In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element.[1]
A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orders that contains the singleton orders and is closed under well-ordered and reverse well-ordered sums.
Laver's theorem (generalizing a conjecture of Roland Fraïssé on countable orders) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order.[2]
The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order on
Q x Z