nLab compact element

Definition

Let $P$ be a poset such that every directed subset of $P$ has a join; that is, $P$ is a dcpo. A compact element, or finite element, of $P$ is a compact object in $P$ regarded as a thin category; that is, homs out of it commute with these directed joins.

In other words, $c \in P$ is compact precisely if for every directed subset $\{d_i\}$ of $P$ we have

$$(c \leq \bigvee_i d_i ) \Leftrightarrow \exists_i (c \leq d_i) \,.$$

Of course, the $\Leftarrow$ part of this is automatic, so the real condition is the $\Rightarrow$ part. In more elementary terms:

• If $c \leq \bigvee D$ for $D$ a directed subset, then $c \leq d$ for some $d \in D$.

In the case where $P$ has a top element $1$, we say that $P$ is compact if $1$ is a compact element.

Examples

• Given a set $X$, the finite elements of its power set are precisely the (Kuratowski)-finite subsets of $X$. (This is the origin of the term ‘finite element’.)

• Given a topological space (or locale) $X$, the compact elements of its frame of open subspaces are precisely the compact open subspaces of $X$. (This is the origin of the term ‘compact element’.)

• If $R$ is a (not necessarily commutative) ring, the lattice $Idl(R)$ of two-sided ideals of $R$ is compact. Indeed, the top element is the ideal generated by $1$, the multiplicative identity, and $1 \in \bigvee_i I_i$ implies $1 \in I_i$ for some index $i$.