compact element

Let $X$ be a set and $𝒯$ 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

$$A\subseteq \bigcup 𝒮,$$

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

$$A\subseteq \bigcup ℱ.$$

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 $\mathrm{\bigvee }S$ exists and $a\mathrm{\le }\mathrm{\bigvee }S$, then there is a finite subset $F\mathrm{\subset }S$ such that $a\mathrm{\le }\mathrm{\bigvee }F$.

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

Here are some other common examples:

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

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

   • $a\vee \mathrm{\infty }=\mathrm{\infty }$, $a\wedge \mathrm{\infty }=a$.

   The resulting set $L=\mathbb{N}\cup \{\mathrm{\infty },a\}$ 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 $a\le \bigvee D$, then there is an element $d\in D$ such that $a\le d$.

• 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: $a\ll a$.