Ultrafilters

Abstract

Filters and ultrafilters—initially introduced by Cartan in the 1930s—are fundamental objects in mathematics and naturally arise in many different contexts. They are featured prominently in Bourbaki’s systematic treatment of general topology as they allow one to capture the central notion of “limit”. The general definition of limit of a filter subsumes, in particular, the usual limit of a sequence (or net). Ultrafilters—a special class of filters—are especially useful, as they always admit a limit provided they are defined on a compact space. Thus, ultrafilters allow one to extend the usual notion of limit to sequences (or nets) that would not have a limit in the usual sense. Such “generalized limits” are frequently used in asymptotic or limiting arguments, where compactness is used in a fundamental way.