Total order

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

${\displaystyle \{a\vee b,a\wedge b\}=\{a,b\}}$ for all a, b.

We then write a ≤ b if and only if ${\displaystyle a=a\wedge b}$. Hence a totally ordered set is a distributive lattice.

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).

A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, the order topology.

When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general).

The order topology induced by a total order may be shown to be hereditarily normal.

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers R is complete but the set of rational numbers Q is not. In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.

There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.
• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.)
• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples.

For any two disjoint total orders ${\displaystyle (A_{1},\leq _{1})}$ and ${\displaystyle (A_{2},\leq _{2})}$, there is a natural order ${\displaystyle \leq _{+}}$ on the set ${\displaystyle A_{1}\cup A_{2}}$, which is called the sum of the two orders or sometimes just ${\displaystyle A_{1}+A_{2}}$:

For ${\displaystyle x,y\in A_{1}\cup A_{2}}$, ${\displaystyle x\leq _{+}y}$ holds if and only if one of the following holds:

1. ${\displaystyle x,y\in A_{1}}$ and ${\displaystyle x\leq _{1}y}$
2. ${\displaystyle x,y\in A_{2}}$ and ${\displaystyle x\leq _{2}y}$
3. ${\displaystyle x\in A_{1}}$ and ${\displaystyle y\in A_{2}}$

Intuitively, this means that the elements of the second set are added on top of the elements of the first set.

More generally, if ${\displaystyle (I,\leq )}$ is a totally ordered index set, and for each ${\displaystyle i\in I}$ the structure ${\displaystyle (A_{i},\leq _{i})}$ is a linear order, where the sets ${\displaystyle A_{i}}$ are pairwise disjoint, then the natural total order on ${\displaystyle \bigcup _{i}A_{i}}$ is defined by

For ${\displaystyle x,y\in \bigcup _{i\in I}A_{i}}$, ${\displaystyle x\leq y}$ holds if:

1. Either there is some ${\displaystyle i\in I}$ with ${\displaystyle x\leq _{i}y}$
2. or there are some ${\displaystyle i<j}$ in ${\displaystyle I}$ with ${\displaystyle x\in A_{i}}$, ${\displaystyle y\in A_{j}}$