In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with more than one element. 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. The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by the lexicographic order on Q × Z {\displaystyle \mathbb {Q} \times \mathbb {Z} } .
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