Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism

Theorem
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$.

Let $H_1$ and $H_2$ be a homeomorphic.

Then it may be the case that there does not exist a homeomorphism $g: T_1 \to T_2$ such that:
 * $g \restriction_{H_1} = f$

where:
 * $g \restriction_{H_1}$ is the restriction of $g$ to $H_1$
 * $f: H_1 \to H_2$ is a homeomorphism.