where , then there is a finite subset of , such that
whenever a subset of such that exists and , then there is a finite subset such that .
If we let to be the collection of closed subsets of , and partial order by inclusion, then becomes a lattice with meet and join defined by set theoretic intersection and union. It is easy to see that an element is a compact element iff is a compact closed subset in .
Here are some other common examples:
Let be a set and the subset lattice (power set) of . The compact elements of are the finite subsets of .
Note in all of the above examples, atoms are compact. However, this is not true in general. Let’s construct one such example. Adjoin the symbol to the lattice of natural numbers (with linear order), so that for all . So is the top element of (and is the bottom element!). Next, adjoin a symbol to , and define the meet and join properties with by
, for all , and
The resulting set is a lattice where is a non-compact atom.
As we have seen from the examples above, compactness is closely associated with the concept of finiteness, a compact element is sometimes called a finite element.
Any finite join of compact elements is compact.
An element in a lattice is compact iff for any directed (http://planetmath.org/DirectedSet) subset of such that exists and , then there is an element such that .
A compact element may be defined in an arbitrary poset : is compact iff is way below itself: .
- 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
|Date of creation||2013-03-22 15:52:50|
|Last modified on||2013-03-22 15:52:50|
|Last modified by||CWoo (3771)|