39:
3785:
of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example,
3914:
many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (for example, Kirman and
Sondermann, 1972). Mihara (1997, 1999) shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.
4267:
2800:
4183:
3580:
3359:
2857:
4306:
3285:
1123:
3141:
2155:
1281:
264:
4109:
4028:
3680:
2961:
2268:
2192:
1493:
1420:
1363:
391:
305:
3220:
3040:
2904:
2697:
2653:
2625:
2480:
2387:
2340:
2292:
1667:). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of
4415:
4188:
2601:
533:
2413:
2077:
Given a maximal ideal of a
Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
617:
503:
698:
2509:
2020:
1971:
1756:
1330:
781:
756:
669:
65:, as it can be extended to the larger nontrivial filter â2, by including also the light green elements. Since â2 cannot be extended any further, it is an ultrafilter.
3320:
3255:
3075:
2996:
2743:
2080:
Given an ultrafilter on a
Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".
2046:
1997:
727:
643:
4670:
Reprinted in K. V. Velupillai, S. Zambelli, and S. Kinsella, ed., Computable
Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.
3627:
1222:
3725:
4442:
4329:
2555:
2235:
2116:
1940:
1892:
1846:
1615:
1524:
938:
203:
156:
126:
4389:
4356:
4076:
4048:
3995:
3860:
3828:
3808:
3755:
3647:
3496:
3456:
3379:
3185:
3161:
3095:
3016:
2928:
2877:
2717:
2673:
2575:
2532:
2456:
2436:
2363:
2312:
2212:
1913:
1869:
1823:
1780:
1733:
1713:
1588:
1564:
1544:
1460:
1440:
1387:
1301:
1242:
1195:
1163:
1143:
1041:
1021:
1001:
961:
915:
849:
821:
801:
579:
411:
350:
330:
225:
3413:
3600:
3516:
3476:
1803:
981:
893:
869:
557:
481:
180:
103:
4856:
4746:
3412:. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in
2748:
5106:
4976:
2059:
3478:
can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element
4784:
4123:
5075:
5045:
4988:
4958:
4898:
4868:
4838:
4804:
4762:
4577:
2055:
3997:
happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on
3521:
417:" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.
5015:
4493:
3687:
4928:
4701:
4362:
characterization theorem) if and only if it contains every cofinite set, that is, every member of the Fréchet filter.
4270:
3325:
2879:. In other words, an ultrafilter may be seen as a family of sets which "locally" resembles a principal ultrafilter.
3907:
2805:
4796:
1652:
3888:
4681:
5207:
2051:
A proof that 1. and 2. are equivalent is also given in (Burris, Sankappanavar, 2012, Corollary 3.13, p.133).
4535:
1620:
4564:
4276:
460:. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.
3260:
1692:
1644:
1591:
1077:
271:
4630:
3110:
2124:
1250:
233:
5182:
4081:
4000:
3652:
2933:
2240:
2164:
1465:
1392:
1335:
363:
277:
5202:
4262:{\displaystyle \left\{x_{1}\right\}\in U,{\text{ or }}\ldots {\text{ or }}\left\{x_{n}\right\}\in U,}
3421:
2483:
4715:
3201:
3021:
2885:
2678:
2634:
2606:
2461:
2368:
2321:
2273:
4650:
3103:
2090:
1639:. On the other hand, the statement that every filter is contained in an ultrafilter does not imply
309:
28:
4394:
2580:
1759:
512:
4791:
3891:
of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.
2161:, is always a Boolean algebra; hence the results of the above section apply. An (ultra)filter on
2392:
4742:
4710:
4645:
3866:
1676:
587:
486:
3939: â Use of filters to describe and characterize all basic topological notions and results.
674:
5197:
4675:
3942:
3877:, that is the large scale geometry of the group. Asymptotic cones are particular examples of
3192:
2488:
2002:
1947:
1738:
1306:
760:
732:
648:
449:
357:
82:
4602:
Kirman, A.; Sondermann, D. (1972). "Arrow's theorem, many agents, and invisible dictators".
4050:
is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter
3786:
pointwise < is not a total ordering. So instead the functions and relations are defined "
3290:
3225:
3045:
2966:
2722:
2025:
1976:
706:
622:
5158:
5137:
3924:
3895:
3835:
3766:
3762:
3605:
1200:
132:
4916:
4696:
3697:
8:
4830:
3936:
3930:
3778:
3758:
3425:
4424:
4311:
3870:
2537:
2217:
2098:
1922:
1874:
1828:
1597:
1506:
920:
185:
138:
108:
4663:
4468:
4374:
4341:
4061:
4033:
3980:
3845:
3813:
3793:
3740:
3632:
3481:
3441:
3364:
3170:
3146:
3080:
3001:
2913:
2862:
2702:
2658:
2560:
2517:
2441:
2421:
2348:
2297:
2197:
1898:
1854:
1808:
1765:
1718:
1698:
1573:
1549:
1529:
1445:
1425:
1372:
1286:
1227:
1180:
1148:
1128:
1026:
1006:
986:
946:
900:
834:
806:
786:
564:
396:
335:
315:
210:
4724:
3787:
5125:
5081:
5071:
5051:
5041:
5021:
5011:
4994:
4984:
4964:
4954:
4934:
4924:
4904:
4894:
4874:
4864:
4844:
4834:
4810:
4800:
4768:
4758:
4615:
4573:
3839:
3781:
from real numbers to sequences of real numbers. This sequence space is regarded as a
3683:
3164:
3098:
2907:
2063:
1624:
414:
353:
24:
5120:
5101:
4667:
4358:
is non-principal if and only if it contains no finite set, that is, (by Nr.3 of the
1567:
1496:
5115:
4780:
4720:
4655:
4611:
4518:
3770:
3585:
3501:
3461:
3431:. This is proved by taking the ultrapower of the set theoretical universe modulo a
1788:
1366:
966:
878:
854:
542:
466:
165:
88:
5154:
5150:
5133:
4822:
3401:
2074:
of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.
1683:
without the axiom of choice, it is possible that every ultrafilter is principal.
1636:
1628:
1500:
1058:
Every ultrafilter falls into exactly one of two categories: principal or free. A
453:
129:
32:
1695:. In this case, ultrafilters are characterized by containing, for each element
5063:
5033:
4886:
4829:. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York:
3691:
38:
5176:
4697:"Arrow's theorem, countably many agents, and more visible invisible dictators"
4522:
5191:
5129:
5055:
5025:
5010:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
4908:
4878:
4510:
3951:
3398:
2158:
2071:
1691:
An important special case of the concept occurs if the considered poset is a
1071:
829:
457:
267:
159:
42:
5164:
4968:
4848:
4814:
1672:
4998:
4938:
4772:
3882:
3774:
3734:
3730:
3409:
3405:
1916:
437:
421:
74:
20:
5085:
4659:
2795:{\displaystyle {\mathcal {U}}\cap {\mathcal {F}}=F_{x}\cap {\mathcal {F}}}
1619:
Every filter on a
Boolean algebra (or more generally, any subset with the
4946:
70:
4359:
3878:
3417:
1169:
of the ultrafilter. Any ultrafilter that is not principal is called a
4754:
3831:
3761:
can be proved this way. In the special case of ultrapowers, one gets
3737:
uses ultrafilters to produce a new model starting from a sequence of
3390:
2119:
228:
54:
4753:. Mathematics and Its Applications. Vol. 13. Dordrecht Boston:
4538:, Burak Kaya, lecture notes, Nesin Mathematics Village, Summer 2019.
3862:
is nonprincipal, then the extension thereby obtained is nontrivial.
1627:) and free ultrafilters therefore exist, but the proofs involve the
3927: â In mathematics, a special subset of a partially ordered set
3838:, this preserves all properties of the reals that can be stated in
3782:
3602:
is again a
Boolean algebra, since in this situation the set of all
3394:
425:
4923:. Translated by CsĂĄszĂĄr, KlĂĄra. Bristol England: Adam Hilger Ltd.
2070:
Given a homomorphism of a
Boolean algebra onto {true, false}, the
1679:
where one can write down an explicit global choice function. In
3898:, non-principal ultrafilters are used to define a rule (called a
46:
3873:
of a group. This construction yields a rigorous way to consider
4498:. Cambridge Mathematical Textbooks. Cambridge University Press.
445:
4951:
Set Theory: The Third
Millennium Edition, Revised and Expanded
3649:. Especially, when considering the ultrafilters on a powerset
2062:
to the 2-element
Boolean algebra {true, false} (also known as
2054:
Moreover, ultrafilters on a
Boolean algebra can be related to
5168:
4741:
2675:
is an ultrafilter if and only if for any finite collection
4953:. Berlin New York: Springer Science & Business Media.
4799:. Berlin New York: Springer Science & Business Media.
1074:. Consequently, each principal ultrafilter is of the form
1651:), a well-known intermediate point between the axioms of
1422:
is principal if and only if it contains a finite set. If
4751:
Fundamentals of General Topology: Problems and Exercises
4556:
4178:{\displaystyle \left\{x_{1},\ldots ,x_{n}\right\}\in U,}
2084:
3956:
Pages displaying short descriptions of redirect targets
3947:
Pages displaying short descriptions of redirect targets
1244:; it is a principal ultrafilter only if it is maximal.
4487:
4485:
4483:
4481:
3588:
3504:
3464:
1791:
1495:
is hence non-principal if and only if it contains the
1049:
969:
881:
857:
545:
469:
168:
91:
4427:
4397:
4377:
4344:
4314:
4279:
4191:
4126:
4084:
4064:
4058:
an arbitrary set"; i.e. they write "(ultra)filter on
4036:
4003:
3983:
3848:
3816:
3796:
3743:
3700:
3655:
3635:
3608:
3524:
3484:
3444:
3367:
3328:
3293:
3263:
3228:
3204:
3173:
3149:
3113:
3083:
3048:
3024:
3004:
2969:
2936:
2916:
2888:
2865:
2808:
2751:
2725:
2705:
2681:
2661:
2637:
2609:
2583:
2563:
2540:
2520:
2491:
2464:
2444:
2424:
2395:
2371:
2351:
2324:
2300:
2276:
2243:
2220:
2200:
2167:
2127:
2101:
2028:
2005:
1979:
1950:
1925:
1901:
1877:
1857:
1831:
1811:
1768:
1741:
1721:
1701:
1600:
1576:
1552:
1532:
1509:
1468:
1448:
1428:
1395:
1375:
1338:
1309:
1289:
1253:
1230:
1203:
1183:
1151:
1131:
1080:
1029:
1009:
989:
949:
923:
903:
837:
809:
789:
763:
735:
709:
677:
651:
625:
590:
567:
515:
489:
399:
366:
338:
318:
280:
236:
213:
188:
182:
that cannot be enlarged to a bigger proper filter on
141:
111:
3869:, non-principal ultrafilters are used to define the
1715:
of the Boolean algebra, exactly one of the elements
4863:. New Jersey: World Scientific Publishing Company.
4562:
4478:
3575:{\displaystyle D_{x}=\left\{U\in G:x\in U\right\}.}
1686:
1283:a principal ultrafilter consists of all subsets of
420:Ultrafilters have many applications in set theory,
5144:
4436:
4409:
4383:
4350:
4323:
4300:
4261:
4177:
4103:
4070:
4042:
4022:
3989:
3973:
3971:
3854:
3822:
3802:
3749:
3719:
3674:
3641:
3621:
3594:
3574:
3510:
3490:
3470:
3450:
3373:
3353:
3314:
3279:
3249:
3214:
3179:
3155:
3135:
3089:
3069:
3034:
3010:
2990:
2955:
2922:
2898:
2871:
2851:
2794:
2737:
2711:
2691:
2667:
2647:
2619:
2595:
2569:
2549:
2526:
2503:
2474:
2450:
2430:
2407:
2381:
2357:
2334:
2306:
2286:
2262:
2229:
2206:
2186:
2149:
2110:
2040:
2014:
1991:
1965:
1934:
1907:
1886:
1863:
1840:
1817:
1797:
1774:
1750:
1727:
1707:
1609:
1582:
1558:
1538:
1518:
1487:
1454:
1434:
1414:
1381:
1357:
1324:
1295:
1275:
1236:
1216:
1189:
1157:
1137:
1117:
1035:
1015:
995:
975:
955:
932:
909:
887:
863:
843:
815:
795:
775:
750:
721:
692:
663:
637:
611:
573:
551:
527:
497:
475:
405:
385:
344:
324:
299:
258:
219:
197:
174:
150:
120:
97:
19:This article is about the mathematical concept in
5005:
4563:Burris, Stanley N.; Sankappanavar, H. P. (2012).
431:
5189:
5178:"Mathematical Logic 15, The Ultrafilter Theorem"
4601:
4515:Snapshots of Modern Mathematics from Oberwolfach
3954: â A generalization of a sequence of points
3933: â Family of sets representing "large" sets
4552:. Springer Monographs in Mathematics. Springer.
4462:
4460:
3968:
3354:{\displaystyle X\setminus A\in {\mathcal {F}},}
1546:is finite, every ultrafilter is principal. If
5006:Narici, Lawrence; Beckenstein, Edward (2011).
4491:
3629:is a base for a compact Hausdorff topology on
1848:then the following statements are equivalent:
5107:Bulletin of the American Mathematical Society
5102:"Ultrafilters: some old and some new results"
4855:
4466:
3773:can be constructed as an ultraproduct of the
2852:{\displaystyle F_{x}=\{Y\subseteq X:x\in Y\}}
4541:
4457:
4365:
2846:
2822:
1224:is a filter, called the principal filter at
1112:
1094:
3222:that is not an ultrafilter, one can define
1369:is of this form. Therefore, an ultrafilter
4631:"Arrow's Theorem and Turing computability"
4271:Ultrafilter (set theory)#Characterizations
2194:is often called just an "(ultra)filter on
1570:is not an ultrafilter on the power set of
5119:
5032:
4714:
4649:
4547:
4508:
1663:theory augmented by the axiom of choice (
494:
490:
5145:Comfort, W. W.; Negrepontis, S. (1974),
5062:
5038:Handbook of Analysis and Its Foundations
4885:
4779:
4688:
4622:
4529:
3424:is incompatible with the existence of a
37:
5099:
4915:
4821:
4743:Arkhangel'skii, Alexander Vladimirovich
4492:Davey, B. A.; Priestley, H. A. (1990).
3420:ultrafilters are used to show that the
2859:is the principal ultrafilter seeded by
1070:) ultrafilter is a filter containing a
274:and hence a poset, and ultrafilters on
5190:
4694:
4628:
4511:"Ultrafilter methods in combinatorics"
3435:-complete, non-principal ultrafilter.
1623:) is contained in an ultrafilter (see
4983:. New York: John Wiley and Sons Ltd.
4975:
4592:Kanamori, The Higher infinite, p. 49.
3902:) for aggregating the preferences of
3167:or false almost everywhere. However,
2085:Ultrafilter on the power set of a set
4945:
4301:{\displaystyle \left\{x_{i}\right\}}
1675:showed that this can be done in the
4861:Convergence Foundations Of Topology
3280:{\displaystyle A\in {\mathcal {F}}}
2318:The empty set is not an element of
1118:{\displaystyle F_{p}=\{x:p\leq x\}}
1052:Types and existence of ultrafilters
13:
5093:
4536:"Ultrafilters and how to use them"
4495:Introduction to Lattices and Order
4467:Alex Kruckman (November 7, 2012).
4269:by the characterization Nr.7 from
4087:
4078:" to abbreviate "(ultra)filter of
4006:
3875:looking at the group from infinity
3757:-indexed models; for example, the
3658:
3343:
3272:
3207:
3143:and every property of elements of
3136:{\displaystyle {\mathcal {P}}(X),}
3116:
3027:
2939:
2891:
2787:
2764:
2754:
2684:
2640:
2612:
2467:
2374:
2327:
2279:
2246:
2170:
2150:{\displaystyle {\mathcal {P}}(X),}
2130:
2006:
1742:
1643:. Indeed, it is equivalent to the
1471:
1398:
1341:
1276:{\displaystyle {\mathcal {P}}(X),}
1256:
369:
283:
259:{\displaystyle {\mathcal {P}}(X),}
239:
14:
5219:
5040:. San Diego, CA: Academic Press.
4702:Journal of Mathematical Economics
4401:
4104:{\displaystyle {\mathcal {P}}(X)}
4023:{\displaystyle {\mathcal {P}}(X)}
3675:{\displaystyle {\mathcal {P}}(S)}
3332:
2956:{\displaystyle {\mathcal {P}}(X)}
2587:
2263:{\displaystyle {\mathcal {P}}(X)}
2187:{\displaystyle {\mathcal {P}}(X)}
1488:{\displaystyle {\mathcal {P}}(X)}
1415:{\displaystyle {\mathcal {P}}(X)}
1358:{\displaystyle {\mathcal {P}}(X)}
1145:of the given poset. In this case
386:{\displaystyle {\mathcal {P}}(X)}
300:{\displaystyle {\mathcal {P}}(X)}
4981:Introduction to General Topology
4517:. Marta Maggioni, Sophia Jahns.
4474:. Berkeley Math Toolbox Seminar.
1687:Ultrafilter on a Boolean algebra
1590:but it is an ultrafilter on the
393:. In this view, every subset of
57:â14 colored dark green. It is a
49:of 210, ordered by the relation
5121:10.1090/S0002-9904-1977-14316-4
4735:
4595:
4586:
3765:of structures. For example, in
3458:of all ultrafilters of a poset
3384:
1247:For ultrafilters on a powerset
483:is a set, partially ordered by
31:. For the physical device, see
5070:. London: Macdonald & Co.
4786:General Topology: Chapters 1â4
4502:
4371:Properties 1 and 3 imply that
4333:
4120:To see the "if" direction: If
4114:
4098:
4092:
4017:
4011:
3906:many individuals. Contrary to
3710:
3702:
3669:
3663:
3414:Stone's representation theorem
3303:
3297:
3238:
3232:
3215:{\displaystyle {\mathcal {F}}}
3191:, and hence does not define a
3127:
3121:
3058:
3052:
3035:{\displaystyle {\mathcal {U}}}
2979:
2973:
2950:
2944:
2899:{\displaystyle {\mathcal {U}}}
2882:An equivalent form of a given
2692:{\displaystyle {\mathcal {F}}}
2648:{\displaystyle {\mathcal {U}}}
2620:{\displaystyle {\mathcal {U}}}
2475:{\displaystyle {\mathcal {U}}}
2382:{\displaystyle {\mathcal {U}}}
2335:{\displaystyle {\mathcal {U}}}
2287:{\displaystyle {\mathcal {U}}}
2257:
2251:
2181:
2175:
2141:
2135:
1482:
1476:
1409:
1403:
1352:
1346:
1267:
1261:
432:Ultrafilters on partial orders
380:
374:
294:
288:
250:
244:
1:
4725:10.1016/S0304-4068(98)00061-5
4566:A Course in Universal Algebra
4450:
3945: â Maximal proper filter
3908:Arrow's impossibility theorem
3410:ultraproducts and ultrapowers
1303:that contain a given element
1177:) ultrafilter. For arbitrary
4616:10.1016/0022-0531(72)90106-8
4410:{\displaystyle X\setminus A}
4308:is the principal element of
4054:a partial ordered set" vs. "
4030:or an (ultra)filter just on
3397:, especially in relation to
2596:{\displaystyle X\setminus A}
1621:finite intersection property
1442:is infinite, an ultrafilter
528:{\displaystyle F\subseteq P}
7:
4893:. Boston: Allyn and Bacon.
4859:; Mynard, Frédéric (2016).
3918:
3688:StoneâÄech compactification
1653:ZermeloâFraenkel set theory
1645:Boolean prime ideal theorem
10:
5224:
5147:The theory of ultrafilters
4680:: CS1 maint: postscript (
4604:Journal of Economic Theory
2408:{\displaystyle B\supset A}
2389:then so is every superset
2214:". Given an arbitrary set
2088:
943:there is no proper filter
619:there exists some element
332:. An ultrafilter on a set
18:
5008:Topological Vector Spaces
4548:Halbeisen, L. J. (2012).
4523:10.14760/SNAP-2021-006-EN
4509:Goldbring, Isaac (2021).
3889:Gödel's ontological proof
3830:is an ultrafilter on the
3582:This is most useful when
3422:axiom of constructibility
2294:consisting of subsets of
1805:is a Boolean algebra and
612:{\displaystyle x,y\in F,}
498:{\displaystyle \,\leq \,}
227:is an arbitrary set, its
4797:ĂlĂ©ments de mathĂ©matique
4550:Combinatorial Set Theory
3961:
2091:Ultrafilter (set theory)
693:{\displaystyle z\leq y,}
29:Ultrafilter (set theory)
5100:Comfort, W. W. (1977).
4469:"Notes on Ultrafilters"
3900:social welfare function
3408:in the construction of
2631:Equivalently, a family
2504:{\displaystyle A\cap B}
2095:Given an arbitrary set
2015:{\displaystyle \lnot x}
1966:{\displaystyle x\in P,}
1751:{\displaystyle \lnot x}
1592:finiteâcofinite algebra
1325:{\displaystyle x\in X.}
776:{\displaystyle x\leq y}
751:{\displaystyle y\in P,}
664:{\displaystyle z\leq x}
352:may be considered as a
310:ultrafilters on the set
105:is a certain subset of
4695:Mihara, H. R. (1999).
4629:Mihara, H. R. (1997).
4438:
4411:
4385:
4352:
4325:
4302:
4263:
4179:
4105:
4072:
4044:
4024:
3991:
3867:geometric group theory
3856:
3824:
3804:
3751:
3721:
3676:
3643:
3623:
3596:
3576:
3512:
3492:
3472:
3452:
3375:
3355:
3316:
3315:{\displaystyle m(A)=0}
3281:
3251:
3250:{\displaystyle m(A)=1}
3216:
3181:
3157:
3137:
3091:
3071:
3070:{\displaystyle m(A)=0}
3036:
3012:
2992:
2991:{\displaystyle m(A)=1}
2957:
2924:
2900:
2873:
2853:
2796:
2739:
2738:{\displaystyle x\in X}
2713:
2693:
2669:
2649:
2621:
2597:
2571:
2551:
2528:
2505:
2476:
2452:
2432:
2409:
2383:
2359:
2336:
2308:
2288:
2264:
2231:
2208:
2188:
2151:
2112:
2042:
2041:{\displaystyle \in F.}
2016:
1993:
1992:{\displaystyle x\in F}
1967:
1936:
1909:
1888:
1865:
1842:
1825:is a proper filter on
1819:
1799:
1776:
1758:(the latter being the
1752:
1729:
1709:
1677:constructible universe
1611:
1584:
1560:
1540:
1520:
1489:
1456:
1436:
1416:
1383:
1359:
1326:
1297:
1277:
1238:
1218:
1191:
1159:
1139:
1119:
1037:
1023:is a proper subset of
1017:
997:
983:that properly extends
977:
957:
934:
911:
889:
865:
845:
817:
797:
777:
752:
723:
722:{\displaystyle x\in F}
694:
665:
639:
638:{\displaystyle z\in F}
613:
575:
553:
529:
499:
477:
413:is either considered "
407:
387:
346:
326:
301:
260:
221:
199:
176:
152:
122:
99:
66:
23:. For ultrafilters on
4660:10.1007/s001990050157
4439:
4412:
4386:
4353:
4326:
4303:
4264:
4180:
4106:
4073:
4045:
4025:
3992:
3943:The ultrafilter lemma
3857:
3834:of the sequences; by
3825:
3805:
3763:elementary extensions
3752:
3722:
3677:
3644:
3624:
3622:{\displaystyle D_{x}}
3597:
3577:
3513:
3493:
3473:
3453:
3381:undefined elsewhere.
3376:
3356:
3317:
3282:
3252:
3217:
3182:
3158:
3138:
3092:
3072:
3037:
3013:
2993:
2958:
2925:
2901:
2874:
2854:
2797:
2740:
2714:
2694:
2670:
2650:
2622:
2598:
2572:
2552:
2529:
2506:
2477:
2453:
2433:
2410:
2384:
2360:
2337:
2309:
2289:
2265:
2232:
2209:
2189:
2152:
2113:
2043:
2017:
1994:
1968:
1937:
1910:
1889:
1871:is an ultrafilter on
1866:
1843:
1820:
1800:
1777:
1753:
1730:
1710:
1612:
1585:
1566:is infinite then the
1561:
1541:
1521:
1490:
1457:
1437:
1417:
1384:
1360:
1327:
1298:
1278:
1239:
1219:
1217:{\displaystyle F_{p}}
1192:
1160:
1140:
1120:
1038:
1018:
998:
978:
958:
935:
912:
890:
866:
846:
818:
798:
778:
753:
724:
695:
666:
640:
614:
576:
554:
530:
500:
478:
450:partially ordered set
408:
388:
347:
327:
302:
261:
222:
200:
177:
153:
123:
100:
83:partially ordered set
41:
16:Maximal proper filter
5208:Nonstandard analysis
5149:, Berlin, New York:
4425:
4395:
4375:
4342:
4312:
4277:
4189:
4124:
4082:
4062:
4034:
4001:
3981:
3925:Filter (mathematics)
3896:social choice theory
3846:
3814:
3794:
3767:nonstandard analysis
3741:
3720:{\displaystyle |S|.}
3698:
3653:
3633:
3606:
3586:
3522:
3502:
3482:
3462:
3442:
3365:
3326:
3291:
3261:
3226:
3202:
3195:in the usual sense.
3171:
3147:
3111:
3081:
3046:
3022:
3002:
2967:
2934:
2914:
2886:
2863:
2806:
2749:
2723:
2703:
2679:
2659:
2635:
2607:
2581:
2561:
2538:
2518:
2489:
2462:
2442:
2422:
2393:
2369:
2349:
2322:
2298:
2274:
2241:
2218:
2198:
2165:
2125:
2099:
2026:
2003:
1977:
1948:
1923:
1899:
1875:
1855:
1829:
1809:
1789:
1766:
1739:
1719:
1699:
1598:
1574:
1550:
1530:
1507:
1466:
1446:
1426:
1393:
1373:
1336:
1332:Each ultrafilter on
1307:
1287:
1251:
1228:
1201:
1181:
1149:
1129:
1078:
1027:
1007:
1003:(that is, such that
987:
967:
947:
921:
901:
879:
855:
835:
807:
787:
761:
733:
707:
675:
649:
623:
588:
565:
543:
513:
487:
467:
397:
364:
336:
316:
278:
234:
211:
186:
166:
139:
109:
89:
3937:Filters in topology
3931:Filter (set theory)
3779:domain of discourse
3759:compactness theorem
3426:measurable cardinal
428:and combinatorics.
307:are usually called
4792:Topologie Générale
4437:{\displaystyle U.}
4434:
4407:
4381:
4348:
4324:{\displaystyle U.}
4321:
4298:
4259:
4175:
4101:
4068:
4040:
4020:
3987:
3852:
3820:
3800:
3747:
3717:
3672:
3639:
3619:
3592:
3572:
3508:
3488:
3468:
3448:
3371:
3351:
3312:
3277:
3247:
3212:
3189:countably additive
3177:
3153:
3133:
3087:
3067:
3032:
3008:
2988:
2953:
2920:
2896:
2869:
2849:
2792:
2735:
2709:
2689:
2665:
2645:
2617:
2593:
2577:or its complement
2567:
2550:{\displaystyle X,}
2547:
2524:
2501:
2472:
2448:
2428:
2405:
2379:
2355:
2332:
2304:
2284:
2260:
2237:an ultrafilter on
2230:{\displaystyle X,}
2227:
2204:
2184:
2147:
2111:{\displaystyle X,}
2108:
2064:2-valued morphisms
2038:
2012:
1989:
1963:
1935:{\displaystyle P,}
1932:
1905:
1887:{\displaystyle P,}
1884:
1861:
1841:{\displaystyle P,}
1838:
1815:
1795:
1772:
1760:Boolean complement
1748:
1725:
1705:
1610:{\displaystyle X.}
1607:
1580:
1556:
1536:
1519:{\displaystyle X.}
1516:
1485:
1452:
1432:
1412:
1379:
1355:
1322:
1293:
1273:
1234:
1214:
1187:
1155:
1135:
1115:
1033:
1013:
993:
973:
953:
933:{\displaystyle P,}
930:
907:
885:
861:
841:
813:
793:
773:
748:
719:
690:
661:
635:
609:
571:
549:
525:
495:
473:
403:
383:
342:
322:
297:
256:
217:
198:{\displaystyle P.}
195:
172:
151:{\displaystyle P;}
148:
121:{\displaystyle P,}
118:
95:
67:
27:specifically, see
5077:978-0-356-02077-8
5047:978-0-12-622760-4
4990:978-0-85226-444-7
4960:978-3-540-44085-7
4900:978-0-697-06889-7
4870:978-981-4571-52-4
4840:978-0-387-90972-1
4806:978-3-540-64241-1
4781:Bourbaki, Nicolas
4764:978-90-277-1355-1
4579:978-0-9880552-0-9
4384:{\displaystyle A}
4351:{\displaystyle U}
4230:
4222:
4071:{\displaystyle X}
4043:{\displaystyle X}
3990:{\displaystyle X}
3855:{\displaystyle U}
3840:first-order logic
3823:{\displaystyle U}
3803:{\displaystyle U}
3771:hyperreal numbers
3750:{\displaystyle X}
3684:topological space
3642:{\displaystyle G}
3491:{\displaystyle x}
3451:{\displaystyle G}
3374:{\displaystyle m}
3180:{\displaystyle m}
3165:almost everywhere
3156:{\displaystyle X}
3099:finitely additive
3090:{\displaystyle m}
3018:is an element of
3011:{\displaystyle A}
2923:{\displaystyle m}
2908:2-valued morphism
2872:{\displaystyle x}
2712:{\displaystyle X}
2668:{\displaystyle X}
2603:is an element of
2570:{\displaystyle A}
2527:{\displaystyle A}
2451:{\displaystyle B}
2431:{\displaystyle A}
2365:is an element of
2358:{\displaystyle A}
2307:{\displaystyle X}
2207:{\displaystyle X}
1908:{\displaystyle F}
1864:{\displaystyle F}
1818:{\displaystyle F}
1775:{\displaystyle x}
1728:{\displaystyle x}
1708:{\displaystyle x}
1635:) in the form of
1625:ultrafilter lemma
1583:{\displaystyle X}
1559:{\displaystyle X}
1539:{\displaystyle X}
1455:{\displaystyle U}
1435:{\displaystyle X}
1382:{\displaystyle U}
1296:{\displaystyle X}
1237:{\displaystyle p}
1190:{\displaystyle p}
1167:principal element
1158:{\displaystyle p}
1138:{\displaystyle p}
1125:for some element
1036:{\displaystyle F}
1016:{\displaystyle U}
996:{\displaystyle U}
956:{\displaystyle F}
910:{\displaystyle U}
844:{\displaystyle U}
816:{\displaystyle F}
796:{\displaystyle y}
574:{\displaystyle F}
415:almost everything
406:{\displaystyle X}
354:finitely additive
345:{\displaystyle X}
325:{\displaystyle X}
220:{\displaystyle X}
5215:
5203:Families of sets
5179:
5161:
5141:
5123:
5089:
5059:
5029:
5002:
4972:
4942:
4921:General topology
4912:
4882:
4852:
4827:General Topology
4823:Dixmier, Jacques
4818:
4776:
4729:
4728:
4718:
4692:
4686:
4685:
4679:
4671:
4653:
4635:
4626:
4620:
4619:
4599:
4593:
4590:
4584:
4583:
4571:
4560:
4554:
4553:
4545:
4539:
4533:
4527:
4526:
4506:
4500:
4499:
4489:
4476:
4475:
4473:
4464:
4444:
4443:
4441:
4440:
4435:
4416:
4414:
4413:
4408:
4390:
4388:
4387:
4382:
4369:
4363:
4357:
4355:
4354:
4349:
4337:
4331:
4330:
4328:
4327:
4322:
4307:
4305:
4304:
4299:
4297:
4293:
4292:
4273:. That is, some
4268:
4266:
4265:
4260:
4249:
4245:
4244:
4231:
4228:
4223:
4220:
4209:
4205:
4204:
4184:
4182:
4181:
4176:
4165:
4161:
4160:
4159:
4141:
4140:
4118:
4112:
4110:
4108:
4107:
4102:
4091:
4090:
4077:
4075:
4074:
4069:
4049:
4047:
4046:
4041:
4029:
4027:
4026:
4021:
4010:
4009:
3996:
3994:
3993:
3988:
3975:
3957:
3948:
3861:
3859:
3858:
3853:
3829:
3827:
3826:
3821:
3809:
3807:
3806:
3801:
3788:pointwise modulo
3777:, extending the
3756:
3754:
3753:
3748:
3733:construction in
3726:
3724:
3723:
3718:
3713:
3705:
3682:, the resulting
3681:
3679:
3678:
3673:
3662:
3661:
3648:
3646:
3645:
3640:
3628:
3626:
3625:
3620:
3618:
3617:
3601:
3599:
3598:
3593:
3581:
3579:
3578:
3573:
3568:
3564:
3534:
3533:
3517:
3515:
3514:
3509:
3497:
3495:
3494:
3489:
3477:
3475:
3474:
3469:
3457:
3455:
3454:
3449:
3434:
3430:
3389:Ultrafilters on
3380:
3378:
3377:
3372:
3360:
3358:
3357:
3352:
3347:
3346:
3321:
3319:
3318:
3313:
3286:
3284:
3283:
3278:
3276:
3275:
3256:
3254:
3253:
3248:
3221:
3219:
3218:
3213:
3211:
3210:
3186:
3184:
3183:
3178:
3162:
3160:
3159:
3154:
3142:
3140:
3139:
3134:
3120:
3119:
3096:
3094:
3093:
3088:
3077:otherwise. Then
3076:
3074:
3073:
3068:
3041:
3039:
3038:
3033:
3031:
3030:
3017:
3015:
3014:
3009:
2997:
2995:
2994:
2989:
2962:
2960:
2959:
2954:
2943:
2942:
2929:
2927:
2926:
2921:
2905:
2903:
2902:
2897:
2895:
2894:
2878:
2876:
2875:
2870:
2858:
2856:
2855:
2850:
2818:
2817:
2801:
2799:
2798:
2793:
2791:
2790:
2781:
2780:
2768:
2767:
2758:
2757:
2744:
2742:
2741:
2736:
2719:, there is some
2718:
2716:
2715:
2710:
2698:
2696:
2695:
2690:
2688:
2687:
2674:
2672:
2671:
2666:
2654:
2652:
2651:
2646:
2644:
2643:
2626:
2624:
2623:
2618:
2616:
2615:
2602:
2600:
2599:
2594:
2576:
2574:
2573:
2568:
2556:
2554:
2553:
2548:
2533:
2531:
2530:
2525:
2510:
2508:
2507:
2502:
2481:
2479:
2478:
2473:
2471:
2470:
2458:are elements of
2457:
2455:
2454:
2449:
2437:
2435:
2434:
2429:
2414:
2412:
2411:
2406:
2388:
2386:
2385:
2380:
2378:
2377:
2364:
2362:
2361:
2356:
2341:
2339:
2338:
2333:
2331:
2330:
2313:
2311:
2310:
2305:
2293:
2291:
2290:
2285:
2283:
2282:
2269:
2267:
2266:
2261:
2250:
2249:
2236:
2234:
2233:
2228:
2213:
2211:
2210:
2205:
2193:
2191:
2190:
2185:
2174:
2173:
2156:
2154:
2153:
2148:
2134:
2133:
2117:
2115:
2114:
2109:
2047:
2045:
2044:
2039:
2021:
2019:
2018:
2013:
1998:
1996:
1995:
1990:
1972:
1970:
1969:
1964:
1941:
1939:
1938:
1933:
1914:
1912:
1911:
1906:
1893:
1891:
1890:
1885:
1870:
1868:
1867:
1862:
1847:
1845:
1844:
1839:
1824:
1822:
1821:
1816:
1804:
1802:
1801:
1796:
1781:
1779:
1778:
1773:
1757:
1755:
1754:
1749:
1734:
1732:
1731:
1726:
1714:
1712:
1711:
1706:
1671:; for example,
1616:
1614:
1613:
1608:
1589:
1587:
1586:
1581:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1501:cofinite subsets
1494:
1492:
1491:
1486:
1475:
1474:
1461:
1459:
1458:
1453:
1441:
1439:
1438:
1433:
1421:
1419:
1418:
1413:
1402:
1401:
1388:
1386:
1385:
1380:
1367:principal filter
1364:
1362:
1361:
1356:
1345:
1344:
1331:
1329:
1328:
1323:
1302:
1300:
1299:
1294:
1282:
1280:
1279:
1274:
1260:
1259:
1243:
1241:
1240:
1235:
1223:
1221:
1220:
1215:
1213:
1212:
1196:
1194:
1193:
1188:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1090:
1089:
1054:
1053:
1042:
1040:
1039:
1034:
1022:
1020:
1019:
1014:
1002:
1000:
999:
994:
982:
980:
979:
974:
962:
960:
959:
954:
939:
937:
936:
931:
916:
914:
913:
908:
894:
892:
891:
886:
870:
868:
867:
862:
850:
848:
847:
842:
822:
820:
819:
814:
802:
800:
799:
794:
782:
780:
779:
774:
757:
755:
754:
749:
728:
726:
725:
720:
699:
697:
696:
691:
670:
668:
667:
662:
644:
642:
641:
636:
618:
616:
615:
610:
580:
578:
577:
572:
558:
556:
555:
550:
534:
532:
531:
526:
504:
502:
501:
496:
482:
480:
479:
474:
412:
410:
409:
404:
392:
390:
389:
384:
373:
372:
351:
349:
348:
343:
331:
329:
328:
323:
306:
304:
303:
298:
287:
286:
265:
263:
262:
257:
243:
242:
226:
224:
223:
218:
204:
202:
201:
196:
181:
179:
178:
173:
157:
155:
154:
149:
127:
125:
124:
119:
104:
102:
101:
96:
59:principal filter
5223:
5222:
5218:
5217:
5216:
5214:
5213:
5212:
5188:
5187:
5177:
5151:Springer-Verlag
5096:
5094:Further reading
5078:
5064:Schubert, Horst
5048:
5034:Schechter, Eric
5018:
4991:
4961:
4931:
4901:
4887:Dugundji, James
4871:
4857:Dolecki, Szymon
4841:
4831:Springer-Verlag
4807:
4765:
4747:Ponomarev, V.I.
4738:
4733:
4732:
4716:10.1.1.199.1970
4693:
4689:
4673:
4672:
4638:Economic Theory
4633:
4627:
4623:
4600:
4596:
4591:
4587:
4580:
4569:
4561:
4557:
4546:
4542:
4534:
4530:
4507:
4503:
4490:
4479:
4471:
4465:
4458:
4453:
4448:
4447:
4426:
4423:
4422:
4421:be elements of
4396:
4393:
4392:
4376:
4373:
4372:
4370:
4366:
4343:
4340:
4339:
4338:
4334:
4313:
4310:
4309:
4288:
4284:
4280:
4278:
4275:
4274:
4240:
4236:
4232:
4227:
4219:
4200:
4196:
4192:
4190:
4187:
4186:
4155:
4151:
4136:
4132:
4131:
4127:
4125:
4122:
4121:
4119:
4115:
4086:
4085:
4083:
4080:
4079:
4063:
4060:
4059:
4035:
4032:
4031:
4005:
4004:
4002:
3999:
3998:
3982:
3979:
3978:
3976:
3969:
3964:
3955:
3946:
3921:
3871:asymptotic cone
3847:
3844:
3843:
3815:
3812:
3811:
3795:
3792:
3791:
3742:
3739:
3738:
3709:
3701:
3699:
3696:
3695:
3694:of cardinality
3657:
3656:
3654:
3651:
3650:
3634:
3631:
3630:
3613:
3609:
3607:
3604:
3603:
3587:
3584:
3583:
3542:
3538:
3529:
3525:
3523:
3520:
3519:
3503:
3500:
3499:
3483:
3480:
3479:
3463:
3460:
3459:
3443:
3440:
3439:
3432:
3428:
3404:spaces, and in
3387:
3366:
3363:
3362:
3342:
3341:
3327:
3324:
3323:
3292:
3289:
3288:
3271:
3270:
3262:
3259:
3258:
3227:
3224:
3223:
3206:
3205:
3203:
3200:
3199:
3187:is usually not
3172:
3169:
3168:
3163:is either true
3148:
3145:
3144:
3115:
3114:
3112:
3109:
3108:
3082:
3079:
3078:
3047:
3044:
3043:
3026:
3025:
3023:
3020:
3019:
3003:
3000:
2999:
2968:
2965:
2964:
2938:
2937:
2935:
2932:
2931:
2915:
2912:
2911:
2890:
2889:
2887:
2884:
2883:
2864:
2861:
2860:
2813:
2809:
2807:
2804:
2803:
2786:
2785:
2776:
2772:
2763:
2762:
2753:
2752:
2750:
2747:
2746:
2724:
2721:
2720:
2704:
2701:
2700:
2683:
2682:
2680:
2677:
2676:
2660:
2657:
2656:
2639:
2638:
2636:
2633:
2632:
2611:
2610:
2608:
2605:
2604:
2582:
2579:
2578:
2562:
2559:
2558:
2539:
2536:
2535:
2534:is a subset of
2519:
2516:
2515:
2490:
2487:
2486:
2482:then so is the
2466:
2465:
2463:
2460:
2459:
2443:
2440:
2439:
2423:
2420:
2419:
2394:
2391:
2390:
2373:
2372:
2370:
2367:
2366:
2350:
2347:
2346:
2326:
2325:
2323:
2320:
2319:
2299:
2296:
2295:
2278:
2277:
2275:
2272:
2271:
2245:
2244:
2242:
2239:
2238:
2219:
2216:
2215:
2199:
2196:
2195:
2169:
2168:
2166:
2163:
2162:
2129:
2128:
2126:
2123:
2122:
2100:
2097:
2096:
2093:
2087:
2027:
2024:
2023:
2004:
2001:
2000:
1978:
1975:
1974:
1949:
1946:
1945:
1924:
1921:
1920:
1900:
1897:
1896:
1876:
1873:
1872:
1856:
1853:
1852:
1830:
1827:
1826:
1810:
1807:
1806:
1790:
1787:
1786:
1767:
1764:
1763:
1740:
1737:
1736:
1720:
1717:
1716:
1700:
1697:
1696:
1693:Boolean algebra
1689:
1629:axiom of choice
1599:
1596:
1595:
1575:
1572:
1571:
1551:
1548:
1547:
1531:
1528:
1527:
1508:
1505:
1504:
1470:
1469:
1467:
1464:
1463:
1447:
1444:
1443:
1427:
1424:
1423:
1397:
1396:
1394:
1391:
1390:
1374:
1371:
1370:
1365:that is also a
1340:
1339:
1337:
1334:
1333:
1308:
1305:
1304:
1288:
1285:
1284:
1255:
1254:
1252:
1249:
1248:
1229:
1226:
1225:
1208:
1204:
1202:
1199:
1198:
1182:
1179:
1178:
1150:
1147:
1146:
1130:
1127:
1126:
1085:
1081:
1079:
1076:
1075:
1056:
1051:
1050:
1028:
1025:
1024:
1008:
1005:
1004:
988:
985:
984:
968:
965:
964:
948:
945:
944:
922:
919:
918:
917:is a filter on
902:
899:
898:
880:
877:
876:
856:
853:
852:
836:
833:
832:
808:
805:
804:
788:
785:
784:
762:
759:
758:
734:
731:
730:
708:
705:
704:
676:
673:
672:
650:
647:
646:
624:
621:
620:
589:
586:
585:
566:
563:
562:
544:
541:
540:
514:
511:
510:
488:
485:
484:
468:
465:
464:
434:
398:
395:
394:
368:
367:
365:
362:
361:
337:
334:
333:
317:
314:
313:
282:
281:
279:
276:
275:
272:Boolean algebra
238:
237:
235:
232:
231:
212:
209:
208:
187:
184:
183:
167:
164:
163:
140:
137:
136:
110:
107:
106:
90:
87:
86:
36:
33:ultrafiltration
17:
12:
11:
5:
5221:
5211:
5210:
5205:
5200:
5186:
5185:
5174:
5162:
5142:
5114:(4): 417â455.
5095:
5092:
5091:
5090:
5076:
5060:
5046:
5030:
5017:978-1584888666
5016:
5003:
4989:
4973:
4959:
4943:
4929:
4913:
4899:
4883:
4869:
4853:
4839:
4819:
4805:
4777:
4763:
4737:
4734:
4731:
4730:
4709:(3): 267â277.
4687:
4651:10.1.1.200.520
4644:(2): 257â276.
4621:
4610:(2): 267â277.
4594:
4585:
4578:
4555:
4540:
4528:
4501:
4477:
4455:
4454:
4452:
4449:
4446:
4445:
4433:
4430:
4420:
4406:
4403:
4400:
4380:
4364:
4347:
4332:
4320:
4317:
4296:
4291:
4287:
4283:
4258:
4255:
4252:
4248:
4243:
4239:
4235:
4229: or
4226:
4221: or
4218:
4215:
4212:
4208:
4203:
4199:
4195:
4174:
4171:
4168:
4164:
4158:
4154:
4150:
4147:
4144:
4139:
4135:
4130:
4113:
4100:
4097:
4094:
4089:
4067:
4039:
4019:
4016:
4013:
4008:
3986:
3966:
3965:
3963:
3960:
3959:
3958:
3949:
3940:
3934:
3928:
3920:
3917:
3876:
3851:
3819:
3799:
3746:
3716:
3712:
3708:
3704:
3692:discrete space
3671:
3668:
3665:
3660:
3638:
3616:
3612:
3595:{\textstyle P}
3591:
3571:
3567:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3541:
3537:
3532:
3528:
3511:{\textstyle P}
3507:
3487:
3471:{\textstyle P}
3467:
3447:
3393:are useful in
3386:
3383:
3370:
3350:
3345:
3340:
3337:
3334:
3331:
3311:
3308:
3305:
3302:
3299:
3296:
3274:
3269:
3266:
3246:
3243:
3240:
3237:
3234:
3231:
3209:
3190:
3176:
3152:
3132:
3129:
3126:
3123:
3118:
3106:
3101:, and hence a
3086:
3066:
3063:
3060:
3057:
3054:
3051:
3029:
3007:
2987:
2984:
2981:
2978:
2975:
2972:
2952:
2949:
2946:
2941:
2919:
2893:
2868:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2816:
2812:
2789:
2784:
2779:
2775:
2771:
2766:
2761:
2756:
2734:
2731:
2728:
2708:
2699:of subsets of
2686:
2664:
2655:of subsets of
2642:
2629:
2628:
2614:
2592:
2589:
2586:
2566:
2546:
2543:
2523:
2512:
2500:
2497:
2494:
2469:
2447:
2427:
2416:
2404:
2401:
2398:
2376:
2354:
2343:
2329:
2303:
2281:
2259:
2256:
2253:
2248:
2226:
2223:
2203:
2183:
2180:
2177:
2172:
2146:
2143:
2140:
2137:
2132:
2107:
2104:
2089:Main article:
2086:
2083:
2082:
2081:
2078:
2075:
2066:) as follows:
2056:maximal ideals
2049:
2048:
2037:
2034:
2031:
2011:
2008:
1988:
1985:
1982:
1962:
1959:
1956:
1953:
1942:
1931:
1928:
1904:
1894:
1883:
1880:
1860:
1837:
1834:
1814:
1798:{\textstyle P}
1794:
1771:
1747:
1744:
1724:
1704:
1688:
1685:
1606:
1603:
1579:
1568:Fréchet filter
1555:
1535:
1515:
1512:
1497:Fréchet filter
1484:
1481:
1478:
1473:
1451:
1431:
1411:
1408:
1405:
1400:
1378:
1354:
1351:
1348:
1343:
1321:
1318:
1315:
1312:
1292:
1272:
1269:
1266:
1263:
1258:
1233:
1211:
1207:
1186:
1168:
1165:is called the
1154:
1134:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1088:
1084:
1055:
1048:
1047:
1046:
1045:
1044:
1032:
1012:
992:
976:{\textstyle P}
972:
952:
941:
929:
926:
906:
888:{\textstyle P}
884:
864:{\textstyle P}
860:
840:
826:
825:
824:
812:
792:
772:
769:
766:
747:
744:
741:
738:
718:
715:
712:
701:
689:
686:
683:
680:
660:
657:
654:
634:
631:
628:
608:
605:
602:
599:
596:
593:
582:
570:
552:{\textstyle P}
548:
524:
521:
518:
493:
476:{\textstyle P}
472:
458:proper filters
433:
430:
402:
382:
379:
376:
371:
341:
321:
312:
296:
293:
290:
285:
270:, is always a
255:
252:
249:
246:
241:
216:
194:
191:
175:{\textstyle P}
171:
147:
144:
117:
114:
98:{\textstyle P}
94:
64:
60:
15:
9:
6:
4:
3:
2:
5220:
5209:
5206:
5204:
5201:
5199:
5196:
5195:
5193:
5184:
5180:
5175:
5173:
5171:
5166:
5163:
5160:
5156:
5152:
5148:
5143:
5139:
5135:
5131:
5127:
5122:
5117:
5113:
5109:
5108:
5103:
5098:
5097:
5087:
5083:
5079:
5073:
5069:
5065:
5061:
5057:
5053:
5049:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5013:
5009:
5004:
5000:
4996:
4992:
4986:
4982:
4978:
4974:
4970:
4966:
4962:
4956:
4952:
4948:
4944:
4940:
4936:
4932:
4930:0-85274-275-4
4926:
4922:
4918:
4917:CsĂĄszĂĄr, Ăkos
4914:
4910:
4906:
4902:
4896:
4892:
4888:
4884:
4880:
4876:
4872:
4866:
4862:
4858:
4854:
4850:
4846:
4842:
4836:
4832:
4828:
4824:
4820:
4816:
4812:
4808:
4802:
4798:
4794:
4793:
4788:
4787:
4782:
4778:
4774:
4770:
4766:
4760:
4756:
4752:
4748:
4744:
4740:
4739:
4726:
4722:
4717:
4712:
4708:
4704:
4703:
4698:
4691:
4683:
4677:
4669:
4665:
4661:
4657:
4652:
4647:
4643:
4639:
4632:
4625:
4617:
4613:
4609:
4605:
4598:
4589:
4581:
4575:
4568:
4567:
4559:
4551:
4544:
4537:
4532:
4524:
4520:
4516:
4512:
4505:
4497:
4496:
4488:
4486:
4484:
4482:
4470:
4463:
4461:
4456:
4431:
4428:
4418:
4404:
4398:
4378:
4368:
4361:
4345:
4336:
4318:
4315:
4294:
4289:
4285:
4281:
4272:
4256:
4253:
4250:
4246:
4241:
4237:
4233:
4224:
4216:
4213:
4210:
4206:
4201:
4197:
4193:
4172:
4169:
4166:
4162:
4156:
4152:
4148:
4145:
4142:
4137:
4133:
4128:
4117:
4095:
4065:
4057:
4053:
4037:
4014:
3984:
3974:
3972:
3967:
3953:
3952:Universal net
3950:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3922:
3916:
3913:
3909:
3905:
3901:
3897:
3892:
3890:
3886:
3884:
3883:metric spaces
3880:
3874:
3872:
3868:
3863:
3849:
3841:
3837:
3833:
3817:
3797:
3789:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3744:
3736:
3732:
3727:
3714:
3706:
3693:
3689:
3685:
3666:
3636:
3614:
3610:
3589:
3569:
3565:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3539:
3535:
3530:
3526:
3505:
3485:
3465:
3445:
3436:
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3400:
3396:
3392:
3382:
3368:
3348:
3338:
3335:
3329:
3309:
3306:
3300:
3294:
3267:
3264:
3244:
3241:
3235:
3229:
3198:For a filter
3196:
3194:
3188:
3174:
3166:
3150:
3130:
3124:
3105:
3102:
3100:
3084:
3064:
3061:
3055:
3049:
3005:
2985:
2982:
2976:
2970:
2947:
2917:
2910:, a function
2909:
2880:
2866:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2819:
2814:
2810:
2782:
2777:
2773:
2769:
2759:
2732:
2729:
2726:
2706:
2662:
2590:
2584:
2564:
2544:
2541:
2521:
2513:
2498:
2495:
2492:
2485:
2445:
2425:
2417:
2402:
2399:
2396:
2352:
2344:
2317:
2316:
2315:
2301:
2254:
2224:
2221:
2201:
2178:
2160:
2159:set inclusion
2144:
2138:
2121:
2105:
2102:
2092:
2079:
2076:
2073:
2072:inverse image
2069:
2068:
2067:
2065:
2061:
2060:homomorphisms
2057:
2052:
2035:
2032:
2029:
2009:
1986:
1983:
1980:
1960:
1957:
1954:
1951:
1943:
1929:
1926:
1918:
1902:
1895:
1881:
1878:
1858:
1851:
1850:
1849:
1835:
1832:
1812:
1792:
1783:
1769:
1761:
1745:
1722:
1702:
1694:
1684:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1617:
1604:
1601:
1593:
1577:
1569:
1553:
1533:
1513:
1510:
1502:
1498:
1479:
1449:
1429:
1406:
1376:
1368:
1349:
1319:
1316:
1313:
1310:
1290:
1270:
1264:
1245:
1231:
1209:
1205:
1184:
1176:
1175:non-principal
1172:
1166:
1152:
1132:
1109:
1106:
1103:
1100:
1097:
1091:
1086:
1082:
1073:
1072:least element
1069:
1065:
1061:
1030:
1010:
990:
970:
950:
942:
927:
924:
904:
897:
896:
882:
874:
871:is called an
858:
838:
831:
830:proper subset
827:
810:
790:
783:implies that
770:
767:
764:
745:
742:
739:
736:
716:
713:
710:
702:
687:
684:
681:
678:
658:
655:
652:
632:
629:
626:
606:
603:
600:
597:
594:
591:
583:
568:
561:
560:
546:
538:
522:
519:
516:
508:
507:
506:
491:
470:
463:Formally, if
461:
459:
455:
451:
447:
443:
439:
429:
427:
423:
418:
416:
400:
377:
359:
355:
339:
319:
311:
308:
291:
273:
269:
268:set inclusion
253:
247:
230:
214:
205:
192:
189:
169:
161:
160:proper filter
145:
142:
134:
131:
115:
112:
92:
85:(or "poset")
84:
80:
76:
72:
62:
61:, but not an
58:
56:
52:
51:is divisor of
48:
44:
43:Hasse diagram
40:
34:
30:
26:
22:
5198:Order theory
5169:
5146:
5111:
5105:
5067:
5037:
5007:
4980:
4977:Joshi, K. D.
4950:
4947:Jech, Thomas
4920:
4890:
4860:
4826:
4790:
4785:
4750:
4736:Bibliography
4706:
4700:
4690:
4676:cite journal
4641:
4637:
4624:
4607:
4603:
4597:
4588:
4565:
4558:
4549:
4543:
4531:
4514:
4504:
4494:
4367:
4335:
4116:
4055:
4051:
3911:
3903:
3899:
3893:
3887:
3864:
3836:ĆoĆ' theorem
3775:real numbers
3735:model theory
3731:ultraproduct
3728:
3437:
3406:model theory
3388:
3385:Applications
3197:
2881:
2630:
2557:then either
2484:intersection
2094:
2053:
2050:
1917:prime filter
1784:
1690:
1680:
1668:
1664:
1660:
1656:
1648:
1640:
1637:Zorn's lemma
1632:
1618:
1246:
1174:
1170:
1067:
1063:
1059:
1057:
872:
581:is nonempty,
536:
535:is called a
462:
441:
438:order theory
435:
422:model theory
419:
206:
78:
75:order theory
71:mathematical
68:
50:
21:order theory
5165:Ultrafilter
3879:ultralimits
2963:defined as
2314:such that:
2157:ordered by
873:ultrafilter
442:ultrafilter
356:0-1-valued
266:ordered by
158:that is, a
81:on a given
79:ultrafilter
63:ultrafilter
53:, with the
5192:Categories
4451:References
3904:infinitely
3418:set theory
3391:power sets
2745:such that
1659:) and the
1197:, the set
703:for every
645:such that
584:for every
456:among all
5130:0002-9904
5056:175294365
5026:144216834
4909:395340485
4879:945169917
4783:(1989) .
4755:D. Reidel
4711:CiteSeerX
4646:CiteSeerX
4402:∖
4251:∈
4225:…
4211:∈
4167:∈
4146:…
3832:index set
3559:∈
3547:∈
3402:Hausdorff
3339:∈
3333:∖
3268:∈
2841:∈
2829:⊆
2783:∩
2760:∩
2730:∈
2588:∖
2496:∩
2400:⊃
2270:is a set
2120:power set
2030:∈
2007:¬
1984:∈
1955:∈
1944:for each
1743:¬
1314:∈
1107:≤
1060:principal
768:≤
740:∈
714:∈
682:≤
656:≤
630:∈
601:∈
520:⊆
509:a subset
492:≤
229:power set
128:namely a
73:field of
55:upper set
5068:Topology
5066:(1968).
5036:(1996).
4979:(1983).
4969:50422939
4949:(2006).
4919:(1978).
4891:Topology
4889:(1966).
4849:10277303
4825:(1984).
4815:18588129
4749:(1984).
4668:15398169
3919:See also
3912:finitely
3810:, where
3783:superset
3438:The set
3395:topology
3361:leaving
452:that is
426:topology
47:divisors
5183:YouTube
5167:at the
5159:0396267
5138:0454893
4999:9218750
4939:4146011
4795:].
4773:9944489
4417:cannot
3686:is the
3399:compact
3193:measure
3104:content
1973:either
1068:trivial
454:maximal
358:measure
130:maximal
69:In the
45:of the
5157:
5136:
5128:
5086:463753
5084:
5074:
5054:
5044:
5024:
5014:
4997:
4987:
4967:
4957:
4937:
4927:
4907:
4897:
4877:
4867:
4847:
4837:
4813:
4803:
4771:
4761:
4713:
4666:
4648:
4576:
3769:, the
3518:, let
2802:where
803:is in
537:filter
505:then
446:subset
133:filter
4789:[
4664:S2CID
4634:(PDF)
4570:(PDF)
4472:(PDF)
4360:above
4185:then
3962:Notes
3842:. If
3690:of a
3416:. In
2906:is a
1915:is a
1673:Gödel
1066:, or
1064:fixed
448:of a
444:is a
440:, an
77:, an
5126:ISSN
5082:OCLC
5072:ISBN
5052:OCLC
5042:ISBN
5022:OCLC
5012:ISBN
4995:OCLC
4985:ISBN
4965:OCLC
4955:ISBN
4935:OCLC
4925:ISBN
4905:OCLC
4895:ISBN
4875:OCLC
4865:ISBN
4845:OCLC
4835:ISBN
4811:OCLC
4801:ISBN
4769:OCLC
4759:ISBN
4682:link
4574:ISBN
4419:both
4391:and
3910:for
3729:The
3287:and
3042:and
2438:and
2118:its
2058:and
1999:or (
1735:and
1649:BPIT
1173:(or
1171:free
1062:(or
823:too;
729:and
671:and
25:sets
5181:on
5172:Lab
5116:doi
4721:doi
4656:doi
4612:doi
4519:doi
3977:If
3894:In
3881:of
3865:In
3498:of
3322:if
3257:if
3107:on
3097:is
2998:if
2930:on
2514:If
2418:If
2345:If
1919:on
1785:If
1782:):
1762:of
1669:ZFC
1665:ZFC
1594:of
1526:If
1503:of
1499:of
1462:on
1389:on
963:on
940:and
895:if
875:on
851:of
700:and
559:if
539:on
436:In
360:on
207:If
162:on
135:on
5194::
5155:MR
5153:,
5134:MR
5132:.
5124:.
5112:83
5110:.
5104:.
5080:.
5050:.
5020:.
4993:.
4963:.
4933:.
4903:.
4873:.
4843:.
4833:.
4809:.
4767:.
4757:.
4745:;
4719:.
4707:32
4705:.
4699:.
4678:}}
4674:{{
4662:.
4654:.
4642:10
4640:.
4636:.
4606:.
4572:.
4513:.
4480:^
4459:^
4111:".
4056:on
4052:of
3970:^
3885:.
3790:"
2022:)
1681:ZF
1661:ZF
1657:ZF
1641:AC
1633:AC
1043:).
828:a
424:,
5170:n
5140:.
5118::
5088:.
5058:.
5028:.
5001:.
4971:.
4941:.
4911:.
4881:.
4851:.
4817:.
4775:.
4727:.
4723::
4684:)
4658::
4618:.
4614::
4608:5
4582:.
4525:.
4521::
4432:.
4429:U
4405:A
4399:X
4379:A
4346:U
4319:.
4316:U
4295:}
4290:i
4286:x
4282:{
4257:,
4254:U
4247:}
4242:n
4238:x
4234:{
4217:,
4214:U
4207:}
4202:1
4198:x
4194:{
4173:,
4170:U
4163:}
4157:n
4153:x
4149:,
4143:,
4138:1
4134:x
4129:{
4099:)
4096:X
4093:(
4088:P
4066:X
4038:X
4018:)
4015:X
4012:(
4007:P
3985:X
3850:U
3818:U
3798:U
3745:X
3715:.
3711:|
3707:S
3703:|
3670:)
3667:S
3664:(
3659:P
3637:G
3615:x
3611:D
3590:P
3570:.
3566:}
3562:U
3556:x
3553::
3550:G
3544:U
3540:{
3536:=
3531:x
3527:D
3506:P
3486:x
3466:P
3446:G
3433:Îș
3429:Îș
3369:m
3349:,
3344:F
3336:A
3330:X
3310:0
3307:=
3304:)
3301:A
3298:(
3295:m
3273:F
3265:A
3245:1
3242:=
3239:)
3236:A
3233:(
3230:m
3208:F
3175:m
3151:X
3131:,
3128:)
3125:X
3122:(
3117:P
3085:m
3065:0
3062:=
3059:)
3056:A
3053:(
3050:m
3028:U
3006:A
2986:1
2983:=
2980:)
2977:A
2974:(
2971:m
2951:)
2948:X
2945:(
2940:P
2918:m
2892:U
2867:x
2847:}
2844:Y
2838:x
2835::
2832:X
2826:Y
2823:{
2820:=
2815:x
2811:F
2788:F
2778:x
2774:F
2770:=
2765:F
2755:U
2733:X
2727:x
2707:X
2685:F
2663:X
2641:U
2627:.
2613:U
2591:A
2585:X
2565:A
2545:,
2542:X
2522:A
2511:.
2499:B
2493:A
2468:U
2446:B
2426:A
2415:.
2403:A
2397:B
2375:U
2353:A
2342:.
2328:U
2302:X
2280:U
2258:)
2255:X
2252:(
2247:P
2225:,
2222:X
2202:X
2182:)
2179:X
2176:(
2171:P
2145:,
2142:)
2139:X
2136:(
2131:P
2106:,
2103:X
2036:.
2033:F
2010:x
1987:F
1981:x
1961:,
1958:P
1952:x
1930:,
1927:P
1903:F
1882:,
1879:P
1859:F
1836:,
1833:P
1813:F
1793:P
1770:x
1746:x
1723:x
1703:x
1655:(
1647:(
1631:(
1605:.
1602:X
1578:X
1554:X
1534:X
1514:.
1511:X
1483:)
1480:X
1477:(
1472:P
1450:U
1430:X
1410:)
1407:X
1404:(
1399:P
1377:U
1353:)
1350:X
1347:(
1342:P
1320:.
1317:X
1311:x
1291:X
1271:,
1268:)
1265:X
1262:(
1257:P
1232:p
1210:p
1206:F
1185:p
1153:p
1133:p
1113:}
1110:x
1104:p
1101::
1098:x
1095:{
1092:=
1087:p
1083:F
1031:F
1011:U
991:U
971:P
951:F
928:,
925:P
905:U
883:P
859:P
839:U
811:F
791:y
771:y
765:x
746:,
743:P
737:y
717:F
711:x
688:,
685:y
679:z
659:x
653:z
633:F
627:z
607:,
604:F
598:y
595:,
592:x
569:F
547:P
523:P
517:F
471:P
401:X
381:)
378:X
375:(
370:P
340:X
320:X
295:)
292:X
289:(
284:P
254:,
251:)
248:X
245:(
240:P
215:X
193:.
190:P
170:P
146:;
143:P
116:,
113:P
93:P
35:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
