Definition:Lower Topology

Definition
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology.

Then $T$ has lower topology
 * $\left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$ is sub-basis of $T$

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