A topological lattice is a lattice equipped with a topology such that the meet and join operations from (with the product topology) to are continuous.

Let be a net in . We say that converges to if is eventually in any open neighborhood of , and we write .

Remarks

• If and are nets, indexed by respectively, then and are nets, both indexed by . This is clear, and is stated in preparation for the proposition below.
• If and , then and .
  Proof. Let's show the first convergence, and the other one follows similarly. The function is a continuous function, being the composition of two continuous functions. If is open, then is open. As , there is an such that for all , which means that . By the same token, for each , the function is a continuous function. Since is open, is open. As , there is a such that for all , or , for all and . Hence .
• For any net , the set is a sublattice of .
  Proof. If , then . So . Similarly .

There are two approaches to finding examples of topological lattices. One way is to start with a topological space such that is partially ordered, then find two continuous binary operations on to form the meet and join operations of a lattice. The real numbers , with operations defined by and , is one such an example. This can be easily generalized to the space of real-valued continuous functions, since, given any two real-valued continuous functions and ,

and

and

The second approach is to start with a general lattice and define a topology on the subsets of the underlying set of , with the hope that both and are continuous under . The obvious example using this second approach is to take the discrete topology of the underlying set. Another way is to impose conditions, such as requiring that the lattice be meet and join continuous. Of course, finding a topology on the underlying set of a lattice may not guarantee a topological lattice.