Lower Topology is Unique

Theorem
Let $T_1 = \left({S, \preceq, \tau_1}\right)$ and $T_2 = \left({S, \preceq, \tau_2}\right)$ be relational structures with lower topologies.

Then $\tau_1 = \tau_2$.

Proof
Define $B = \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$

Thus