|
|
|
Revision difference : topological lattice |
| Version 5 |
Version 4 |
| 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. |
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 continous 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 continous functions, since, given any two real-valued continuous functions $f$ and $g$, $f\vee g:=\max(f,g)$ and $f\wedge g:=\min(f,g)$ are well-defined real-valued continuous functions as well.
|
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 continous 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 continous functions, since, given any two real-valued continuous functions $f$ and $g$, $f\vee g:=\max(f,g)$ and $f\wedge g:=\min(f,g)$ are well-defined real-valued continuous functions as well (of course, it is enough to say that the absolute value of a continous function $f$ is continous).
|
|
|
| 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. |
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. |
|
|
|
|
