in the whole set if and only if it is unbounded in the whole set or it has a maximum which is also maximum of the whole set. A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type. See also Tree (set theory), generalization Ordinal number Well-founded set Well partial order Prewellordering Directed set References Binary relations Order theory Ordinal numbers Wellfoundedness