Infima

Infima, plural of infimum, is a concept in order theory and real analysis describing the greatest lower bound of a subset within a given ordered set. If S is a subset of a partially ordered set P, a element l ∈ P is a lower bound of S if l ≤ s for every s in S. The infimum of S is the greatest such lower bound: it is a lower bound and any other lower bound is at most as large as it. When an infimum exists, it is denoted inf S. If the infimum also lies in S, it is the minimum of S; otherwise the infimum may not be an element of S.

Existence of infima depends on the ambient structure. In a complete lattice, every nonempty subset has an infimum. In the real numbers with the usual order, every nonempty set that is bounded below has an infimum; the infimum need not be contained in the set itself (the minimum may fail to exist). The concept is dual to the supremum (least upper bound); while the infimum is the greatest lower bound, the supremum is the least upper bound.

Examples help clarify the idea. The infimum of the open interval (0, 1) is 0, even though 0 is not an element of the interval. The set {1/n : n ∈ N} has infimum 0, which is not attained by any member of the set. Conversely, if a set contains its infimum, that infimum is also the minimum.

Infima are fundamental in optimization, analysis, and lattice theory, where they define the meet operation and underpin arguments about limits, convergence, and bounds.