Path-Connectedness is Preserved under Homeomorphism
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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.
$\blacksquare$