General Topology/Order topology and semicontinuity

order topology Let be poset. The order topology on is the topology which has as a subbasis the sets . 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. We have . 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 . The sets form an open cover of , where range over . By compactness, we may find a finite subcover . But , so that maps every point in into the latter interval.