Definition (order topology):
Let be poset. The order topology on is the topology which has as a subbasis the sets
Proposition (half-open intervals on lattices form a topology base):
Let be a lattice. Then the sets
form a -system; in particular, they form a topology base.
Proof: We have
Theorem (Weierstraß-type theorem):
Let be a compact topological space, and let be a lattice. Let be continuous with respect to the order topology on . Then is bounded in .
Proof: The sets
form an open cover of , where range over . By compactness, we may find a finite subcover
so that maps every point in into the latter interval.