# compact element

Let $X$ be a set and $\mathcal{T}$ be a topology on $X$, a well-known concept is that of a compact set: a set $A$ is compact if every open cover of $A$ has a finite subcover. Another way of putting this, symbolically, is that if

 $A\subseteq\bigcup\mathcal{S},$

where $\mathcal{S}\subset\mathcal{T}$, then there is a finite subset $\mathcal{F}$ of $\mathcal{S}$, such that

 $A\subseteq\bigcup\mathcal{F}.$

A more general concept, derived from above, is that of a compact element in a lattice. Let $L$ be a lattice and $a\in L$. Then $a$ is said to be compact if

whenever a subset $S$ of $L$ such that $\bigvee S$ exists and $a\leq\bigvee S$, then there is a finite subset $F\subset S$ such that $a\leq\bigvee F$.

If we let $\mathcal{D}$ to be the collection of closed subsets of $X$, and partial order $\mathcal{D}$ by inclusion, then $\mathcal{D}$ becomes a lattice with meet and join defined by set theoretic intersection and union. It is easy to see that an element $A\in\mathcal{D}$ is a compact element iff $D$ is a compact closed subset in $X$.

Here are some other common examples:

1. 1.

Let $C$ be a set and $2^{C}$ the subset lattice (power set) of $C$. The compact elements of $2^{C}$ are the finite subsets of $C$.

2. 2.

Let $V$ be a vector space and $L(V)$ be the subspace lattice of $V$. Then the compact elements of $L(V)$ are exactly the finite dimensional subspaces of $V$.

3. 3.

Let $G$ be a group and $L(G)$ the subgroup lattice of $G$. Then the compact elements are the finitely generated subgroups of $G$.

4. 4.

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 $\infty$ to the lattice $\mathbb{N}$ of natural numbers (with linear order), so that $n<\infty$ for all $n\in\mathbb{N}$. So $\infty$ is the top element of $\mathbb{N}\cup\{\infty\}$ (and $1$ is the bottom element!). Next, adjoin a symbol $a$ to $\mathbb{N}\cup\{\infty\}$, and define the meet and join properties with $a$ by

• $a\vee n=\infty$, $a\wedge n=1$ for all $n\in\mathbb{N}$, and

• $a\vee\infty=\infty$, $a\wedge\infty=a$.

The resulting set $L=\mathbb{N}\cup\{\infty,a\}$ is a lattice where $a$ is a non-compact atom.

Remarks.

• 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 $a$ in a lattice $L$ is compact iff for any directed (http://planetmath.org/DirectedSet) subset $D$ of $L$ such that $\bigvee D$ exists and $a\leq\bigvee D$, then there is an element $d\in D$ such that $a\leq d$.

• As the last example indicates, not all atoms are compact. However, in an algebraic lattice, atoms are compact. The first three examples are all instances of algebraic lattices.

• A compact element may be defined in an arbitrary poset $P$: $a\in P$ is compact iff $a$ is way below itself: $a\ll a$.

## References

• 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).
Title compact element CompactElement 2013-03-22 15:52:50 2013-03-22 15:52:50 CWoo (3771) CWoo (3771) 17 CWoo (3771) Definition msc 06B23 finite element