Total order

A total order is a reflexive, antisymmetric, transitive relation in which every two elements are comparable.

The topology generated by open intervals and open initial and final rays in a total order. For a dense order without endpoints, a subset is dense in this topology exactly when it meets every nonempty open interval.

Every pairwise disjoint collection of nonempty open intervals is countable. This is one of the defining properties of a Suslin line.

A subset of a total order meeting every nonempty open interval between two distinct points. A countable order-dense subset witnesses separability in the order topology of a dense order.

Every nonempty bounded-above subset of a total order has a least upper bound. In the real line this is the supremum property used to extend maps defined on an order-dense subset.

The completion of a total order by its nonprincipal proper cuts, supplying least upper bounds. For a dense order it contains the original order densely, and preserves the interval countable chain condition and separability or nonseparability.

An order isomorphism is a bijection preserving and reflecting the order relation. It transfers well-foundedness and order type between a coded order and its ordinal interpretation.

The order sum $A+B$ is the disjoint union of two total orders, preserving each order and placing every element of $A$ before every element of $B$. For example $\mathbb{N}+\mathbb{Z}$ has a least element but also has elements with infinitely many predecessors.

An order automorphism of a total order $(I,<)$ is a bijection $\pi :I\to I$ such that $i<j$ if and only if $\pi (i)<\pi (j)$. Composition and inverses are again order automorphisms. Transporting the indices of an order-indiscernible sequence by such a map preserves every first-order formula on increasing finite tuples.

An order-dense subset with no nonidentity order automorphism. A transfinite diagonal construction marks an included point and an excluded image for every nonidentity real-line order automorphism, yielding such a subset of size $2^{{\aleph }_0}$.

An order automorphism of an order-dense subset $D$ of the real line extends uniquely by $\tilde{\phi }(x)=\sup \{\phi (d):d\in D,d<x\}$. Order completeness supplies the extension, and the lower cuts in $D$ give uniqueness.

A strict total order is an irreflexive and transitive relation $<$ for which exactly one of $x<y$ and $y<x$ holds whenever $x\ne y$. It corresponds to a total order through $x\le y$ if and only if $x<y$ or $x=y$.

A well-order is a total order in which every nonempty subset has a least element.

A decidable well-order on the natural numbers is a well-order whose comparison relation is computable. Comparing two codes is an effective finite task; the proof that the relation has no infinite descending chain is a separate mathematical assertion. Computable Cantor normal form notation gives such an order of type epsilon zero.

The empty order is the unique total order on the empty set.

An element is greatest when every element of the ordered set is less than or equal to it.

An initial segment contains every element below each of its elements.