topological lattice

A topological lattice is a latticeMathworldPlanetmath L equipped with a topologyMathworldPlanetmath 𝒯 such that the meet and join operationsMathworldPlanetmath from LΓ—L (with the product topology) to L are continuousMathworldPlanetmathPlanetmath.

Let (xi)i∈I be a net in L. We say that (xi) convergesPlanetmathPlanetmath to x∈L if (xi) is eventually in any open neighborhood of x, and we write xiβ†’x.


  • β€’

    If (xi) and (yj) are nets, indexed by I,J respectively, then (xi∧yj) and (xi∨yj) are nets, both indexed by IΓ—J. This is clear, and is stated in preparation for the propositionPlanetmathPlanetmath below.

  • β€’

    If xiβ†’x and yjβ†’y, then xi∧yjβ†’x∧y and xi∨yjβ†’x∨y.


    Let’s show the first convergence, and the other one follows similarly. The function f:x↦(x,y)↦x∧y is a continuous function, being the composition of two continuous functions. If x∧y∈U is open, then x∈f-1⁒(U) is open. As xiβ†’x, there is an i0∈I such that xi∈f-1⁒(U) for all iβ‰₯i0, which means that xi∧y=f⁒(xi)∈U. By the same token, for each i∈I, the function gi:y↦(xi,y)↦xi∧y is a continuous function. Since xi∧y∈U is open, y∈g-1⁒(U) is open. As yjβ†’y, there is a j0∈J such that yj∈g-1⁒(U) for all jβ‰₯j0, or xi∧yj=gi⁒(yj)∈U, for all iβ‰₯i0 and jβ‰₯j0. Hence xi∧yjβ†’x∧y. ∎

  • β€’

    For any net (xi), the set A={a∈L∣xiβ†’a} is a sublattice of L.


    If a,b∈A, then xi=xi∧xiβ†’a∧b. So a∧b∈A. Similarly a∨b∈A. ∎

There are two approaches to finding examples of topological lattices. One way is to start with a topological space X such that X is partially ordered, then find two continuous binary operations on X to form the meet and join operations of a lattice. The real numbers ℝ, with operations defined by a∧b=inf⁑{a,b} and a∨b=sup⁑{a,b}, 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 f and g,

f∨g:=max⁑(f,g)⁒ and ⁒f∧g:=min⁑(f,g)

are well-defined real-valued continuous functions as well (in fact, it is enough to say that for any continuous function f, its absolute valueMathworldPlanetmathPlanetmathPlanetmath |f| is also continuous, so that


and thus

max⁑(f,g)=max⁑(f-g,0)+g⁒ and ⁒min⁑(f,g)=f+g-max⁑(f,g)

are both continuous as well).

The second approach is to start with a general lattice L and define a topology 𝒯 on the subsets of the underlying set of L, with the hope that both ∨ and ∧ are continuous under 𝒯. 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.

Title topological lattice
Canonical name TopologicalLattice
Date of creation 2013-03-22 15:47:26
Last modified on 2013-03-22 15:47:26
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 19
Author CWoo (3771)
Entry type Definition
Classification msc 06F30
Classification msc 06B30
Classification msc 54H12
Related topic OrderedVectorSpace