.
Theorem 31.2: Every Peano space is arcwise connected.
I try to prove it in details .