Path-Connectedness is Preserved under Homeomorphism

Theorem
Let $\struct {T_1, \tau_1}, \struct {T_2, \tau_2}$ be topological spaces.

Let $\phi: T_1 \to T_2$ be a homeomorphism.

Let $S \subseteq T_1$ be a subset of $T_1$.

Let $S$ be path-connected in $\struct {T_1, \tau_1}$.

Then $\phi \sqbrk S$ is path-connected in $\struct {T_2, \tau_2}$.

That is, path-connectedness is a topological property.

Proof
By definition of homeomorphism, $\phi$ is a continuous mapping.

The result now follows from Continuous Image of Path-Connected Set is Path-Connected.