compact element
Let $X$ be a set and $\mathcal{T}$ be a topology^{} on $X$, a wellknown 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 $\mathrm{\bigvee}S$ 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 $\mathcal{D}$ 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.
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

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

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$a\vee \mathrm{\infty}=\mathrm{\infty}$, $a\wedge \mathrm{\infty}=a$.
The resulting set $L=\mathbb{N}\cup \{\mathrm{\infty},a\}$ is a lattice where $a$ is a noncompact atom.

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Remarks.

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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.

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Any finite join of compact elements is compact.

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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 $a\le d$.

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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.

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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 

Canonical name  CompactElement 
Date of creation  20130322 15:52:50 
Last modified on  20130322 15:52:50 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  17 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B23 
Synonym  finite element 