General Topology/Order topology and semicontinuity

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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.