1. Examples

(1) $[0,1] \subseteq$   R

(2) Pow$(X)$

(3) Any finite lattice.

(4) Subsp$(V)$, for any vector space $V$.

(5) Subgp$(G)$, for any group $G$.

(6) ${\cal I} (L)$, the lattice of ideals of a lattice $L$, together with the empty set if $L$ has no zero element.

(7) The family of closed subsets of a topological space $X$.

(7$'$) The family of open subsets of a topological space $X$.

(8) Any closure system ${\cal C}$ on a set $X$.

This means a family ${\cal C}$ of subsets of $X$ such that

(i) ${\cal C}$ is closed under arbitrary intersections;

(ii) $X \in {\cal C}$.

(9) The family of closed sets under a closure operator on a set $X$.

This means a map $A \rightarrow \overline A$ of Pow$(X) \rightarrow$   Pow$(X)$ such that

(a) $A \subseteq \overline A$

(b) $A \subseteq B$ implies $\overline A \subseteq \overline B$.

(c) $\overline {\overline A} = \overline A$

$A$ is closed when $\overline A = A$.

(10) The family of closed subspaces of a Hilbert space.

(11) A ``conditionally complete'' lattice with $0,1$ adjoined (if needed).

A lattice $L$ is said to be conditionally complete if any subset bounded above has a least upper bound and any subset bounded below has a greatest lower bound.

(12) A well ordered set with top element.

A chain $C$ is said to be well ordered if every nonempty subset in $C$ has a least element.

(13) Any complete sublattice of a complete lattice.

(14) CC$(P)$, the ``completion by cuts'' of a partially ordered set $P$. (See §)

(15) Part$(X)$, the lattice of partitions of a set $X$, or equivalently, the lattice of equivalence relations on $X$.

Partitions are usually compared as equivalence relations: $\theta \subseteq \psi$ when every $\theta$-block is a $\psi$-block. Under this ordering, the largest partition is the one that consists of one block containing all elements, and the smallest partition is the one where all blocks are singletons, or in other words, the equality relation.

Sometimes in analysis the ordering of partitions is by refinement, which is the reverse: the finest partition (with singleton blocks) is the largest.