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| Title of object: |
topological lattice |
| Canonical Name: |
TopologicalLattice |
| Type: |
Definition |
| Created on: |
2006-03-20 18:39:52 |
| Modified on: |
2007-01-27 12:33:54 |
| Classification: |
msc:06B30, msc:06F30 |
Preamble:
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Content:
A \emph{topological lattice} is a lattice $L$ equipped with a topology $\mathcal{T}$ such that the meet and join operations from $L\times L$ (with the product topology) to $L$ are continuous.
There are two approaches in 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 $\mathbb{R}$, with operations defined by $a\wedge b= \operatorname{inf}\lbrace a,b\rbrace$ and $a\vee b =\operatorname{sup}\lbrace a,b\rbrace$, 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 $g$,
$$f\vee g:=\max(f,g)\mbox{ and }f\wedge g:=\min(f,g)$$
are well-defined real-valued continuous functions as well (in fact, it is enough to say that for any continuous function $f$, its absolute value $|f|$ is also continuous, so that $$\max(f,0)=\frac{1}{2}(f+|f|),$$ and thus $$\max(f,g)= \max(f-g,0)+g\mbox{ and }\min(f,g)=f+g-\max(f,g)$$ are both continuous as well).
The second approach is to start with a general lattice $L$ and define a topology $\mathcal{T}$ on the subsets of the underlying set of $L$, with the hope that both $\vee$ and $\wedge$ are continuous under $\mathcal{T}$. The obvious example using this second approach is to take the discrete topology of the underlying set. Of course, finding a topology on the underlying set of a lattice may not guarantee a topological lattice. |
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