Infimum

Infimum, meaning greatest lower bound, is a fundamental concept in order theory and analysis. For a subset A of a partially ordered set P, the infimum of A is an element m in P satisfying two conditions: m ≤ a for every a in A, and for any t with t ≤ a for all a in A, we have t ≤ m. In other words, m is the largest element that is not greater than any member of A.

Existence depends on the ambient set. In the real numbers, if A is nonempty and bounded below, then inf(A) exists in R. If A is unbounded below, the infimum is often described as −∞ in the extended real line. If A is empty, the infimum is defined only in the extended real line as +∞; it is not a real number.

The infimum may or may not be attained by a member of A. If there exists a ∈ A with a = inf(A), then inf(A) is the minimum of A. Otherwise, inf(A) is not attained within A, though it remains the greatest lower bound.

Relation to supremum: infimum is the greatest lower bound, while supremum is the least upper bound. They mirror each other under the duality between lower and upper bounds. In subsets of the real numbers, inf(A) is always a lower bound of A, and inf(A) ≤ a for every a in A.

Examples illustrate the concept: for A = (0,1), inf(A) = 0 (not attained); for A = {x ∈ R : x > 2}, inf(A) = 2; for A = {1/n : n ∈ N}, inf(A) = 0.