Ultrafilter (set theory)

rdf:langString Ultrafilter (set theory)

rdf:langString Proposition

rdf:langString The /principle/theorem

rdf:langString July 2016

rdf:langString ultrafilter

rdf:langString A function m can certainly be defined in that way. However, this is pointless unless such an m can be shown to have some useful properties . They should be stated here.

rdf:langString Ultrafilter

rdf:langString In the mathematical field of set theory, an ultrafilter is a maximal proper filter: it is a filter on a given non-empty set which is a certain type of non-empty family of subsets of that is not equal to the power set of (such filters are called proper) and that is also "maximal" in that there does not exist any other proper filter on that contains it as a proper subset. Said differently, a proper filter is called an ultrafilter if there exists exactly one proper filter that contains it as a subset, that proper filter (necessarily) being itself. More formally, an ultrafilter on is a proper filter that is also a maximal filter on with respect to set inclusion, meaning that there does not exist any proper filter on that contains as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set and the partial order is subset inclusion Ultrafilters have many applications in set theory, model theory, and topology.

rdf:langString Every proper filter on a set is contained in some ultrafilter on

rdf:langString If is an ultrafilter on then the following are equivalent: is fixed, or equivalently, not free. is principal. Some element of is a finite set. Some element of is a singleton set. is principal at some point of which means for some does contain the Fréchet filter on as a subset. is sequential.