Category of open sets

The category of open sets $$\operatorname{Op}(X)$$ of a topological space $$X$$ is the poset category of the inclusion poset of $$X$$, i.e. the poset given by $$U \leq V$$ if $$U \subseteq V$$.

In other words, the objects of $$\operatorname{Op}(X)$$ are the open sets of X, and the morphisms are the inclusion functions $$i : U \hookrightarrow V$$.