Scattered order

 ( Order Theory ) 

In mathematical order theory, a scattered order is a linear order that contains no dense order subset with more than one element.

A characterization due to Felix 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 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.Harzheim, Theorem 6.17, p. 201;

The order topology of a scattered order is scattered space. The converse implication does not hold, as witnessed by the lexicographic order on $\backslash mathbb\; Q\backslash times\backslash mathbb\; Z$.