topological lattice
A topological lattice is a lattice^{} $L$ equipped with a topology^{} $\mathrm{\pi \x9d\x92\u2015}$ such that the meet and join operations^{} from $L\Gamma \x97L$ (with the product topology) to $L$ are continuous^{}.
Let ${({x}_{i})}_{i\beta \x88\x88I}$ be a net in $L$. We say that $({x}_{i})$ converges^{} to $x\beta \x88\x88L$ if $({x}_{i})$ is eventually in any open neighborhood of $x$, and we write ${x}_{i}\beta \x86\x92x$.
Remarks

β’
If $({x}_{i})$ and $({y}_{j})$ are nets, indexed by $I,J$ respectively, then $({x}_{i}\beta \x88\S {y}_{j})$ and $({x}_{i}\beta \x88\xa8{y}_{j})$ are nets, both indexed by $I\Gamma \x97J$. This is clear, and is stated in preparation for the proposition^{} below.

β’
If ${x}_{i}\beta \x86\x92x$ and ${y}_{j}\beta \x86\x92y$, then ${x}_{i}\beta \x88\S {y}_{j}\beta \x86\x92x\beta \x88\S y$ and ${x}_{i}\beta \x88\xa8{y}_{j}\beta \x86\x92x\beta \x88\xa8y$.
Proof.
Letβs show the first convergence, and the other one follows similarly. The function $f:x\beta \x86\xa6(x,y)\beta \x86\xa6x\beta \x88\S y$ is a continuous function, being the composition of two continuous functions. If $x\beta \x88\S y\beta \x88\x88U$ is open, then $x\beta \x88\x88{f}^{1}\beta \x81\u2019(U)$ is open. As ${x}_{i}\beta \x86\x92x$, there is an ${i}_{0}\beta \x88\x88I$ such that ${x}_{i}\beta \x88\x88{f}^{1}\beta \x81\u2019(U)$ for all $i\beta \x89\u20af{i}_{0}$, which means that ${x}_{i}\beta \x88\S y=f\beta \x81\u2019({x}_{i})\beta \x88\x88U$. By the same token, for each $i\beta \x88\x88I$, the function ${g}_{i}:y\beta \x86\xa6({x}_{i},y)\beta \x86\xa6{x}_{i}\beta \x88\S y$ is a continuous function. Since ${x}_{i}\beta \x88\S y\beta \x88\x88U$ is open, $y\beta \x88\x88{g}^{1}\beta \x81\u2019(U)$ is open. As ${y}_{j}\beta \x86\x92y$, there is a ${j}_{0}\beta \x88\x88J$ such that ${y}_{j}\beta \x88\x88{g}^{1}\beta \x81\u2019(U)$ for all $j\beta \x89\u20af{j}_{0}$, or ${x}_{i}\beta \x88\S {y}_{j}={g}_{i}\beta \x81\u2019({y}_{j})\beta \x88\x88U$, for all $i\beta \x89\u20af{i}_{0}$ and $j\beta \x89\u20af{j}_{0}$. Hence ${x}_{i}\beta \x88\S {y}_{j}\beta \x86\x92x\beta \x88\S y$. β

β’
For any net $({x}_{i})$, the set $A=\{a\beta \x88\x88L\beta \x88\pounds {x}_{i}\beta \x86\x92a\}$ is a sublattice of $L$.
Proof.
If $a,b\beta \x88\x88A$, then ${x}_{i}={x}_{i}\beta \x88\S {x}_{i}\beta \x86\x92a\beta \x88\S b$. So $a\beta \x88\S b\beta \x88\x88A$. Similarly $a\beta \x88\xa8b\beta \x88\x88A$. β
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 $\mathrm{\beta \x84\x9d}$, with operations defined by $a\beta \x88\S b=\mathrm{inf}\beta \x81\u2018\{a,b\}$ and $a\beta \x88\xa8b=\mathrm{sup}\beta \x81\u2018\{a,b\}$, 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 $g$,
$$f\beta \x88\xa8g:=\mathrm{max}\beta \x81\u2018(f,g)\beta \x81\u2019\text{\Beta and\Beta}\beta \x81\u2019f\beta \x88\S g:=\mathrm{min}\beta \x81\u2018(f,g)$$ 
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
$$\mathrm{max}\beta \x81\u2018(f,0)=\frac{1}{2}\beta \x81\u2019(f+f),$$ 
and thus
$$\mathrm{max}\beta \x81\u2018(f,g)=\mathrm{max}\beta \x81\u2018(fg,0)+g\beta \x81\u2019\text{\Beta and\Beta}\beta \x81\u2019\mathrm{min}\beta \x81\u2018(f,g)=f+g\mathrm{max}\beta \x81\u2018(f,g)$$ 
are both continuous as well).
The second approach is to start with a general lattice $L$ and define a topology $\mathrm{\pi \x9d\x92\u2015}$ on the subsets of the underlying set of $L$, with the hope that both $\beta \x88\xa8$ and $\beta \x88\S $ are continuous under $\mathrm{\pi \x9d\x92\u2015}$. 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  20130322 15:47:26 
Last modified on  20130322 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 