In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element (Harzheim 2005:193ff.)
A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orders which contains the singleton orders and is closed under well-ordered and reverse well-ordered sums.
Laver's theorem (generalizing Fraïssé's conjecture) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order (Harzheim 2005:265).
The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order on