# General Topology/Order topology and semicontinuity

**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

- .

But

- ,

so that maps every point in into the latter interval.