Homeomorphism may Exist between Non-Comparable Topologies

Theorem
Let $S$ be a set.

Let $T_1 = \struct {S, \tau_1}$ and $T_2 = \struct {S, \tau_2}$ be topological spaces defined on the underlying set $S$.

Let $\tau_1$ and $\tau_2$ be non-comparable.

Then it may possibly be the case that $T_1$ and $T_2$ are homeomorphic.

Proof
A counterexample is demonstrated in Homeomorphic Non-Comparable Particular Point Topologies.