Ultilimit

Ultilimit is a term used in mathematical analysis to denote a generalized limit of a sequence defined with respect to an ultrafilter. For a sequence (a_n) in a topological space X and a non-principal ultrafilter U on the natural numbers, the Ultilimit of (a_n) along U is a point x ∈ X such that for every neighborhood V of x, the set {n ∈ N : a_n ∈ V} ∈ U. If X is compact Hausdorff, such an Ultilimit exists for every U, and different ultrafilters may yield different Ultilimits for non-convergent sequences. When the sequence converges to L in the usual sense, every Ultilimit along any ultrafilter equals L.

Origin and context: The concept builds on ultrafilter theory and the idea of limits along ultrafilters, closely related to the Stone-Čech compactification βN. It is used in analysis to study limiting behavior of sequences that do not converge ordinarily, and in model theory where ultralimit constructions underpin nonstandard analysis.

Properties and uses: In compact spaces, ultralimits exist for all ultrafilters; the limit is unique for a fixed ultrafilter but can vary with different ultrafilters. Ultilimits commute with continuous maps: if f: X → Y is continuous, lim_U f(a_n) = f(lim_U a_n). They provide a convenient tool for proving convergence statements or constructing counterexamples by selecting appropriate ultrafilters.

Examples: If X = R with the usual topology and a_n alternates between 1 and -1, the ultralimit along an ultrafilter U can be 1 or -1 depending on U.