Poset

We say is a poset if is a Partial Order on . The Topology on is given by base open sets of the form

for some . In particular a set in is open if and only if it is downwards closed.

Chain Antichain Upper Bound Least Upper Bound Complete Poset Upward Closure Knaster-Tarski fixed point theorem Zorn’s Lemma

Compatible Incompatible Strong Antichain Maximal Strong Antichain Filter Base Filter Ultrafilter Chain Condition