4795:
4011:
3973:
3966:
3881:
3874:
3867:
3792:
3785:
3778:
3765:
3758:
3751:
3697:
3690:
3683:
3670:
3663:
3585:
3542:
3535:
3514:
3507:
3500:
3368:
3361:
3297:
3276:
3222:
3215:
3201:
3147:
3140:
3133:
3126:
3079:
3072:
3065:
3058:
3051:
3044:
3037:
2942:
2899:
2892:
2864:
2857:
2850:
2779:
2772:
2765:
2758:
2751:
2725:
2718:
2711:
2704:
2697:
2690:
2645:
2638:
2631:
2624:
2617:
2610:
2580:
2573:
2566:
2559:
2552:
2545:
2538:
2531:
4729:
3980:
3888:
4025:
4018:
3959:
3952:
3945:
3926:
3919:
3860:
3853:
3846:
3830:
3735:
3656:
3637:
3599:
3592:
3572:
3565:
3558:
3528:
3521:
3493:
3486:
3479:
3460:
3453:
3446:
3439:
3432:
3425:
3418:
3411:
3404:
3382:
3375:
3354:
3347:
3340:
3333:
3326:
3304:
3290:
3283:
3269:
3262:
3255:
3248:
3229:
3208:
3194:
3187:
3180:
3173:
3154:
3119:
3112:
3105:
3098:
3030:
3023:
3016:
2994:
2987:
2935:
2885:
2732:
2683:
2676:
2652:
2603:
2596:
2524:
2517:
1334:, specifying which points belong to which lines. An incidence structure can be specified by a family of sets (even if two distinct lines contain the same set of points), the sets of points belonging to each line, and any family of sets can be interpreted as an incidence structure in this way.
1311:, joined face to face. In an abstract simplicial complex, each simplex is represented simply as the set of its vertices. Any family of finite sets without repetitions in which the subsets of any set in the family also belong to the family forms an abstract simplicial complex.
1129:
868:
1296:, each of which may be an arbitrary set. The hyperedges of a hypergraph form a family of sets, and any family of sets can be interpreted as a hypergraph that has the union of the sets as its vertices.
752:
3824:
3729:
3631:
4499:
1188:
2491:
2425:
2376:
2458:
4442:
4062:
2149:
1278:
Certain types of objects from other areas of mathematics are equivalent to families of sets, in that they can be described purely as a collection of sets of objects of some type:
1269:
944:
1239:
4353:
2327:
922:
643:
4384:
4318:
1860:
1156:
4466:
4091:
2975:
2844:
2812:
2178:
2068:
1833:
1809:
1779:
1212:
1072:
4287:
4228:
2929:
2298:
2210:
1510:
1390:
343:
4202:
4158:
2269:
2240:
2090:
1016:
561:
375:
4003:
3911:
1434:
1991:
1981:
1694:
1463:
506:
418:
147:
1958:
1934:
1716:
1667:
1635:
1615:
1595:
1575:
1530:
1487:
1410:
984:
663:
581:
527:
481:
460:
440:
395:
311:
287:
249:
219:
199:
179:
120:
96:
69:
1077:
5252:
1353:. A block code can also be described as a family of sets, by describing each codeword as the set of positions at which it contains a 1.
221:. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a
2110:
757:
672:
5402:
4712:
4941:
4761:
3799:
3704:
3606:
1750:
5269:
1669:
lies in only finitely many members of the family. If every point of a cover lies in exactly one member, the cover is a
4471:
4694:
4676:
1161:
2466:
1542:
1043:, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a
5247:
2384:
2335:
5127:
2433:
2103:
5021:
4900:
4733:
4389:
5264:
4235:
4040:
2497:
2127:
1244:
929:
5257:
4895:
4858:
1912:
1783:
1739:
1300:
1217:
1044:
4332:
2306:
1036:
4912:
873:
589:
5407:
4946:
4838:
4826:
4821:
4617:, p.34. The University Series in Undergraduate Mathematics, 1960. Litton Educational Publishing, Inc.
4362:
4296:
2096:
1838:
1734:
1342:
1134:
4447:
4072:
2950:
2819:
2787:
2159:
2049:
1814:
1790:
1760:
1193:
1053:
4754:
4266:
4213:
2908:
2277:
2194:
1495:
509:
463:
5366:
5284:
5159:
5111:
4925:
4848:
1645:
4526: – Collection of sets in mathematics that can be defined based on a property of its members
261:
concerns the largest and smallest examples of families of sets satisfying certain restrictions.
5318:
5199:
5011:
4831:
4547:
1363:
316:
5234:
5204:
5148:
5068:
5048:
5026:
4187:
4143:
4093:
2248:
2219:
2180:
2075:
1350:
1287:
992:
533:
351:
5308:
5298:
5132:
5063:
5016:
4956:
4843:
4577:
4529:
3988:
3936:
3896:
3837:
1883:
1438:
1419:
1307:, a shape formed by unions of line segments, triangles, tetrahedra, and higher-dimensional
1895:
is a set family such that any minimal subfamily with empty intersection has bounded size.
8:
5303:
5214:
5122:
5117:
4931:
4873:
4811:
4747:
3549:
2023:
1999:
1546:
1315:
1124:{\displaystyle \cup {\mathcal {F}}:={\textstyle \bigcup \limits _{F\in {\mathcal {F}}}}F}
258:
1963:
1744:
1676:
1445:
488:
400:
129:
5226:
5221:
5006:
4961:
4868:
4523:
4324:
2663:
2035:
2016:
1943:
1919:
1896:
1701:
1670:
1652:
1620:
1600:
1580:
1560:
1515:
1472:
1395:
1304:
1026:
969:
955:
648:
566:
512:
466:
445:
425:
380:
296:
272:
234:
222:
204:
184:
164:
105:
81:
54:
48:
5083:
4920:
4883:
4853:
4784:
4708:
4690:
4672:
4101:
1719:
1357:
76:
36:
4251:
2588:
5371:
5361:
5346:
5341:
5209:
4863:
4544: – Algebraic concept in measure theory, also referred to as an algebra of sets
1553:
1346:
4556: – Collection of objects, each associated with an element from some index set
5240:
5178:
4996:
4816:
4517:
2007:
1937:
1904:
1864:
1327:
1030:
1022:
5376:
5173:
5154:
5058:
5000:
4936:
4878:
4553:
4356:
4290:
3742:
2742:
1878:
947:
1532:
itself, and is closed under arbitrary set unions and finite set intersections.
5396:
5381:
5183:
5097:
5092:
4559:
4541:
3314:
2980:
2871:
5351:
5331:
5326:
5144:
5073:
5031:
4890:
4794:
4571:
3086:
3004:
2188:
1891:
4587:
3392:
5356:
4991:
2031:
2012:
1989:
is an abstract simplicial complex with an additional property called the
1754:
is a family that is the union of countably many locally finite families.
1345:
of 0s and 1s, all the same length. When each pair of codewords has large
1040:
161:. Additionally, a family of sets may be defined as a function from a set
40:
24:
5336:
5107:
4770:
4593:
4535:
3470:
3239:
3164:
1900:
1728:
1338:
1292:
1283:
253:
229:
20:
1882:
is a set family in which none of the sets contains any of the others.
5139:
5102:
5053:
4951:
4565:
4562: – Family closed under complements and countable disjoint unions
4257:
3644:
2506:
1490:
987:
346:
290:
2027:
2011:
is a set family closed under arbitrary intersections and unions of
1723:
1639:
666:
44:
1985:
1597:
belongs to some member of the family. A subfamily of a cover of
1308:
201:, in which case the sets of the family are indexed by members of
149:
More generally, a collection of any sets whatsoever is called a
5164:
4986:
4728:
72:
3979:
3887:
954:
family of sets. That is, it is not itself a set but instead a
35:) can mean, depending upon the context, any of the following:
5036:
4803:
4739:
4550: – Expression denoting a set of sets in formal semantics
1742:
that intersects only finitely many members of the family. A
863:{\displaystyle A_{1}=\{a,b,c\},A_{2}=\{1,2\},A_{3}=\{1,2\},}
4574: – Family closed under unions and relative complements
747:{\displaystyle F=\left\{A_{1},A_{2},A_{3},A_{4}\right\},}
4689:(5th ed.), Upper Saddle River, NJ: Prentice Hall,
1095:
4474:
4450:
4392:
4365:
4335:
4299:
4269:
4216:
4190:
4146:
4075:
4043:
3991:
3899:
3802:
3707:
3609:
2953:
2911:
2822:
2790:
2469:
2436:
2387:
2338:
2309:
2280:
2251:
2222:
2197:
2162:
2130:
2078:
2052:
1966:
1946:
1922:
1841:
1817:
1793:
1763:
1704:
1679:
1655:
1623:
1603:
1583:
1563:
1518:
1498:
1475:
1465:
which is a family of sets (whose elements are called
1448:
1422:
1398:
1366:
1247:
1220:
1196:
1164:
1137:
1080:
1056:
995:
972:
932:
876:
760:
675:
651:
592:
569:
536:
515:
491:
469:
448:
428:
403:
383:
354:
319:
299:
275:
237:
207:
187:
167:
132:
108:
84:
57:
4596: – Family of sets closed under countable unions
4520: – Identities and relationships involving sets
3819:{\displaystyle \varnothing \not \in {\mathcal {F}}}
3724:{\displaystyle \varnothing \not \in {\mathcal {F}}}
3626:{\displaystyle \varnothing \not \in {\mathcal {F}}}
1341:consists of a set of codewords, each of which is a
4493:
4460:
4436:
4378:
4347:
4312:
4281:
4222:
4196:
4152:
4085:
4056:
3997:
3905:
3818:
3723:
3625:
2969:
2923:
2838:
2806:
2485:
2452:
2419:
2370:
2321:
2292:
2263:
2234:
2204:
2172:
2143:
2084:
2062:
1975:
1952:
1928:
1854:
1827:
1803:
1773:
1710:
1688:
1661:
1629:
1609:
1589:
1569:
1524:
1504:
1481:
1457:
1428:
1404:
1384:
1303:is a combinatorial abstraction of the notion of a
1286:, also called a set system, is formed by a set of
1263:
1233:
1206:
1182:
1150:
1123:
1066:
1010:
978:
938:
916:
862:
746:
657:
637:
575:
555:
521:
500:
475:
454:
434:
412:
389:
369:
337:
305:
281:
243:
213:
193:
173:
141:
114:
90:
63:
4538: – Ring closed under countable intersections
5394:
4568: – Family of sets closed under intersection
4494:{\displaystyle {\mathcal {F}}\neq \varnothing .}
1871:
1183:{\displaystyle \cup \varnothing =\varnothing .}
2486:{\displaystyle \varnothing \in {\mathcal {F}}}
1543:List of partition topics § Set partitions
4755:
4702:
4638:
2104:
1886:bounds the maximum size of a Sperner family.
4532: – Symmetric arrangement of finite sets
1021:Any family of sets without repetitions is a
908:
890:
854:
842:
823:
811:
792:
774:
629:
599:
2420:{\displaystyle A_{1}\cup A_{2}\cup \cdots }
2371:{\displaystyle A_{1}\cap A_{2}\cap \cdots }
16:Any collection of sets, or subsets of a set
4762:
4748:
4590: – Algebraic structure of set algebra
4582:Set of sets that do not contain themselves
2111:
2097:
1907:of bounded dimension form Helly families.
2453:{\displaystyle \Omega \in {\mathcal {F}}}
2198:
4703:Roberts, Fred S.; Tesman, Barry (2009),
1536:
4707:(2nd ed.), Boca Raton: CRC Press,
4684:
4626:
4437:{\displaystyle A,B,A_{1},A_{2},\ldots }
1241:and also a family over any superset of
5395:
269:The set of all subsets of a given set
4743:
4666:
4650:
4329:is a semiring where every complement
4057:{\displaystyle {\mathcal {F}}\colon }
2144:{\displaystyle {\mathcal {F}}\colon }
1940:; that is, every subset of a set in
1936:(consisting of finite sets) that is
1868:is a particular type of refinement.
1412:is a set (whose elements are called
1264:{\displaystyle \cup {\mathcal {F}}.}
939:{\displaystyle \operatorname {Ord} }
645:An example of a family of sets over
4022:
4015:
3977:
3956:
3949:
3942:
3923:
3916:
3885:
3857:
3850:
3843:
3827:
3732:
3653:
3634:
3596:
3589:
3569:
3562:
3555:
3525:
3518:
3490:
3483:
3476:
3457:
3450:
3443:
3436:
3429:
3422:
3415:
3408:
3401:
3379:
3372:
3351:
3344:
3337:
3330:
3323:
3301:
3287:
3280:
3266:
3259:
3252:
3245:
3226:
3205:
3191:
3184:
3177:
3170:
3151:
3116:
3109:
3102:
3095:
3027:
3020:
3013:
2991:
2984:
2946:
2932:
2903:
2882:
2815:
2783:
2729:
2680:
2673:
2649:
2600:
2593:
2521:
2514:
2022:Other examples of set families are
1751:countably locally finite collection
1273:
1234:{\displaystyle \cup {\mathcal {F}}}
1097:
13:
4580: – Paradox in set theory (or
4477:
4453:
4368:
4348:{\displaystyle \Omega \setminus A}
4336:
4302:
4191:
4147:
4078:
4046:
4029:
4008:
3970:
3963:
3930:
3878:
3871:
3864:
3811:
3796:
3789:
3782:
3775:
3772:
3769:
3762:
3755:
3748:
3716:
3701:
3694:
3687:
3680:
3677:
3674:
3667:
3660:
3618:
3603:
3582:
3579:
3576:
3539:
3532:
3511:
3504:
3497:
3464:
3386:
3365:
3358:
3308:
3294:
3273:
3233:
3219:
3212:
3198:
3158:
3144:
3137:
3130:
3123:
3076:
3069:
3062:
3055:
3048:
3041:
3034:
2998:
2939:
2896:
2889:
2861:
2854:
2847:
2776:
2769:
2762:
2755:
2748:
2736:
2722:
2715:
2708:
2701:
2694:
2687:
2656:
2642:
2635:
2628:
2621:
2614:
2607:
2577:
2570:
2563:
2556:
2549:
2542:
2535:
2528:
2478:
2445:
2437:
2322:{\displaystyle \Omega \setminus A}
2310:
2165:
2133:
2079:
2055:
1844:
1820:
1796:
1766:
1253:
1226:
1199:
1140:
1109:
1086:
1059:
1045:system of distinct representatives
996:
355:
320:
14:
5419:
4721:
4485:
4339:
4273:
2313:
2284:
1738:if each point in the space has a
1174:
1168:
1131:denotes the union of all sets in
4793:
4727:
4023:
4016:
4009:
3978:
3971:
3964:
3957:
3950:
3943:
3924:
3917:
3886:
3879:
3872:
3865:
3858:
3851:
3844:
3828:
3790:
3783:
3776:
3763:
3756:
3749:
3733:
3695:
3688:
3681:
3668:
3661:
3654:
3635:
3597:
3590:
3583:
3570:
3563:
3556:
3540:
3533:
3526:
3519:
3512:
3505:
3498:
3491:
3484:
3477:
3458:
3451:
3444:
3437:
3430:
3423:
3416:
3409:
3402:
3380:
3373:
3366:
3359:
3352:
3345:
3338:
3331:
3324:
3302:
3295:
3288:
3281:
3274:
3267:
3260:
3253:
3246:
3227:
3220:
3213:
3206:
3199:
3192:
3185:
3178:
3171:
3152:
3145:
3138:
3131:
3124:
3117:
3110:
3103:
3096:
3077:
3070:
3063:
3056:
3049:
3042:
3035:
3028:
3021:
3014:
2992:
2985:
2940:
2933:
2897:
2890:
2883:
2862:
2855:
2848:
2777:
2770:
2763:
2756:
2749:
2730:
2723:
2716:
2709:
2702:
2695:
2688:
2681:
2674:
2650:
2643:
2636:
2629:
2622:
2615:
2608:
2601:
2594:
2578:
2571:
2564:
2557:
2550:
2543:
2536:
2529:
2522:
2515:
1722:, a cover whose members are all
1018:if it has no repeated members.
917:{\displaystyle A_{4}=\{a,b,1\}.}
638:{\displaystyle S=\{a,b,c,1,2\}.}
228:A finite family of subsets of a
4379:{\displaystyle {\mathcal {F}}.}
4313:{\displaystyle {\mathcal {F}}.}
1855:{\displaystyle {\mathcal {C}}.}
1835:is contained in some member of
1151:{\displaystyle {\mathcal {F}},}
966:Any family of subsets of a set
4769:
4644:
4632:
4620:
4607:
4461:{\displaystyle {\mathcal {F}}}
4086:{\displaystyle {\mathcal {F}}}
2970:{\displaystyle A_{i}\nearrow }
2964:
2839:{\displaystyle A_{i}\nearrow }
2833:
2807:{\displaystyle A_{i}\searrow }
2801:
2173:{\displaystyle {\mathcal {F}}}
2063:{\displaystyle {\mathcal {F}}}
1828:{\displaystyle {\mathcal {F}}}
1804:{\displaystyle {\mathcal {C}}}
1774:{\displaystyle {\mathcal {F}}}
1379:
1367:
1207:{\displaystyle {\mathcal {F}}}
1067:{\displaystyle {\mathcal {F}}}
1005:
999:
548:
542:
364:
358:
329:
323:
1:
4660:
1872:Special types of set families
1290:together with another set of
961:
181:, known as the index set, to
5403:Basic concepts in set theory
4685:Brualdi, Richard A. (2010),
4282:{\displaystyle B\setminus A}
4223:{\displaystyle \varnothing }
2924:{\displaystyle A\subseteq B}
2293:{\displaystyle B\setminus A}
2205:{\displaystyle \,\supseteq }
1551:A family of sets is said to
1505:{\displaystyle \varnothing }
7:
4671:, Oxford: Clarendon Press,
4511:
4024:
4017:
3958:
3951:
3944:
3925:
3918:
3859:
3852:
3845:
3829:
3734:
3655:
3636:
3598:
3591:
3571:
3564:
3557:
3527:
3520:
3492:
3485:
3478:
3459:
3452:
3445:
3438:
3431:
3424:
3417:
3410:
3403:
3381:
3374:
3353:
3346:
3339:
3332:
3325:
3303:
3289:
3282:
3268:
3261:
3254:
3247:
3228:
3207:
3193:
3186:
3179:
3172:
3153:
3118:
3111:
3104:
3097:
3029:
3022:
3015:
2993:
2986:
2934:
2884:
2731:
2682:
2675:
2651:
2602:
2595:
2523:
2516:
1913:abstract simplicial complex
1301:abstract simplicial complex
1074:is any family of sets then
264:
10:
5424:
5253:von Neumann–Bernays–Gödel
4687:Introductory Combinatorics
4444:are arbitrary elements of
4246:
4010:
3972:
3965:
3880:
3873:
3866:
3791:
3784:
3777:
3764:
3757:
3750:
3696:
3689:
3682:
3669:
3662:
3584:
3541:
3534:
3513:
3506:
3499:
3367:
3360:
3296:
3275:
3221:
3214:
3200:
3146:
3139:
3132:
3125:
3078:
3071:
3064:
3057:
3050:
3043:
3036:
2941:
2898:
2891:
2863:
2856:
2849:
2778:
2771:
2764:
2757:
2750:
2724:
2717:
2710:
2703:
2696:
2689:
2644:
2637:
2630:
2623:
2616:
2609:
2579:
2572:
2565:
2558:
2551:
2544:
2537:
2530:
2043:
1540:
986:is itself a subset of the
5317:
5280:
5192:
5082:
5054:One-to-one correspondence
4970:
4911:
4802:
4791:
4777:
4667:Biggs, Norman L. (1985),
4639:Roberts & Tesman 2009
1385:{\displaystyle (X,\tau )}
1214:of sets is a family over
397:is a family of sets over
4600:
4560:λ-system (Dynkin system)
1787:another (coarser) cover
1643:. A family is called a
1617:that is also a cover of
583:form a family of sets.
338:{\displaystyle \wp (S).}
23:and related branches of
4468:and it is assumed that
4263:where every complement
4197:{\displaystyle \Omega }
4153:{\displaystyle \Omega }
4037:Is necessarily true of
2264:{\displaystyle A\cup B}
2235:{\displaystyle A\cap B}
2124:Is necessarily true of
2085:{\displaystyle \Omega }
1646:point-finite collection
1489:that contains both the
1349:, it can be used as an
1037:Hall's marriage theorem
1011:{\displaystyle \wp (S)}
556:{\displaystyle S^{(k)}}
370:{\displaystyle \wp (S)}
5012:Constructible universe
4839:Constructibility (V=L)
4548:Generalized quantifier
4495:
4462:
4438:
4380:
4349:
4314:
4283:
4224:
4198:
4154:
4087:
4058:
3999:
3907:
3820:
3725:
3627:
2971:
2925:
2840:
2808:
2487:
2454:
2421:
2372:
2323:
2294:
2265:
2236:
2206:
2174:
2145:
2086:
2064:
1977:
1954:
1930:
1856:
1829:
1805:
1775:
1732:. A family is called
1712:
1690:
1663:
1631:
1611:
1591:
1571:
1526:
1506:
1483:
1459:
1430:
1406:
1386:
1265:
1235:
1208:
1184:
1152:
1125:
1068:
1012:
980:
940:
918:
864:
748:
659:
639:
577:
557:
523:
502:
477:
456:
436:
414:
391:
371:
339:
307:
283:
245:
215:
195:
175:
143:
116:
92:
65:
5235:Principia Mathematica
5069:Transfinite induction
4928:(i.e. set difference)
4705:Applied Combinatorics
4496:
4463:
4439:
4381:
4355:is equal to a finite
4350:
4315:
4289:is equal to a finite
4284:
4225:
4199:
4155:
4088:
4059:
4000:
3998:{\displaystyle \cap }
3908:
3906:{\displaystyle \cup }
3821:
3726:
3628:
2972:
2926:
2841:
2809:
2488:
2455:
2422:
2373:
2324:
2295:
2266:
2237:
2207:
2175:
2146:
2087:
2065:
2015:(with respect to the
2002:is a family of sets.
1992:augmentation property
1978:
1955:
1931:
1857:
1830:
1806:
1776:
1713:
1691:
1664:
1632:
1612:
1592:
1572:
1537:Covers and topologies
1527:
1507:
1484:
1460:
1431:
1429:{\displaystyle \tau }
1407:
1387:
1351:error-correcting code
1326:, and an (arbitrary)
1318:consists of a set of
1266:
1236:
1209:
1185:
1158:where in particular,
1153:
1126:
1069:
1013:
981:
941:
919:
865:
749:
660:
640:
578:
558:
524:
503:
478:
462:elements is called a
457:
437:
415:
392:
372:
340:
308:
284:
246:
216:
196:
176:
144:
117:
93:
66:
5309:Burali-Forti paradox
5064:Set-builder notation
5017:Continuum hypothesis
4957:Symmetric difference
4736:at Wikimedia Commons
4669:Discrete Mathematics
4530:Combinatorial design
4472:
4448:
4390:
4363:
4333:
4297:
4267:
4214:
4188:
4144:
4073:
4041:
3989:
3897:
3800:
3705:
3607:
2951:
2909:
2820:
2788:
2467:
2434:
2385:
2336:
2307:
2278:
2249:
2220:
2195:
2160:
2128:
2076:
2050:
2024:independence systems
1964:
1944:
1920:
1839:
1815:
1791:
1761:
1702:
1677:
1653:
1621:
1601:
1581:
1561:
1516:
1496:
1473:
1446:
1420:
1396:
1364:
1245:
1218:
1194:
1162:
1135:
1078:
1054:
993:
970:
930:
874:
758:
673:
649:
590:
567:
534:
513:
489:
467:
446:
426:
401:
381:
352:
317:
297:
273:
235:
205:
185:
165:
130:
106:
82:
55:
5270:Tarski–Grothendieck
2036:bornological spaces
1811:if every member of
1547:Filters in topology
1360:consists of a pair
1316:incidence structure
669:sense) is given by
259:extremal set theory
4859:Limitation of size
4524:Class (set theory)
4491:
4458:
4434:
4376:
4345:
4310:
4279:
4220:
4194:
4150:
4083:
4054:
3995:
3903:
3816:
3721:
3623:
2967:
2921:
2836:
2804:
2483:
2450:
2417:
2368:
2319:
2290:
2261:
2232:
2202:
2170:
2141:
2082:
2060:
2017:inclusion relation
1976:{\displaystyle F.}
1973:
1950:
1926:
1852:
1825:
1801:
1771:
1708:
1689:{\displaystyle X.}
1686:
1659:
1649:if every point of
1627:
1607:
1587:
1577:if every point of
1567:
1522:
1502:
1479:
1458:{\displaystyle X,}
1455:
1426:
1402:
1382:
1332:incidence relation
1305:simplicial complex
1261:
1231:
1204:
1180:
1148:
1121:
1116:
1115:
1064:
1008:
976:
936:
914:
860:
744:
655:
635:
573:
553:
519:
501:{\displaystyle S.}
498:
473:
452:
432:
413:{\displaystyle S.}
410:
387:
367:
335:
313:and is denoted by
303:
279:
241:
211:
191:
171:
142:{\displaystyle S.}
139:
112:
88:
61:
5390:
5389:
5299:Russell's paradox
5248:Zermelo–Fraenkel
5149:Dedekind-infinite
5022:Diagonal argument
4921:Cartesian product
4785:Set (mathematics)
4732:Media related to
4714:978-1-4200-9982-9
4578:Russell's paradox
4506:
4505:
1953:{\displaystyle F}
1929:{\displaystyle F}
1884:Sperner's theorem
1720:topological space
1711:{\displaystyle X}
1662:{\displaystyle X}
1630:{\displaystyle X}
1610:{\displaystyle X}
1590:{\displaystyle X}
1570:{\displaystyle X}
1525:{\displaystyle X}
1482:{\displaystyle X}
1405:{\displaystyle X}
1358:topological space
1096:
1029:of all sets (the
979:{\displaystyle S}
658:{\displaystyle S}
576:{\displaystyle S}
522:{\displaystyle k}
476:{\displaystyle k}
455:{\displaystyle k}
435:{\displaystyle S}
390:{\displaystyle S}
306:{\displaystyle S}
282:{\displaystyle S}
257:. The subject of
251:is also called a
244:{\displaystyle S}
214:{\displaystyle I}
194:{\displaystyle F}
174:{\displaystyle I}
115:{\displaystyle S}
100:family of subsets
91:{\displaystyle S}
64:{\displaystyle F}
5415:
5408:Families of sets
5372:Bertrand Russell
5362:John von Neumann
5347:Abraham Fraenkel
5342:Richard Dedekind
5304:Suslin's problem
5215:Cantor's theorem
4932:De Morgan's laws
4797:
4764:
4757:
4750:
4741:
4740:
4731:
4717:
4699:
4681:
4654:
4648:
4642:
4636:
4630:
4624:
4618:
4615:Naive Set Theory
4611:
4500:
4498:
4497:
4492:
4481:
4480:
4467:
4465:
4464:
4459:
4457:
4456:
4443:
4441:
4440:
4435:
4427:
4426:
4414:
4413:
4385:
4383:
4382:
4377:
4372:
4371:
4354:
4352:
4351:
4346:
4319:
4317:
4316:
4311:
4306:
4305:
4288:
4286:
4285:
4280:
4260:
4248:Additionally, a
4242:
4231:
4230:
4229:
4227:
4226:
4221:
4205:
4204:
4203:
4201:
4200:
4195:
4179:
4178:
4170:
4169:
4161:
4160:
4159:
4157:
4156:
4151:
4133:
4132:
4124:
4123:
4115:
4114:
4106:
4097:
4096:
4092:
4090:
4089:
4084:
4082:
4081:
4065:
4064:
4063:
4061:
4060:
4055:
4050:
4049:
4027:
4026:
4020:
4019:
4013:
4012:
4006:
4004:
4002:
4001:
3996:
3985:(even arbitrary
3982:
3975:
3974:
3968:
3967:
3961:
3960:
3954:
3953:
3947:
3946:
3939:
3928:
3927:
3921:
3920:
3914:
3912:
3910:
3909:
3904:
3893:(even arbitrary
3890:
3883:
3882:
3876:
3875:
3869:
3868:
3862:
3861:
3855:
3854:
3848:
3847:
3840:
3832:
3831:
3825:
3823:
3822:
3817:
3815:
3814:
3794:
3793:
3787:
3786:
3780:
3779:
3767:
3766:
3760:
3759:
3753:
3752:
3745:
3737:
3736:
3730:
3728:
3727:
3722:
3720:
3719:
3699:
3698:
3692:
3691:
3685:
3684:
3672:
3671:
3665:
3664:
3658:
3657:
3650:
3647:
3639:
3638:
3632:
3630:
3629:
3624:
3622:
3621:
3601:
3600:
3594:
3593:
3587:
3586:
3574:
3573:
3567:
3566:
3560:
3559:
3552:
3544:
3543:
3537:
3536:
3530:
3529:
3523:
3522:
3516:
3515:
3509:
3508:
3502:
3501:
3495:
3494:
3488:
3487:
3481:
3480:
3473:
3462:
3461:
3455:
3454:
3448:
3447:
3441:
3440:
3434:
3433:
3427:
3426:
3420:
3419:
3413:
3412:
3406:
3405:
3398:
3395:
3384:
3383:
3377:
3376:
3370:
3369:
3363:
3362:
3356:
3355:
3349:
3348:
3342:
3341:
3335:
3334:
3328:
3327:
3320:
3319:
3306:
3305:
3299:
3298:
3292:
3291:
3285:
3284:
3278:
3277:
3271:
3270:
3264:
3263:
3257:
3256:
3250:
3249:
3242:
3231:
3230:
3224:
3223:
3217:
3216:
3210:
3209:
3203:
3202:
3196:
3195:
3189:
3188:
3182:
3181:
3175:
3174:
3167:
3156:
3155:
3149:
3148:
3142:
3141:
3135:
3134:
3128:
3127:
3121:
3120:
3114:
3113:
3107:
3106:
3100:
3099:
3092:
3091:
3090:(Measure theory)
3081:
3080:
3074:
3073:
3067:
3066:
3060:
3059:
3053:
3052:
3046:
3045:
3039:
3038:
3032:
3031:
3025:
3024:
3018:
3017:
3010:
3009:
2996:
2995:
2989:
2988:
2976:
2974:
2973:
2968:
2963:
2962:
2944:
2943:
2937:
2936:
2930:
2928:
2927:
2922:
2901:
2900:
2894:
2893:
2887:
2886:
2879:
2878:
2874:
2866:
2865:
2859:
2858:
2852:
2851:
2845:
2843:
2842:
2837:
2832:
2831:
2813:
2811:
2810:
2805:
2800:
2799:
2781:
2780:
2774:
2773:
2767:
2766:
2760:
2759:
2753:
2752:
2745:
2734:
2733:
2727:
2726:
2720:
2719:
2713:
2712:
2706:
2705:
2699:
2698:
2692:
2691:
2685:
2684:
2678:
2677:
2671:
2669:
2666:
2654:
2653:
2647:
2646:
2640:
2639:
2633:
2632:
2626:
2625:
2619:
2618:
2612:
2611:
2605:
2604:
2598:
2597:
2591:
2582:
2581:
2575:
2574:
2568:
2567:
2561:
2560:
2554:
2553:
2547:
2546:
2540:
2539:
2533:
2532:
2526:
2525:
2519:
2518:
2512:
2509:
2500:
2493:
2492:
2490:
2489:
2484:
2482:
2481:
2460:
2459:
2457:
2456:
2451:
2449:
2448:
2427:
2426:
2424:
2423:
2418:
2410:
2409:
2397:
2396:
2378:
2377:
2375:
2374:
2369:
2361:
2360:
2348:
2347:
2329:
2328:
2326:
2325:
2320:
2300:
2299:
2297:
2296:
2291:
2271:
2270:
2268:
2267:
2262:
2242:
2241:
2239:
2238:
2233:
2213:
2211:
2209:
2208:
2203:
2184:
2183:
2179:
2177:
2176:
2171:
2169:
2168:
2152:
2151:
2150:
2148:
2147:
2142:
2137:
2136:
2113:
2106:
2099:
2092:
2091:
2089:
2088:
2083:
2069:
2067:
2066:
2061:
2059:
2058:
2041:
2040:
1982:
1980:
1979:
1974:
1959:
1957:
1956:
1951:
1935:
1933:
1932:
1927:
1916:is a set family
1905:Euclidean spaces
1861:
1859:
1858:
1853:
1848:
1847:
1834:
1832:
1831:
1826:
1824:
1823:
1810:
1808:
1807:
1802:
1800:
1799:
1780:
1778:
1777:
1772:
1770:
1769:
1745:σ-locally finite
1717:
1715:
1714:
1709:
1695:
1693:
1692:
1687:
1668:
1666:
1665:
1660:
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1531:
1529:
1528:
1523:
1511:
1509:
1508:
1503:
1488:
1486:
1485:
1480:
1464:
1462:
1461:
1456:
1435:
1433:
1432:
1427:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1347:Hamming distance
1274:Related concepts
1270:
1268:
1267:
1262:
1257:
1256:
1240:
1238:
1237:
1232:
1230:
1229:
1213:
1211:
1210:
1205:
1203:
1202:
1189:
1187:
1186:
1181:
1157:
1155:
1154:
1149:
1144:
1143:
1130:
1128:
1127:
1122:
1117:
1114:
1113:
1112:
1090:
1089:
1073:
1071:
1070:
1065:
1063:
1062:
1017:
1015:
1014:
1009:
985:
983:
982:
977:
945:
943:
942:
937:
923:
921:
920:
915:
886:
885:
869:
867:
866:
861:
838:
837:
807:
806:
770:
769:
753:
751:
750:
745:
740:
736:
735:
734:
722:
721:
709:
708:
696:
695:
664:
662:
661:
656:
644:
642:
641:
636:
582:
580:
579:
574:
562:
560:
559:
554:
552:
551:
528:
526:
525:
520:
507:
505:
504:
499:
482:
480:
479:
474:
461:
459:
458:
453:
441:
439:
438:
433:
419:
417:
416:
411:
396:
394:
393:
388:
376:
374:
373:
368:
344:
342:
341:
336:
312:
310:
309:
304:
288:
286:
285:
280:
250:
248:
247:
242:
220:
218:
217:
212:
200:
198:
197:
192:
180:
178:
177:
172:
148:
146:
145:
140:
121:
119:
118:
113:
97:
95:
94:
89:
70:
68:
67:
62:
51:. A collection
5423:
5422:
5418:
5417:
5416:
5414:
5413:
5412:
5393:
5392:
5391:
5386:
5313:
5292:
5276:
5241:New Foundations
5188:
5078:
4997:Cardinal number
4980:
4966:
4907:
4798:
4789:
4773:
4768:
4724:
4715:
4697:
4679:
4663:
4658:
4657:
4649:
4645:
4637:
4633:
4625:
4621:
4612:
4608:
4603:
4518:Algebra of sets
4514:
4508:
4501:
4476:
4475:
4473:
4470:
4469:
4452:
4451:
4449:
4446:
4445:
4422:
4418:
4409:
4405:
4391:
4388:
4387:
4386:
4367:
4366:
4364:
4361:
4360:
4334:
4331:
4330:
4320:
4301:
4300:
4298:
4295:
4294:
4268:
4265:
4264:
4258:
4239:
4237:
4234:
4215:
4212:
4211:
4209:
4208:
4189:
4186:
4185:
4183:
4182:
4176:
4174:
4173:
4167:
4165:
4164:
4145:
4142:
4141:
4139:
4137:
4136:
4130:
4128:
4127:
4121:
4119:
4118:
4112:
4110:
4109:
4103:
4100:
4077:
4076:
4074:
4071:
4070:
4068:
4067:
4066:
4045:
4044:
4042:
4039:
4038:
4036:
4035:
3990:
3987:
3986:
3984:
3983:
3938:Closed Topology
3937:
3898:
3895:
3894:
3892:
3891:
3838:
3810:
3809:
3801:
3798:
3797:
3743:
3715:
3714:
3706:
3703:
3702:
3648:
3645:
3617:
3616:
3608:
3605:
3604:
3550:
3471:
3396:
3393:
3317:
3315:
3240:
3165:
3089:
3087:
3007:
3005:
2978:
2958:
2954:
2952:
2949:
2948:
2910:
2907:
2906:
2905:
2877:(Dynkin System)
2876:
2875:
2872:
2827:
2823:
2821:
2818:
2817:
2795:
2791:
2789:
2786:
2785:
2743:
2667:
2664:
2662:
2587:
2507:
2505:
2496:
2477:
2476:
2468:
2465:
2464:
2463:
2444:
2443:
2435:
2432:
2431:
2430:
2405:
2401:
2392:
2388:
2386:
2383:
2382:
2381:
2356:
2352:
2343:
2339:
2337:
2334:
2333:
2332:
2308:
2305:
2304:
2303:
2279:
2276:
2275:
2274:
2250:
2247:
2246:
2245:
2221:
2218:
2217:
2216:
2196:
2193:
2192:
2190:
2187:
2164:
2163:
2161:
2158:
2157:
2155:
2154:
2153:
2132:
2131:
2129:
2126:
2125:
2123:
2122:
2117:
2077:
2074:
2073:
2054:
2053:
2051:
2048:
2047:
2044:
2008:convexity space
1965:
1962:
1961:
1945:
1942:
1941:
1938:downward closed
1921:
1918:
1917:
1897:Helly's theorem
1874:
1865:star refinement
1843:
1842:
1840:
1837:
1836:
1819:
1818:
1816:
1813:
1812:
1795:
1794:
1792:
1789:
1788:
1765:
1764:
1762:
1759:
1758:
1703:
1700:
1699:
1678:
1675:
1674:
1654:
1651:
1650:
1622:
1619:
1618:
1602:
1599:
1598:
1582:
1579:
1578:
1562:
1559:
1558:
1549:
1539:
1517:
1514:
1513:
1497:
1494:
1493:
1474:
1471:
1470:
1447:
1444:
1443:
1421:
1418:
1417:
1397:
1394:
1393:
1365:
1362:
1361:
1328:binary relation
1276:
1252:
1251:
1246:
1243:
1242:
1225:
1224:
1219:
1216:
1215:
1198:
1197:
1195:
1192:
1191:
1163:
1160:
1159:
1139:
1138:
1136:
1133:
1132:
1108:
1107:
1100:
1094:
1085:
1084:
1079:
1076:
1075:
1058:
1057:
1055:
1052:
1051:
994:
991:
990:
971:
968:
967:
964:
948:ordinal numbers
931:
928:
927:
881:
877:
875:
872:
871:
833:
829:
802:
798:
765:
761:
759:
756:
755:
730:
726:
717:
713:
704:
700:
691:
687:
686:
682:
674:
671:
670:
650:
647:
646:
591:
588:
587:
568:
565:
564:
541:
537:
535:
532:
531:
514:
511:
510:
490:
487:
486:
468:
465:
464:
447:
444:
443:
427:
424:
423:
402:
399:
398:
382:
379:
378:
377:of a given set
353:
350:
349:
318:
315:
314:
298:
295:
294:
274:
271:
270:
267:
236:
233:
232:
206:
203:
202:
186:
183:
182:
166:
163:
162:
131:
128:
127:
107:
104:
103:
83:
80:
79:
56:
53:
52:
17:
12:
11:
5:
5421:
5411:
5410:
5405:
5388:
5387:
5385:
5384:
5379:
5377:Thoralf Skolem
5374:
5369:
5364:
5359:
5354:
5349:
5344:
5339:
5334:
5329:
5323:
5321:
5315:
5314:
5312:
5311:
5306:
5301:
5295:
5293:
5291:
5290:
5287:
5281:
5278:
5277:
5275:
5274:
5273:
5272:
5267:
5262:
5261:
5260:
5245:
5244:
5243:
5231:
5230:
5229:
5218:
5217:
5212:
5207:
5202:
5196:
5194:
5190:
5189:
5187:
5186:
5181:
5176:
5171:
5162:
5157:
5152:
5142:
5137:
5136:
5135:
5130:
5125:
5115:
5105:
5100:
5095:
5089:
5087:
5080:
5079:
5077:
5076:
5071:
5066:
5061:
5059:Ordinal number
5056:
5051:
5046:
5041:
5040:
5039:
5034:
5024:
5019:
5014:
5009:
5004:
4994:
4989:
4983:
4981:
4979:
4978:
4975:
4971:
4968:
4967:
4965:
4964:
4959:
4954:
4949:
4944:
4939:
4937:Disjoint union
4934:
4929:
4923:
4917:
4915:
4909:
4908:
4906:
4905:
4904:
4903:
4898:
4887:
4886:
4884:Martin's axiom
4881:
4876:
4871:
4866:
4861:
4856:
4851:
4849:Extensionality
4846:
4841:
4836:
4835:
4834:
4829:
4824:
4814:
4808:
4806:
4800:
4799:
4792:
4790:
4788:
4787:
4781:
4779:
4775:
4774:
4767:
4766:
4759:
4752:
4744:
4738:
4737:
4723:
4722:External links
4720:
4719:
4718:
4713:
4700:
4695:
4682:
4677:
4662:
4659:
4656:
4655:
4643:
4631:
4619:
4605:
4604:
4602:
4599:
4598:
4597:
4591:
4585:
4575:
4569:
4563:
4557:
4554:Indexed family
4551:
4545:
4539:
4533:
4527:
4521:
4513:
4510:
4504:
4503:
4490:
4487:
4484:
4479:
4455:
4433:
4430:
4425:
4421:
4417:
4412:
4408:
4404:
4401:
4398:
4395:
4375:
4370:
4357:disjoint union
4344:
4341:
4338:
4327:
4309:
4304:
4291:disjoint union
4278:
4275:
4272:
4254:
4244:
4243:
4232:
4219:
4206:
4193:
4180:
4171:
4162:
4149:
4134:
4125:
4116:
4107:
4098:
4080:
4053:
4048:
4032:
4031:
4028:
4021:
4014:
4007:
3994:
3976:
3969:
3962:
3955:
3948:
3941:
3933:
3932:
3929:
3922:
3915:
3902:
3884:
3877:
3870:
3863:
3856:
3849:
3842:
3834:
3833:
3826:
3813:
3808:
3805:
3795:
3788:
3781:
3774:
3771:
3768:
3761:
3754:
3747:
3744:Filter subbase
3739:
3738:
3731:
3718:
3713:
3710:
3700:
3693:
3686:
3679:
3676:
3673:
3666:
3659:
3652:
3641:
3640:
3633:
3620:
3615:
3612:
3602:
3595:
3588:
3581:
3578:
3575:
3568:
3561:
3554:
3546:
3545:
3538:
3531:
3524:
3517:
3510:
3503:
3496:
3489:
3482:
3475:
3467:
3466:
3463:
3456:
3449:
3442:
3435:
3428:
3421:
3414:
3407:
3400:
3389:
3388:
3385:
3378:
3371:
3364:
3357:
3350:
3343:
3336:
3329:
3322:
3311:
3310:
3307:
3300:
3293:
3286:
3279:
3272:
3265:
3258:
3251:
3244:
3236:
3235:
3232:
3225:
3218:
3211:
3204:
3197:
3190:
3183:
3176:
3169:
3161:
3160:
3157:
3150:
3143:
3136:
3129:
3122:
3115:
3108:
3101:
3094:
3083:
3082:
3075:
3068:
3061:
3054:
3047:
3040:
3033:
3026:
3019:
3012:
3008:(Order theory)
3001:
3000:
2997:
2990:
2983:
2966:
2961:
2957:
2945:
2938:
2931:
2920:
2917:
2914:
2902:
2895:
2888:
2881:
2868:
2867:
2860:
2853:
2846:
2835:
2830:
2826:
2814:
2803:
2798:
2794:
2782:
2775:
2768:
2761:
2754:
2747:
2744:Monotone class
2739:
2738:
2735:
2728:
2721:
2714:
2707:
2700:
2693:
2686:
2679:
2672:
2659:
2658:
2655:
2648:
2641:
2634:
2627:
2620:
2613:
2606:
2599:
2592:
2584:
2583:
2576:
2569:
2562:
2555:
2548:
2541:
2534:
2527:
2520:
2513:
2502:
2501:
2494:
2480:
2475:
2472:
2461:
2447:
2442:
2439:
2428:
2416:
2413:
2408:
2404:
2400:
2395:
2391:
2379:
2367:
2364:
2359:
2355:
2351:
2346:
2342:
2330:
2318:
2315:
2312:
2301:
2289:
2286:
2283:
2272:
2260:
2257:
2254:
2243:
2231:
2228:
2225:
2214:
2201:
2185:
2167:
2140:
2135:
2119:
2118:
2116:
2115:
2108:
2101:
2093:
2081:
2057:
1972:
1969:
1949:
1925:
1879:Sperner family
1873:
1870:
1867:
1851:
1846:
1822:
1798:
1786:
1768:
1753:
1747:
1737:
1735:locally finite
1731:
1707:
1685:
1682:
1658:
1648:
1642:
1626:
1606:
1586:
1566:
1556:
1538:
1535:
1534:
1533:
1521:
1501:
1478:
1454:
1451:
1441:
1425:
1401:
1381:
1378:
1375:
1372:
1369:
1354:
1335:
1312:
1297:
1275:
1272:
1260:
1255:
1250:
1228:
1223:
1201:
1179:
1176:
1173:
1170:
1167:
1147:
1142:
1120:
1111:
1106:
1103:
1099:
1093:
1088:
1083:
1061:
1007:
1004:
1001:
998:
975:
963:
960:
935:
913:
910:
907:
904:
901:
898:
895:
892:
889:
884:
880:
859:
856:
853:
850:
847:
844:
841:
836:
832:
828:
825:
822:
819:
816:
813:
810:
805:
801:
797:
794:
791:
788:
785:
782:
779:
776:
773:
768:
764:
743:
739:
733:
729:
725:
720:
716:
712:
707:
703:
699:
694:
690:
685:
681:
678:
654:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
572:
550:
547:
544:
540:
518:
497:
494:
472:
451:
431:
409:
406:
386:
366:
363:
360:
357:
334:
331:
328:
325:
322:
302:
289:is called the
278:
266:
263:
240:
210:
190:
170:
151:family of sets
138:
135:
124:family of sets
111:
87:
60:
15:
9:
6:
4:
3:
2:
5420:
5409:
5406:
5404:
5401:
5400:
5398:
5383:
5382:Ernst Zermelo
5380:
5378:
5375:
5373:
5370:
5368:
5367:Willard Quine
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5324:
5322:
5320:
5319:Set theorists
5316:
5310:
5307:
5305:
5302:
5300:
5297:
5296:
5294:
5288:
5286:
5283:
5282:
5279:
5271:
5268:
5266:
5265:Kripke–Platek
5263:
5259:
5256:
5255:
5254:
5251:
5250:
5249:
5246:
5242:
5239:
5238:
5237:
5236:
5232:
5228:
5225:
5224:
5223:
5220:
5219:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5197:
5195:
5191:
5185:
5182:
5180:
5177:
5175:
5172:
5170:
5168:
5163:
5161:
5158:
5156:
5153:
5150:
5146:
5143:
5141:
5138:
5134:
5131:
5129:
5126:
5124:
5121:
5120:
5119:
5116:
5113:
5109:
5106:
5104:
5101:
5099:
5096:
5094:
5091:
5090:
5088:
5085:
5081:
5075:
5072:
5070:
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5042:
5038:
5035:
5033:
5030:
5029:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5002:
4998:
4995:
4993:
4990:
4988:
4985:
4984:
4982:
4976:
4973:
4972:
4969:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4927:
4924:
4922:
4919:
4918:
4916:
4914:
4910:
4902:
4901:specification
4899:
4897:
4894:
4893:
4892:
4889:
4888:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4845:
4842:
4840:
4837:
4833:
4830:
4828:
4825:
4823:
4820:
4819:
4818:
4815:
4813:
4810:
4809:
4807:
4805:
4801:
4796:
4786:
4783:
4782:
4780:
4776:
4772:
4765:
4760:
4758:
4753:
4751:
4746:
4745:
4742:
4735:
4730:
4726:
4725:
4716:
4710:
4706:
4701:
4698:
4696:0-13-602040-2
4692:
4688:
4683:
4680:
4678:0-19-853252-0
4674:
4670:
4665:
4664:
4652:
4647:
4640:
4635:
4628:
4623:
4616:
4610:
4606:
4595:
4592:
4589:
4586:
4583:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4542:Field of sets
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4515:
4509:
4502:
4488:
4482:
4431:
4428:
4423:
4419:
4415:
4410:
4406:
4402:
4399:
4396:
4393:
4373:
4358:
4342:
4328:
4326:
4323:
4307:
4292:
4276:
4270:
4262:
4255:
4253:
4250:
4245:
4241:
4233:
4217:
4207:
4181:
4172:
4168:intersections
4163:
4135:
4126:
4117:
4113:intersections
4108:
4105:
4099:
4095:
4094:closed under:
4051:
4034:
4033:
3992:
3981:
3940:
3935:
3934:
3900:
3889:
3841:
3839:Open Topology
3836:
3835:
3806:
3803:
3746:
3741:
3740:
3711:
3708:
3651:
3649:(Filter base)
3643:
3642:
3613:
3610:
3553:
3548:
3547:
3474:
3469:
3468:
3399:
3391:
3390:
3321:
3313:
3312:
3243:
3238:
3237:
3168:
3163:
3162:
3093:
3085:
3084:
3011:
3003:
3002:
2982:
2959:
2955:
2918:
2915:
2912:
2880:
2870:
2869:
2828:
2824:
2796:
2792:
2746:
2741:
2740:
2670:
2661:
2660:
2590:
2586:
2585:
2511:
2504:
2503:
2499:
2495:
2473:
2470:
2462:
2440:
2429:
2414:
2411:
2406:
2402:
2398:
2393:
2389:
2380:
2365:
2362:
2357:
2353:
2349:
2344:
2340:
2331:
2316:
2302:
2287:
2281:
2273:
2258:
2255:
2252:
2244:
2229:
2226:
2223:
2215:
2212:
2199:
2186:
2182:
2181:closed under:
2138:
2121:
2120:
2114:
2109:
2107:
2102:
2100:
2095:
2094:
2071:
2042:
2039:
2037:
2033:
2029:
2025:
2020:
2018:
2014:
2010:
2009:
2003:
2001:
1996:
1994:
1993:
1988:
1987:
1970:
1967:
1947:
1939:
1923:
1915:
1914:
1908:
1906:
1902:
1898:
1894:
1893:
1887:
1885:
1881:
1880:
1869:
1866:
1863:
1849:
1785:
1782:
1755:
1752:
1749:
1746:
1743:
1741:
1736:
1733:
1730:
1727:
1726:is called an
1725:
1721:
1705:
1696:
1683:
1680:
1672:
1656:
1647:
1644:
1641:
1638:
1624:
1604:
1584:
1564:
1555:
1552:
1548:
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1519:
1499:
1492:
1476:
1468:
1452:
1449:
1440:
1437:
1423:
1415:
1399:
1376:
1373:
1370:
1359:
1355:
1352:
1348:
1344:
1340:
1336:
1333:
1330:, called the
1329:
1325:
1321:
1317:
1313:
1310:
1306:
1302:
1298:
1295:
1294:
1289:
1285:
1281:
1280:
1279:
1271:
1258:
1248:
1221:
1177:
1171:
1165:
1145:
1118:
1104:
1101:
1091:
1081:
1048:
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1042:
1038:
1034:
1032:
1028:
1024:
1019:
1002:
989:
973:
959:
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953:
949:
933:
924:
911:
905:
902:
899:
896:
893:
887:
882:
878:
857:
851:
848:
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839:
834:
830:
826:
820:
817:
814:
808:
803:
799:
795:
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783:
780:
777:
771:
766:
762:
741:
737:
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727:
723:
718:
714:
710:
705:
701:
697:
692:
688:
683:
679:
676:
668:
652:
632:
626:
623:
620:
617:
614:
611:
608:
605:
602:
596:
593:
584:
570:
545:
538:
530:
516:
495:
492:
484:
470:
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429:
420:
407:
404:
384:
361:
348:
332:
326:
300:
292:
276:
262:
260:
256:
255:
238:
231:
226:
224:
208:
188:
168:
160:
156:
152:
136:
133:
125:
109:
101:
85:
78:
74:
58:
50:
46:
42:
38:
34:
30:
26:
22:
5332:Georg Cantor
5327:Paul Bernays
5258:Morse–Kelley
5233:
5166:
5165:Subset
5112:hereditarily
5074:Venn diagram
5043:
5032:ordered pair
4947:Intersection
4891:Axiom schema
4734:Set families
4704:
4686:
4668:
4646:
4634:
4627:Brualdi 2010
4622:
4614:
4609:
4581:
4572:Ring of sets
4507:
4322:
4249:
4247:
4238:Intersection
2045:
2032:antimatroids
2021:
2006:
2004:
1997:
1990:
1984:
1911:
1909:
1899:states that
1892:Helly family
1890:
1888:
1877:
1875:
1756:
1740:neighborhood
1697:
1637:is called a
1550:
1466:
1413:
1331:
1323:
1319:
1291:
1277:
1049:
1035:
1027:proper class
1020:
965:
956:proper class
951:
925:
585:
422:A subset of
421:
268:
252:
227:
223:proper class
158:
154:
150:
123:
99:
98:is called a
32:
28:
18:
5357:Thomas Jech
5200:Alternative
5179:Uncountable
5133:Ultrafilter
4992:Cardinality
4896:replacement
4844:Determinacy
4613:P. Halmos,
4359:of sets in
4325:semialgebra
4293:of sets in
4138:complements
4131:complements
2668:(Semifield)
2665:Semialgebra
1960:is also in
1901:convex sets
1781:is said to
1322:, a set of
1190:Any family
1041:Philip Hall
75:of a given
41:indexed set
25:mathematics
5397:Categories
5352:Kurt Gödel
5337:Paul Cohen
5174:Transitive
4942:Identities
4926:Complement
4913:Operations
4874:Regularity
4812:Adjunction
4771:Set theory
4661:References
4651:Biggs 1985
3472:Dual ideal
3397:(𝜎-Field)
3394:𝜎-Algebra
1729:open cover
1541:See also:
1339:block code
1293:hyperedges
1284:hypergraph
962:Properties
926:The class
254:hypergraph
230:finite set
159:set system
155:set family
33:collection
21:set theory
5285:Paradoxes
5205:Axiomatic
5184:Universal
5160:Singleton
5155:Recursive
5098:Countable
5093:Amorphous
4952:Power set
4869:Power set
4827:dependent
4822:countable
4641:, pg. 692
4629:, pg. 322
4588:σ-algebra
4486:∅
4483:≠
4432:…
4340:∖
4337:Ω
4274:∖
4218:∅
4210:contains
4192:Ω
4184:contains
4175:countable
4166:countable
4148:Ω
4052::
3993:∩
3901:∪
3804:∅
3709:∅
3646:Prefilter
3611:∅
2979:they are
2965:↗
2916:⊆
2873:𝜆-system
2834:↗
2802:↘
2474:∈
2471:∅
2441:∈
2438:Ω
2415:⋯
2412:∪
2399:∪
2366:⋯
2363:∩
2350:∩
2314:∖
2311:Ω
2285:∖
2256:∪
2227:∩
2200:⊇
2139::
2080:Ω
2046:Families
2028:greedoids
1724:open sets
1671:partition
1500:∅
1491:empty set
1467:open sets
1424:τ
1377:τ
1337:A binary
1309:simplices
1249:∪
1222:∪
1175:∅
1169:∅
1166:∪
1105:∈
1098:⋃
1082:∪
1039:, due to
997:℘
988:power set
563:of a set
356:℘
347:power set
321:℘
291:power set
5289:Problems
5193:Theories
5169:Superset
5145:Infinite
4974:Concepts
4854:Infinity
4778:Overview
4653:, pg. 89
4566:π-system
4512:See also
4252:semiring
4240:Property
4129:relative
4104:downward
4102:directed
3807:∉
3712:∉
3614:∉
3316:Algebra
2981:disjoint
2947:only if
2816:only if
2784:only if
2589:Semiring
2189:Directed
1757:A cover
1640:subcover
1439:topology
1288:vertices
1031:universe
1023:subclass
667:multiset
665:(in the
529:-subsets
265:Examples
45:multiset
5227:General
5222:Zermelo
5128:subbase
5110: (
5049:Forcing
5027:Element
4999: (
4977:Methods
4864:Pairing
4261:-system
4069:or, is
3318:(Field)
3241:𝜎-Ring
2904:only if
2510:-system
2156:or, is
2070:of sets
1986:matroid
1469:) over
1025:of the
946:of all
483:-subset
442:having
157:, or a
122:, or a
73:subsets
5118:Filter
5108:Finite
5044:Family
4987:Almost
4832:global
4817:Choice
4804:Axioms
4711:
4693:
4675:
4594:σ-ring
4536:δ-ring
4236:Finite
4177:unions
4122:unions
4120:finite
4111:finite
4030:Never
3931:Never
3551:Filter
3465:Never
3387:Never
3309:Never
3234:Never
3166:δ-Ring
3159:Never
2999:Never
2737:Never
2657:Never
2498:F.I.P.
2034:, and
2013:chains
2000:filter
1998:Every
1784:refine
1557:a set
1545:, and
1416:) and
1414:points
1392:where
1343:string
1320:points
754:where
29:family
5210:Naive
5140:Fuzzy
5103:Empty
5086:types
5037:tuple
5007:Class
5001:large
4962:Union
4879:Union
4601:Notes
4256:is a
3773:Never
3770:Never
3678:Never
3675:Never
3580:Never
3577:Never
3088:Ring
3006:Ring
2072:over
1718:is a
1698:When
1554:cover
1436:is a
1324:lines
952:large
950:is a
126:over
49:class
47:, or
5123:base
4709:ISBN
4691:ISBN
4673:ISBN
1512:and
1033:).
870:and
586:Let
508:The
345:The
31:(or
27:, a
5084:Set
4140:in
2191:by
2019:).
1995:.
1910:An
1903:in
1748:or
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1442:on
1314:An
1299:An
1050:If
934:Ord
485:of
293:of
102:of
77:set
71:of
37:set
19:In
5399::
4321:A
2977:or
2038:.
2030:,
2026:,
2005:A
1983:A
1889:A
1876:A
1862:A
1356:A
1282:A
1092::=
1047:.
958:.
225:.
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5167:·
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5147:(
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4763:e
4756:t
4749:v
4584:)
4489:.
4478:F
4454:F
4429:,
4424:2
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4411:1
4407:A
4403:,
4400:B
4397:,
4394:A
4374:.
4369:F
4343:A
4308:.
4303:F
4277:A
4271:B
4259:π
4079:F
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3812:F
3717:F
3619:F
2960:i
2956:A
2919:B
2913:A
2829:i
2825:A
2797:i
2793:A
2508:π
2479:F
2446:F
2407:2
2403:A
2394:1
2390:A
2358:2
2354:A
2345:1
2341:A
2317:A
2288:A
2282:B
2259:B
2253:A
2230:B
2224:A
2166:F
2134:F
2112:e
2105:t
2098:v
2056:F
1971:.
1968:F
1948:F
1924:F
1850:.
1845:C
1821:F
1797:C
1767:F
1706:X
1684:.
1681:X
1657:X
1625:X
1605:X
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1565:X
1520:X
1477:X
1453:,
1450:X
1400:X
1380:)
1374:,
1371:X
1368:(
1259:.
1254:F
1227:F
1200:F
1178:.
1172:=
1146:,
1141:F
1119:F
1110:F
1102:F
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1060:F
1006:)
1003:S
1000:(
974:S
912:.
909:}
906:1
903:,
900:b
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879:A
858:,
855:}
852:2
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846:1
843:{
840:=
835:3
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827:,
824:}
821:2
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812:{
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800:A
796:,
793:}
790:c
787:,
784:b
781:,
778:a
775:{
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767:1
763:A
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738:}
732:4
728:A
724:,
719:3
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706:2
702:A
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684:{
680:=
677:F
653:S
633:.
630:}
627:2
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600:{
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594:S
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189:F
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86:S
59:F
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