If and , then and .
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 .
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 ,
are well-defined real-valued continuous functions as well (in fact, it is enough to say that for any continuous function , its absolute value is also continuous, so that
are both continuous as well).
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.
|Date of creation||2013-03-22 15:47:26|
|Last modified on||2013-03-22 15:47:26|
|Last modified by||CWoo (3771)|