Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
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For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.
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Cofinal subsets are very important in the theory of directed sets and cofinal subnet is the appropriate generalization of subsequence .
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The cofinality of an ordinal \ alpha is the smallest ordinal \ delta that is the order type of a cofinal subset of \ alpha.
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A more natural definition of a subnet would be to require " B " to be a cofinal subset of " A " and that " h " be the identity map.
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They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of " A " is referred to as the cofinality of " A ".
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For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element which is not in the subset would fail to be " less than " any element of the subset, violating the definition of cofinal.
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