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

where $x^\succeq$ denotes the upper closure of $x$.

Thus: