# Totally Ordered Set

A set is **totally ordered** if

- ifor any pair of elements in the set
*a*and*b*, either*a*≤*b*or*b*≤*a*(totality) - if
*a*≤*b*and*b*≤*c*, then*a*≤*c*(transitivity) - if
*a*≤*b*and*b*≤*a*, then*a*=*b*(antisymmetry)

If a set is totally ordered, then any subset is also totally ordered.

If a set *E* is totally ordered, then we might want to find **least element** or **greatest element** in it. That is, we seek elements in *E* such that

- min(
*E*) ≤*a*for all*a*∈*E* - max(
*E*) ≥*a*for all*a*∈*E*

Such elements might not exist. For example, the set of integers under the customary order does not have a smallest or largest element. Finite, nonempty, totally ordered sets always have least and greatest elements, however.

The antisymmetry property guarantees the uniqueness of a least element or a greatest element if it exists.

A set is **well-ordered** if every non-empty set has a least element. The natural numbers are well-ordered; the integers, rationals, and real numbers are not.

If *X* is totally ordered and *E* ⊆ *X*, then an **upper bound** (resp. **lower bound**) of *E* is an element *x* ∈ *X* such that

*a*≤*x*(resp.*a*≥*x*) for all*a*∈*E*.

An upper bound or lower bound might not exist, in which case *E* is said to be **unbounded**.

Even if the upper bound or lower bound exists, it might not be unique. An upper bound or lower bound might or might not be in the set *E*. At most one upper bound and one lower bound can be in the set *E*.

Although upper bounds and lower bounds are not necessarily unique, the **least upper bound** and **greatest lower bound** are unique if they exist. They can fail to exist even if there are upper bounds and lower bounds.

A totally ordered set is **complete** if every nonempty set with an upper bound has a least upper bound.

# Partially Ordered Set

- inf, sup

# Sequence

- limit (sequence)

# Function

- limit (function)
- right limit, left limit

# Topological Space

- nets