Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces, let $f: X \to Y$ be a homeomorphism, and let $A \subset X$. Equip $A$ with the [subspace topology](/page/Subspace%20Topology) inherited from $X$, and equip $f(A)$ with the subspace topology inherited from $Y$. Then $A$ is connected if and only if $f(A)$ is connected.