About: Filter (set theory)

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In mathematics, a filter on a set is a family of subsets such that: 1. * and 2. * if and ,then 3. * If ,and ,then A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.

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  • In mathematics, a filter on a set is a family of subsets such that: 1. * and 2. * if and ,then 3. * If ,and ,then A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion, see the article Filter (set theory). (en)
  • Rodzina podzbiorów zbioru jest filtrem podzbiorów zbioru jeśli są spełnione następujące warunki: (i) Jeśli i to również (ii) Część wspólna skończonej liczby elementów rodziny należy do (iii) . Z aksjomatu (ii) i (iii) wynika, że przecięcie dowolnej skończonej liczby zbiorów rodziny jest niepuste. Aksjomat (ii) jest równoważny dwóm następującym: (ii1) Jeśli to (ii2) Zbiór należy do . (pl)
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  • If is an ultrafilter on then the following are equivalent: is fixed, or equivalently, not free, meaning is principal, meaning Some element of is a finite set. Some element of is a singleton set. is principal at some point of which means does contain the Fréchet filter on is sequential. (en)
  • Every filter on a set is a subset of some ultrafilter on (en)
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  • Proposition (en)
  • The ultrafilter lemma/principal/theorem (en)
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  • Rodzina podzbiorów zbioru jest filtrem podzbiorów zbioru jeśli są spełnione następujące warunki: (i) Jeśli i to również (ii) Część wspólna skończonej liczby elementów rodziny należy do (iii) . Z aksjomatu (ii) i (iii) wynika, że przecięcie dowolnej skończonej liczby zbiorów rodziny jest niepuste. Aksjomat (ii) jest równoważny dwóm następującym: (ii1) Jeśli to (ii2) Zbiór należy do . (pl)
  • In mathematics, a filter on a set is a family of subsets such that: 1. * and 2. * if and ,then 3. * If ,and ,then A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. (en)
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  • Filter (set theory) (en)
  • Filtr (teoria zbiorów) (pl)
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