Infimum

The infimum of a set in an ordered structure is its greatest lower bound: every element of the set lies above it, and no larger element has that property.

Lower bound: An element below every member of a given set in an ordered structure. An infimum is the largest element among the set’s lower bounds.
Supremum: The least upper bound of a set in an ordered structure, whether or not it belongs to the set. Reversing the order turns an infimum problem into a supremum problem.
Limit inferior: The limit of the infima of successive tails of a sequence, when interpreted in an extended order. It tracks the smallest values that persist arbitrarily far along a sequence.
Least element: An element of an ordered set that is less than or equal to every element of that set. An infimum need not lie in the original set, unlike a least element.
Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Infima are defined through order, including orders that are not total.
Minimum: A member of a set that is less than or equal to every other member. Unlike an infimum, a minimum must belong to the set.
Essential infimum: The greatest lower bound of a measurable function after ignoring sets of measure zero. It adapts infima to settings where negligible exceptions should not matter.
Lower envelope: A function formed from pointwise lower bounds of a family of functions. Pointwise infima provide a standard way to form a lower envelope.
Greatest element: An element of a set that is greater than or equal to every other member. An infimum is a greatest element of the set of lower bounds.
Open interval: A real interval containing its endpoints’ interior points but excluding both endpoints. The infimum of an open interval can be its excluded left endpoint.

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