The principal ultrafilter at is . An ultraproduct by a principal ultrafilter is isomorphic to the factor indexed by .
A nonprincipal ultrafilter contains no finite set. On an infinite set it contains the cofinite filter.
The Stone-Cech compactification can be identified with the space of ultrafilters on . Its basic clopen sets are .
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An ultrafilter is a mathematical concept that arises in the field of set theory and topology, particularly in the context of ordered sets and Boolean algebras. Here's an overview of what an ultrafilter is: 1. **Definition**: An ultrafilter on a set \( X \) is a maximal filter, which is a collection of subsets of \( X \) that satisfies certain properties: - It is non-empty.