Infimum in a sentence

For instance, a lattice is a partially ordered set in which all finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum.

If S contains a least element, then that element is the infimum; otherwise, the infimum does not belong to S (or does not exist).

Advanced : : : : If a set has a smallest element, as in the first example, then the smallest element is the infimum for the set.

As the last three examples show, the infimum of a set does not have to belong to the set.

By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum.

Definition for a set The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set.

Existence of an infimum of a subset of can fail if has no lower bound at all, or if the set of lower bounds does not contain a maximal element.

Hence, it is the infimum of the limit points.

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

Likewise, if the infimum exists, it is unique.

Limit inferior is also called infimum limit, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

On the other hand, if we define infimum to be set intersection, the open sets form a bounded but not complete lattice; in general, arbitrary intersections of open sets are not open.

Similarly a lower bound is said to be a tight lower bound, a greatest lower bound, or an infimum if no greater value is a lower bound.

The infimum is given by the intersection of topologies, and the supremum by the topology generated by the union of topologies.

The infimum is in a precise sense dual to the concept of a supremum.

The least upper bound on this sequence of meets of tails is :: :So the limit infimum contains all subsets which are lower bounds for all except finitely many sets of the sequence.

The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets.

The limits and are allowed, but they do not represent valid values for the index, only the supremum or infimum of such values, respectively.

The number would be a lower bound but not the "greatest lower bound" and hence not the "Infimum".

There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other number which could be used as a lower bound.