Editing Total order (section)

==Examples==

* Any [[subset]] of a totally ordered set {{math|''X''}} is totally ordered for the restriction of the order on {{math|''X''}}. 
* The unique order on the empty set, {{math|∅}}, is a total order.
* Any set of [[cardinal number]]s or [[ordinal number]]s (more strongly, these are [[well-order]]s).
* If {{math|''X''}} is any set and {{math|''f''}}{{math|}} an [[injective function]] from {{math|''X''}} to a totally ordered set then {{math|''f''}} induces a total ordering on {{math|''X''}} by setting {{math|''x''<sub>1</sub> ≤ ''x''<sub>2</sub>}} if and only if {{math|''f''(''x''<sub>1</sub>) ≤ ''f''(''x''<sub>2</sub>)}}.
* The [[lexicographical order]] on the [[Cartesian product]] of a family of totally ordered sets, [[Index set|indexed]] by a [[well-order|well ordered set]], is itself a total order.  
* The set of [[real numbers]] ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered. Hence each subset of the real numbers is totally ordered, such as the [[natural numbers]], [[integers]], and [[rational numbers]]. Each of these can be shown to be the unique (up to an [[order isomorphism]]) "initial example" of a totally ordered set with a certain property, (here, a total order {{math|''A''}} is ''initial'' for a property, if, whenever {{math|''B''}} has the property, there is an order isomorphism from {{math|''A''}} to a subset of {{math|''B''}}):<ref>This definition resembles that of an [[initial object]] of a [[category (mathematics)|category]], but is weaker.</ref>{{citation needed|reason=such non-evident properties must be sourced; see talk page|date=March 2021}}
** The natural numbers form an initial non-empty totally ordered set with no [[upper bound]].
** The integers form an initial non-empty totally ordered set with neither an upper nor a [[lower bound]].
** The rational numbers form an initial totally ordered set which is [[dense set|dense]] in the real numbers. Moreover, the reflexive reduction < is a [[dense order]] on the rational numbers.
** The real numbers form an initial unbounded totally ordered set that is [[connectedness|connected]] in the [[order topology]] (defined below).
* [[Ordered field]]s are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any ''[[Dedekind-complete]]'' ordered field is isomorphic to the real numbers.
* The letters of the alphabet ordered by the standard [[Alphabetical order|dictionary order]], e.g., {{math|''A'' < ''B'' < ''C''}} etc., is a strict total order.