|
|
|
Viewing Version
12
of
'topological lattice'
|
[ view 'topological lattice'
|
back to history
]
| Title of object: |
topological lattice |
| Canonical Name: |
TopologicalLattice |
| Type: |
Definition |
| Created on: |
2006-03-20 18:39:52 |
| Modified on: |
2007-01-31 01:06:35 |
| Classification: |
msc:06B30, msc:06F30 |
Preamble:
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% define commands here |
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.
Let $(x_i)_{i\in I}$ be a net in $L$. We say that $(x_i)$ converges to $x\in L$ if $(x_i)$ is eventually in any open neighborhood of $x$, and we write $x_i\to x$.
% Since meet and join are both continuous, this means that if $(x_i)$ and $(y_i)$ are nets with the same domain $I$, then
% \begin{center}
% $x_i\to x$ and $y_i\to y$ imply $x_i\wedge y_i\to x\wedge y$ and $x_i \vee y_i\to x \vee y$.
% \end{center}
% To see this, say $x_i\wedge y_i\to x\wedge y$, let's break this down into two steps: $x_i\wedge y_i\to x\wedge y_i$ and $x\wedge y_i\to x\wedge y$. note that the maps $f: x\mapsto (x,y)\mapsto x\wedge y$ and $g: y\mapsto (x,y)\mapsto x\wedge y$ are both compositions of two continuous functions, and hence are continuous themselves. Any open set $U$ of $x\wedge y$ corresponds to open sets $f^{-1}(U)$ of $x$, and $g^{-1}(U)$ of $y$. Since $(x_i)$ is eventually in $f^{-1}(U)$, $(f(x_i))=(a\wedge x_i)$, is eventually in $U$, implying $a\wedge x_i \to a\wedge x$.
% In particular, if $x_i\to \bigvee_{i\in I} \lbrace x_i\rbrace$,
\textbf{Remarks}
\begin{itemize}
\item if $(x_i)_{i\in I}$ and $(y_j)_{j\in J}$ are nets, then so are $(x_i\wedge y_j)_{(i,j)\in I\times J}$ and $(x_i\vee y_j)_{(i,j)\in I\times J}$ nets.
\item if $x_i\to x$ and $y_j\to y$, then $x_i\wedge y_j\to x\wedge y$ and $x_i \vee y_j\to x\vee y$.
\item for any net $(x_i)$, the set $A=\lbrace a\in L \mid x_i\to a\rbrace$ is a sublattice of $L$.
\item if $L$ is Hausdorff, then $A$ is at most a singleton.
%\item if $x_i\to x$ where $x=\bigvee x_i$, then $y_j=\bigvee_{i\le j}x_i$ form a net and $y_j\to x$.
%\item if $x_i\to \bigvee x_i$, then $a\wedge x_i\to \bigvee (a\wedge x_i)$ for any $a\in L$.
%\item if $L$ is a complete Hausdorff topological lattice, then $L$ is meet continuous and join continuous.
\end{itemize}
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 $\mathbb{R}$, 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. |
|
|
|
|
|
