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Hometopological lattice
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topological lattice
A 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 $x_{i}\to x$.
Remarks

If $(x_{i})$ and $(y_{j})$ are nets, indexed by $I,J$ respectively, then $(x_{i}\wedge y_{j})$ and $(x_{i}\vee y_{j})$ are nets, both indexed by $I\times J$. This is clear, and is stated in preparation for the proposition below.

If $x_{i}\to x$ and $y_{j}\to y$, then $x_{i}\wedge y_{j}\to x\wedge y$ and $x_{i}\vee y_{j}\to x\vee y$.
Proof.
Let’s show the first convergence, and the other one follows similarly. The function $f:x\mapsto(x,y)\mapsto x\wedge y$ is a continuous function, being the composition of two continuous functions. If $x\wedge y\in U$ is open, then $x\in f^{{1}}(U)$ is open. As $x_{i}\to x$, there is an $i_{0}\in I$ such that $x_{i}\in f^{{1}}(U)$ for all $i\geq i_{0}$, which means that $x_{i}\wedge y=f(x_{i})\in U$. By the same token, for each $i\in I$, the function $g_{i}:y\mapsto(x_{i},y)\mapsto x_{i}\wedge y$ is a continuous function. Since $x_{i}\wedge y\in U$ is open, $y\in g^{{1}}(U)$ is open. As $y_{j}\to y$, there is a $j_{0}\in J$ such that $y_{j}\in g^{{1}}(U)$ for all $j\geq j_{0}$, or $x_{i}\wedge y_{j}=g_{i}(y_{j})\in U$, for all $i\geq i_{0}$ and $j\geq j_{0}$. Hence $x_{i}\wedge y_{j}\to x\wedge y$. ∎

For any net $(x_{i})$, the set $A=\{a\in L\mid x_{i}\to a\}$ is a sublattice of $L$.
Proof.
If $a,b\in A$, then $x_{i}=x_{i}\wedge x_{i}\to a\wedge b$. So $a\wedge b\in A$. Similarly $a\vee b\in A$. ∎
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 $\mathbb{R}$, with operations defined by $a\wedge b=\operatorname{inf}\{a,b\}$ and $a\vee b=\operatorname{sup}\{a,b\}$, is one such an example. This can be easily generalized to the space of realvalued continuous functions, since, given any two realvalued continuous functions $f$ and $g$,
$f\vee g:=\max(f,g)\mbox{ and }f\wedge g:=\min(f,g)$ 
are welldefined realvalued 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
$\max(f,g)=\max(fg,0)+g\mbox{ and }\min(f,g)=f+g\max(f,g)$ 
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. 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.
Mathematics Subject Classification
06F30 no label found06B30 no label found54H12 no label found Forums
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