611:
The proof that the
Hausdorff maximal principle is equivalent to Zorn's lemma is somehow similar to this proof. Indeed, first assume Zorn's lemma. Since a union of a totally ordered set of chains is a chain, the hypothesis of Zorn's lemma (every chain has an upper bound) is satisfied for
1012:
consists of all circular regions with centers at the origin. Another maximal totally ordered sub-collection consists of all circular regions bounded by circles tangent from the right to the y-axis at the origin.
2107:
2472:
3703:
176:
An equivalent form of the
Hausdorff maximal principle is that in every partially ordered set, there exists a maximal chain. (Note if the set is empty, the empty subset is a maximal chain.)
856:
4471:
3613:
3578:
3407:
1242:
4506:
3764:
2585:
2530:
2403:
2175:
911:
4362:
4059:
3897:
3822:
3793:
3642:
2868:
2776:
2368:
1149:
1120:
1059:
3970:
4209:
1569:
2315:
1681:
3868:
3496:
3372:
3145:
3119:
2661:
1443:
1417:
2142:
4526:
2283:
1721:
983:
4149:
4126:
4106:
4030:
3279:
3239:
3165:
809:
660:
635:
586:
481:
389:
344:
319:
294:
202:
2224:
937:
4436:
4389:
4333:
4286:
4236:
4176:
4086:
3997:
3543:
3346:
3219:
3192:
3093:
3045:
2996:
2966:
2939:
2722:
2690:
2615:
2251:
456:
249:
124:
3729:
1371:
963:
49:
In a partially ordered set, a totally ordered subset is also called a chain. Thus, the maximal principle says every chain in the set extends to a maximal chain.
4409:
4306:
4259:
3937:
3917:
3842:
3662:
3516:
3470:
3450:
3430:
3319:
3299:
3259:
3066:
3016:
2912:
2888:
2839:
2819:
2796:
2742:
2635:
2550:
2495:
2335:
2195:
2047:
2023:
2003:
1983:
1963:
1943:
1923:
1883:
1863:
1843:
1823:
1803:
1783:
1763:
1743:
1701:
1649:
1629:
1609:
1589:
1543:
1523:
1503:
1483:
1463:
1391:
1345:
1325:
1305:
1285:
1265:
1200:
1180:
876:
783:
763:
743:
723:
703:
680:
606:
561:
541:
521:
501:
429:
409:
364:
269:
222:
171:
151:
97:
1287:, which is partially ordered by set inclusion. It is nonempty as it contains the empty set and thus by the maximal principle, it contains a maximal chain
5327:
5310:
4840:
179:
This form follows from the original form since the empty set is a chain. Conversely, to deduce the original form from this form, consider the set
4676:
4645:
1122:
under which two points are comparable only if they lie on the same horizontal line. The maximal totally ordered sets are horizontal lines in
5157:
5293:
5152:
4598:
2029:(Zorn's lemma itself also follows from this weak form.) The maximal principle follows from the above since the set of all chains in
1925:
be a nonempty set of subsets of some fixed set, ordered by set inclusion, such that (1) the union of each totally ordered subset of
5365:
5147:
1885:. (So, the difference is that the maximal principle gives a maximal chain while Zorn's lemma gives a maximal element directly.)
57:
4783:
2055:
4865:
1008:
is the collection of all circular regions (interiors of circles) in the plane. One maximal totally ordered sub-collection of
5184:
5104:
2411:
4778:
4969:
4898:
3667:
4872:
4860:
4823:
4773:
4727:
4696:
4803:
4793:
5169:
4669:
814:
5142:
4808:
5360:
5074:
4701:
4441:
3583:
3548:
3377:
5322:
5305:
5234:
4850:
1205:
5370:
5212:
5047:
5038:
4907:
4742:
4706:
4662:
4476:
3734:
2555:
2500:
2373:
2147:
881:
4788:
4338:
4035:
3873:
3798:
3769:
3618:
2844:
2340:
1125:
1096:
1035:
5300:
5259:
5249:
5239:
4984:
4947:
4937:
4917:
4902:
997:
3942:
46:
is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion.
5227:
5138:
5084:
5043:
5033:
4922:
4855:
4818:
4181:
1548:
5266:
5119:
5028:
5018:
4959:
4877:
3766:. (This is the moment we needed to collapse a set to an element by the axiom of choice to define
2750:
2288:
1654:
5339:
5179:
4813:
3847:
3475:
3351:
3124:
3098:
2640:
1422:
1396:
2112:
1893:
The idea of the proof is essentially due to
Zermelo and is to prove the following weak form of
1725:
For the purpose of comparison, here is a proof of the same fact by Zorn's lemma. As above, let
5276:
5254:
5114:
5099:
5079:
4882:
4641:
4594:
4511:
2256:
1706:
968:
4134:
4111:
4091:
4002:
3264:
3224:
3150:
788:
5089:
4942:
2203:
916:
4414:
4367:
4311:
4264:
4214:
4154:
4064:
3975:
3521:
3324:
3197:
3170:
3071:
3023:
2974:
2944:
2917:
2700:
2668:
2593:
2229:
434:
227:
102:
5271:
5054:
4932:
4927:
4912:
4737:
4722:
1898:
53:
32:
4828:
3708:
1894:
1350:
942:
296:
is partially ordered by set inclusion. Thus, by the maximal principle in the above form,
28:
640:
615:
566:
461:
369:
324:
299:
274:
182:
5189:
5174:
5164:
5023:
5001:
4979:
4604:
4394:
4291:
4244:
3922:
3902:
3827:
3647:
3501:
3455:
3435:
3415:
3304:
3284:
3244:
3051:
3001:
2897:
2873:
2824:
2804:
2781:
2727:
2620:
2535:
2480:
2320:
2180:
2032:
2008:
1988:
1968:
1948:
1928:
1908:
1868:
1848:
1828:
1808:
1788:
1768:
1748:
1728:
1686:
1634:
1614:
1594:
1574:
1528:
1508:
1488:
1468:
1448:
1376:
1330:
1310:
1290:
1270:
1250:
1185:
1165:
861:
768:
748:
728:
708:
688:
665:
591:
546:
526:
506:
486:
414:
394:
349:
254:
207:
156:
136:
82:
5354:
5288:
5244:
5222:
5094:
4964:
4952:
4757:
4621:
3432:
is a tower. The conditions 1. and 2. are again straightforward to check. For 3., let
1160:
77:
36:
5109:
4991:
4974:
4892:
4732:
4685:
4633:
5315:
5008:
4887:
4752:
4638:
Real and
Complex Analysis (International Series in Pure and Applied Mathematics)
4584:
127:
40:
20:
3241:
is a tower. The conditions 1. and 2. are straightforward to check. For 3., let
5283:
5217:
5058:
1825:
is also orthonormal by the same argument as above and so is an upper bound of
173:). In general, there may be several maximal chains containing a given chain.
5334:
5207:
5013:
52:
The
Hausdorff maximal principle is one of many statements equivalent to the
5129:
4996:
4747:
996:
is any collection of sets, the relation "is a proper subset of" is a
130:
43:
411:
since a union of a totally ordered set of chains is a chain. Since
153:(i.e., a chain that is not contained in a strictly larger chain in
1267:
be the set of all orthonormal subsets of the given
Hilbert space
2894:
There exists at least one tower; indeed, the set of all sets in
4658:
4654:
60:
without the axiom of choice). The principle is also called the
1347:. We shall show it is a maximal orthonormal subset. First, if
2968:
be the intersection of all towers, which is again a tower.
2798:, where "totally ordered" is with respect to set inclusion.
1505:
and so they are orthogonal to each other (and of course,
2102:{\displaystyle f:{\mathfrak {P}}(P)-\{\emptyset \}\to P}
1159:
By the
Hausdorff maximal principle, we can show every
4514:
4479:
4444:
4417:
4397:
4370:
4341:
4314:
4294:
4267:
4247:
4217:
4184:
4157:
4137:
4114:
4094:
4067:
4038:
4005:
3978:
3945:
3925:
3905:
3876:
3850:
3830:
3801:
3772:
3737:
3711:
3670:
3650:
3621:
3586:
3551:
3524:
3504:
3478:
3458:
3438:
3418:
3380:
3354:
3327:
3307:
3287:
3267:
3247:
3227:
3200:
3173:
3153:
3127:
3101:
3074:
3054:
3026:
3004:
2977:
2947:
2920:
2900:
2876:
2847:
2827:
2807:
2784:
2753:
2730:
2703:
2671:
2643:
2623:
2596:
2558:
2538:
2503:
2483:
2414:
2376:
2343:
2323:
2291:
2259:
2232:
2206:
2183:
2150:
2115:
2058:
2035:
2011:
1991:
1971:
1951:
1931:
1911:
1871:
1851:
1831:
1811:
1791:
1771:
1751:
1731:
1709:
1689:
1657:
1637:
1617:
1597:
1577:
1551:
1531:
1511:
1491:
1471:
1451:
1425:
1399:
1379:
1353:
1333:
1313:
1293:
1273:
1253:
1208:
1188:
1168:
1128:
1099:
1038:
971:
945:
919:
884:
864:
817:
791:
771:
751:
731:
711:
691:
668:
643:
618:
594:
569:
549:
529:
509:
489:
464:
437:
417:
397:
372:
352:
327:
302:
277:
257:
230:
210:
185:
159:
139:
105:
85:
2467:{\displaystyle {\widetilde {C}}=C\cup \{f(C^{*})\}.}
1202:
as follows. (This fact can be stated as saying that
76:
The
Hausdorff maximal principle states that, in any
5200:
5128:
5067:
4837:
4766:
4715:
4520:
4500:
4465:
4430:
4403:
4383:
4356:
4327:
4300:
4280:
4253:
4230:
4203:
4170:
4143:
4120:
4100:
4080:
4053:
4024:
3991:
3964:
3931:
3911:
3891:
3862:
3836:
3816:
3787:
3758:
3723:
3698:{\displaystyle A\subset C\subset {\widetilde {A}}}
3697:
3656:
3636:
3607:
3572:
3537:
3510:
3490:
3464:
3444:
3424:
3401:
3366:
3340:
3313:
3293:
3273:
3253:
3233:
3213:
3186:
3159:
3139:
3113:
3087:
3060:
3039:
3010:
2990:
2960:
2933:
2906:
2882:
2862:
2833:
2813:
2790:
2770:
2736:
2716:
2684:
2655:
2629:
2609:
2579:
2544:
2524:
2489:
2466:
2397:
2362:
2329:
2309:
2277:
2245:
2218:
2189:
2169:
2136:
2101:
2041:
2017:
1997:
1977:
1957:
1937:
1917:
1877:
1857:
1837:
1817:
1797:
1777:
1757:
1737:
1715:
1695:
1675:
1643:
1623:
1603:
1583:
1563:
1537:
1517:
1497:
1477:
1457:
1437:
1411:
1385:
1365:
1339:
1319:
1299:
1279:
1259:
1236:
1194:
1174:
1143:
1114:
1053:
977:
957:
931:
905:
870:
850:
803:
777:
757:
737:
717:
697:
674:
654:
629:
600:
580:
555:
535:
515:
495:
475:
450:
423:
403:
383:
358:
338:
313:
288:
263:
243:
216:
196:
165:
145:
118:
91:
685:Conversely, if the maximal principle holds, then
662:contains a maximal element or a maximal chain in
4593:. Reprinted by Springer-Verlag, New York, 1974.
35:in 1914 (Moore 1982:168). It states that in any
16:Mathematical result or axiom on order relations
1445:. That is, any given two distinct elements in
4670:
4108:. This completes the proof of the claim that
8:
2458:
2436:
2304:
2298:
2090:
2084:
1670:
1664:
851:{\displaystyle {\widetilde {C}}=C\cup \{y\}}
845:
839:
2052:By the axiom of choice, we have a function
27:is an alternate and earlier formulation of
5328:Positive cone of a partially ordered group
4677:
4663:
4655:
4591:. Princeton, NJ: D. Van Nostrand Company.
4513:
4481:
4480:
4478:
4466:{\displaystyle {\widetilde {C}}\subset C}
4446:
4445:
4443:
4422:
4416:
4396:
4375:
4369:
4343:
4342:
4340:
4319:
4313:
4293:
4272:
4266:
4246:
4222:
4216:
4189:
4183:
4162:
4156:
4136:
4113:
4093:
4072:
4066:
4040:
4039:
4037:
4016:
4004:
3983:
3977:
3956:
3944:
3924:
3904:
3878:
3877:
3875:
3849:
3829:
3803:
3802:
3800:
3774:
3773:
3771:
3745:
3744:
3736:
3710:
3684:
3683:
3669:
3649:
3623:
3622:
3620:
3608:{\displaystyle C\subset {\widetilde {A}}}
3594:
3593:
3585:
3573:{\displaystyle {\widetilde {A}}\subset C}
3553:
3552:
3550:
3529:
3523:
3503:
3477:
3457:
3437:
3417:
3402:{\displaystyle {\widetilde {C}}\subset A}
3382:
3381:
3379:
3353:
3332:
3326:
3306:
3286:
3266:
3246:
3226:
3205:
3199:
3178:
3172:
3152:
3126:
3100:
3079:
3073:
3053:
3031:
3025:
3003:
2982:
2976:
2952:
2946:
2925:
2919:
2899:
2875:
2849:
2848:
2846:
2826:
2806:
2783:
2752:
2747:The union of each totally ordered subset
2729:
2708:
2702:
2676:
2670:
2642:
2622:
2601:
2595:
2560:
2559:
2557:
2537:
2505:
2504:
2502:
2482:
2449:
2416:
2415:
2413:
2378:
2377:
2375:
2348:
2342:
2322:
2290:
2258:
2237:
2231:
2205:
2182:
2152:
2151:
2149:
2114:
2066:
2065:
2057:
2034:
2010:
1990:
1970:
1950:
1930:
1910:
1870:
1850:
1830:
1810:
1790:
1770:
1750:
1745:be the set of all orthonormal subsets of
1730:
1708:
1688:
1656:
1636:
1616:
1596:
1576:
1550:
1530:
1510:
1490:
1470:
1450:
1424:
1398:
1378:
1352:
1332:
1312:
1292:
1272:
1252:
1219:
1207:
1187:
1167:
1135:
1131:
1130:
1127:
1106:
1102:
1101:
1098:
1045:
1041:
1040:
1037:
970:
944:
918:
886:
885:
883:
863:
819:
818:
816:
790:
770:
750:
730:
710:
690:
667:
642:
617:
593:
568:
548:
528:
508:
488:
463:
442:
436:
416:
396:
371:
351:
326:
301:
276:
256:
235:
229:
209:
184:
158:
138:
110:
104:
84:
5311:Positive cone of an ordered vector space
4536:
4555:
1182:contains a maximal orthonormal subset
4567:
4543:
2532:. Thus, we are done if we can find a
725:. By the hypothesis of Zorn's lemma,
7:
2497:is a maximal element if and only if
1237:{\displaystyle H\simeq \ell ^{2}(A)}
3999:is the intersection of all towers,
2153:
2067:
483:. Also, since any chain containing
4838:Properties & Types (
4501:{\displaystyle {\widetilde {C}}=C}
4198:
4138:
4115:
4095:
3759:{\displaystyle C={\widetilde {A}}}
3268:
3228:
3154:
2580:{\displaystyle {\widetilde {C}}=C}
2525:{\displaystyle {\widetilde {C}}=C}
2398:{\displaystyle {\widetilde {C}}=C}
2357:
2170:{\displaystyle {\mathfrak {P}}(P)}
2087:
1525:is a subset of the unit sphere in
906:{\displaystyle {\widetilde {C}}=C}
133:) is contained in a maximal chain
14:
5294:Positive cone of an ordered field
2998:is totally ordered. We say a set
5148:Ordered topological vector space
4357:{\displaystyle {\widetilde {C}}}
4054:{\displaystyle {\widetilde {C}}}
3892:{\displaystyle {\widetilde {A}}}
3817:{\displaystyle {\widetilde {A}}}
3788:{\displaystyle {\widetilde {A}}}
3637:{\displaystyle {\widetilde {A}}}
2863:{\displaystyle {\widetilde {C}}}
2363:{\displaystyle C^{*}=\emptyset }
2049:satisfies the above conditions.
1965:and (2) each subset of a set in
1683:is a chain strictly larger than
1144:{\displaystyle \mathbb {R} ^{2}}
1115:{\displaystyle \mathbb {R} ^{2}}
1093:. This is a partial ordering of
1054:{\displaystyle \mathbb {R} ^{2}}
563:is in fact a maximal element of
3965:{\displaystyle U\subset T_{0}}
3664:. In the second case, we have
2455:
2442:
2164:
2158:
2125:
2119:
2093:
2078:
2072:
1231:
1225:
1032:) are two points of the plane
1:
5105:Series-parallel partial order
4204:{\displaystyle T_{0}=\Gamma }
1564:{\displaystyle B\supsetneq A}
4784:Cantor's isomorphism theorem
62:Hausdorff maximality theorem
4824:Szpilrajn extension theorem
4799:Hausdorff maximal principle
4774:Boolean prime ideal theorem
2771:{\displaystyle T'\subset T}
2310:{\displaystyle C\cup \{x\}}
1865:contains a maximal element
1676:{\displaystyle Q\cup \{B\}}
588:; i.e., a maximal chain in
58:Zermelo–Fraenkel set theory
25:Hausdorff maximal principle
5387:
5170:Topological vector lattice
4601:(Springer-Verlag edition).
3863:{\displaystyle C\subset A}
3491:{\displaystyle A\subset C}
3367:{\displaystyle A\subset C}
3167:be the set of all sets in
3140:{\displaystyle C\subset A}
3114:{\displaystyle A\subset C}
2656:{\displaystyle T\subset F}
1438:{\displaystyle T\subset S}
1412:{\displaystyle S\subset T}
4692:
4616:Zermelo's axiom of choice
2137:{\displaystyle f(S)\in S}
1845:. Thus, by Zorn's lemma,
705:contains a maximal chain
321:contains a maximal chain
224:containing a given chain
4779:Cantor–Bernstein theorem
4521:{\displaystyle \square }
4151:is a tower contained in
2278:{\displaystyle x\in P-C}
1716:{\displaystyle \square }
978:{\displaystyle \square }
5366:Mathematical principles
5323:Partially ordered group
5143:Specialization preorder
4144:{\displaystyle \Gamma }
4121:{\displaystyle \Gamma }
4101:{\displaystyle \Gamma }
4025:{\displaystyle U=T_{0}}
3939:is a tower. Now, since
3795:.) Either way, we have
3705:, which implies either
3274:{\displaystyle \Gamma }
3234:{\displaystyle \Gamma }
3194:that are comparable in
3160:{\displaystyle \Gamma }
804:{\displaystyle y\geq x}
4809:Kruskal's tree theorem
4804:Knaster–Tarski theorem
4794:Dushnik–Miller theorem
4614:Gregory Moore (1982),
4522:
4502:
4467:
4432:
4405:
4385:
4358:
4329:
4302:
4282:
4255:
4232:
4205:
4172:
4145:
4122:
4102:
4082:
4055:
4026:
3993:
3966:
3933:
3913:
3893:
3864:
3838:
3818:
3789:
3760:
3725:
3699:
3658:
3638:
3609:
3574:
3539:
3512:
3492:
3466:
3446:
3426:
3403:
3368:
3342:
3315:
3295:
3281:be given and then let
3275:
3255:
3235:
3215:
3188:
3161:
3141:
3115:
3089:
3062:
3041:
3012:
2992:
2962:
2935:
2908:
2884:
2864:
2835:
2815:
2792:
2772:
2738:
2718:
2686:
2657:
2631:
2611:
2581:
2546:
2526:
2491:
2468:
2399:
2364:
2331:
2311:
2279:
2247:
2220:
2219:{\displaystyle C\in F}
2191:
2171:
2138:
2103:
2043:
2025:has a maximal element.
2019:
1999:
1979:
1959:
1939:
1919:
1879:
1859:
1839:
1819:
1799:
1779:
1759:
1739:
1717:
1697:
1677:
1645:
1625:
1605:
1585:
1565:
1539:
1519:
1499:
1479:
1465:are contained in some
1459:
1439:
1413:
1387:
1367:
1341:
1321:
1301:
1281:
1261:
1238:
1196:
1176:
1145:
1116:
1055:
979:
959:
933:
932:{\displaystyle y\in C}
907:
878:and so by maximality,
872:
858:is a chain containing
852:
805:
779:
759:
739:
719:
699:
676:
656:
631:
602:
582:
557:
537:
517:
497:
477:
458:, it is an element of
452:
425:
405:
391:, which is a chain in
385:
360:
340:
315:
290:
265:
245:
218:
198:
167:
147:
120:
93:
4523:
4503:
4468:
4433:
4431:{\displaystyle T_{0}}
4406:
4386:
4384:{\displaystyle T_{0}}
4359:
4330:
4328:{\displaystyle T_{0}}
4303:
4283:
4281:{\displaystyle T_{0}}
4256:
4233:
4231:{\displaystyle T_{0}}
4206:
4173:
4171:{\displaystyle T_{0}}
4146:
4123:
4103:
4083:
4081:{\displaystyle T_{0}}
4056:
4027:
3994:
3992:{\displaystyle T_{0}}
3967:
3934:
3914:
3894:
3865:
3839:
3819:
3790:
3761:
3726:
3700:
3659:
3639:
3615:. In the first case,
3610:
3575:
3540:
3538:{\displaystyle T_{0}}
3513:
3493:
3467:
3447:
3427:
3404:
3369:
3343:
3341:{\displaystyle T_{0}}
3316:
3296:
3276:
3256:
3236:
3216:
3214:{\displaystyle T_{0}}
3189:
3187:{\displaystyle T_{0}}
3162:
3142:
3116:
3090:
3088:{\displaystyle T_{0}}
3063:
3042:
3040:{\displaystyle T_{0}}
3013:
2993:
2991:{\displaystyle T_{0}}
2963:
2961:{\displaystyle T_{0}}
2936:
2934:{\displaystyle C_{0}}
2909:
2885:
2865:
2836:
2816:
2793:
2773:
2739:
2719:
2717:{\displaystyle C_{0}}
2687:
2685:{\displaystyle C_{0}}
2658:
2632:
2612:
2610:{\displaystyle C_{0}}
2582:
2547:
2527:
2492:
2469:
2400:
2365:
2332:
2312:
2280:
2248:
2246:{\displaystyle C^{*}}
2221:
2192:
2172:
2139:
2104:
2044:
2020:
2000:
1980:
1960:
1940:
1920:
1880:
1860:
1840:
1820:
1800:
1780:
1760:
1740:
1718:
1698:
1678:
1646:
1626:
1606:
1586:
1566:
1540:
1520:
1500:
1480:
1460:
1440:
1414:
1388:
1368:
1342:
1322:
1302:
1282:
1262:
1239:
1197:
1177:
1146:
1117:
1056:
980:
960:
934:
908:
873:
853:
806:
780:
760:
740:
720:
700:
677:
657:
632:
603:
583:
558:
538:
518:
498:
478:
453:
451:{\displaystyle C_{0}}
426:
406:
386:
361:
341:
316:
291:
266:
246:
244:{\displaystyle C_{0}}
219:
199:
168:
148:
121:
119:{\displaystyle C_{0}}
94:
78:partially ordered set
37:partially ordered set
5301:Ordered vector space
4512:
4477:
4442:
4415:
4395:
4368:
4339:
4312:
4292:
4265:
4245:
4238:is totally ordered.
4215:
4182:
4155:
4135:
4112:
4092:
4065:
4036:
4003:
3976:
3943:
3923:
3903:
3874:
3848:
3828:
3799:
3770:
3735:
3709:
3668:
3648:
3619:
3584:
3549:
3522:
3502:
3476:
3456:
3436:
3416:
3378:
3352:
3325:
3305:
3285:
3265:
3245:
3225:
3198:
3171:
3151:
3125:
3099:
3072:
3052:
3024:
3002:
2975:
2945:
2918:
2898:
2874:
2845:
2825:
2805:
2782:
2751:
2728:
2701:
2669:
2641:
2621:
2594:
2556:
2536:
2501:
2481:
2412:
2374:
2341:
2321:
2289:
2257:
2230:
2204:
2181:
2148:
2113:
2056:
2033:
2009:
1989:
1969:
1949:
1929:
1909:
1869:
1849:
1829:
1809:
1805:, then the union of
1789:
1769:
1749:
1729:
1707:
1687:
1655:
1635:
1615:
1595:
1575:
1549:
1529:
1509:
1489:
1469:
1449:
1423:
1397:
1377:
1351:
1331:
1311:
1291:
1271:
1251:
1244:as Hilbert spaces.)
1206:
1186:
1166:
1126:
1097:
1036:
998:strict partial order
969:
943:
917:
882:
862:
815:
789:
769:
749:
729:
709:
689:
666:
641:
616:
592:
567:
547:
527:
507:
487:
462:
435:
415:
395:
370:
350:
325:
300:
275:
255:
228:
208:
183:
157:
137:
103:
83:
5139:Alexandrov topology
5085:Lexicographic order
5044:Well-quasi-ordering
3724:{\displaystyle A=C}
2971:Now, we shall show
2637:. We call a subset
1703:, a contradiction.
1366:{\displaystyle S,T}
958:{\displaystyle y=x}
745:has an upper bound
5120:Transitive closure
5080:Converse/Transpose
4789:Dilworth's theorem
4518:
4498:
4463:
4428:
4401:
4381:
4354:
4325:
4298:
4278:
4251:
4228:
4201:
4168:
4141:
4118:
4098:
4078:
4051:
4022:
3989:
3962:
3929:
3909:
3889:
3860:
3834:
3814:
3785:
3756:
3721:
3695:
3654:
3634:
3605:
3570:
3535:
3508:
3488:
3462:
3442:
3422:
3399:
3364:
3338:
3311:
3301:be the set of all
3291:
3271:
3251:
3231:
3211:
3184:
3157:
3137:
3111:
3085:
3058:
3037:
3008:
2988:
2958:
2931:
2904:
2880:
2860:
2831:
2811:
2788:
2768:
2734:
2714:
2682:
2653:
2627:
2607:
2577:
2542:
2522:
2487:
2464:
2395:
2360:
2327:
2307:
2275:
2253:be the set of all
2243:
2216:
2187:
2167:
2144:for the power set
2134:
2099:
2039:
2015:
1995:
1975:
1955:
1935:
1915:
1875:
1855:
1835:
1815:
1795:
1775:
1755:
1735:
1713:
1693:
1673:
1641:
1621:
1601:
1581:
1561:
1535:
1515:
1495:
1475:
1455:
1435:
1409:
1383:
1363:
1337:
1317:
1297:
1277:
1257:
1234:
1192:
1172:
1141:
1112:
1051:
975:
955:
929:
903:
868:
848:
801:
775:
755:
735:
715:
695:
672:
655:{\displaystyle P'}
652:
630:{\displaystyle P'}
627:
598:
581:{\displaystyle P'}
578:
553:
533:
513:
493:
476:{\displaystyle P'}
473:
448:
421:
401:
384:{\displaystyle C'}
381:
356:
339:{\displaystyle C'}
336:
314:{\displaystyle P'}
311:
289:{\displaystyle P'}
286:
261:
241:
214:
197:{\displaystyle P'}
194:
163:
143:
116:
89:
68:(Kelley 1955:33).
5348:
5347:
5306:Partially ordered
5115:Symmetric closure
5100:Reflexive closure
4843:
4647:978-0-07-054234-1
4489:
4454:
4404:{\displaystyle C}
4351:
4301:{\displaystyle C}
4254:{\displaystyle C}
4061:is comparable in
4048:
3932:{\displaystyle U}
3912:{\displaystyle U}
3886:
3837:{\displaystyle U}
3811:
3782:
3753:
3692:
3657:{\displaystyle U}
3631:
3602:
3561:
3518:is comparable in
3511:{\displaystyle C}
3465:{\displaystyle U}
3445:{\displaystyle A}
3425:{\displaystyle U}
3390:
3348:such that either
3314:{\displaystyle A}
3294:{\displaystyle U}
3254:{\displaystyle C}
3061:{\displaystyle A}
3011:{\displaystyle C}
2907:{\displaystyle F}
2883:{\displaystyle T}
2857:
2834:{\displaystyle T}
2814:{\displaystyle C}
2791:{\displaystyle T}
2737:{\displaystyle T}
2630:{\displaystyle F}
2568:
2545:{\displaystyle C}
2513:
2490:{\displaystyle C}
2424:
2405:. Otherwise, let
2386:
2330:{\displaystyle F}
2190:{\displaystyle P}
2042:{\displaystyle P}
2018:{\displaystyle F}
1998:{\displaystyle F}
1978:{\displaystyle F}
1958:{\displaystyle F}
1938:{\displaystyle F}
1918:{\displaystyle F}
1878:{\displaystyle A}
1858:{\displaystyle P}
1838:{\displaystyle Q}
1818:{\displaystyle Q}
1798:{\displaystyle P}
1778:{\displaystyle Q}
1758:{\displaystyle H}
1738:{\displaystyle P}
1696:{\displaystyle Q}
1644:{\displaystyle Q}
1624:{\displaystyle B}
1604:{\displaystyle P}
1584:{\displaystyle B}
1538:{\displaystyle H}
1518:{\displaystyle A}
1498:{\displaystyle Q}
1478:{\displaystyle S}
1458:{\displaystyle A}
1386:{\displaystyle Q}
1340:{\displaystyle Q}
1320:{\displaystyle A}
1300:{\displaystyle Q}
1280:{\displaystyle H}
1260:{\displaystyle P}
1195:{\displaystyle A}
1175:{\displaystyle H}
894:
871:{\displaystyle C}
827:
778:{\displaystyle P}
758:{\displaystyle x}
738:{\displaystyle C}
718:{\displaystyle C}
698:{\displaystyle P}
675:{\displaystyle P}
601:{\displaystyle P}
556:{\displaystyle C}
536:{\displaystyle C}
516:{\displaystyle C}
496:{\displaystyle C}
424:{\displaystyle C}
404:{\displaystyle P}
359:{\displaystyle C}
264:{\displaystyle P}
217:{\displaystyle P}
204:of all chains in
166:{\displaystyle P}
146:{\displaystyle C}
92:{\displaystyle P}
5378:
5090:Linear extension
4839:
4819:Mirsky's theorem
4679:
4672:
4665:
4656:
4651:
4609:General topology
4592:
4589:Naive set theory
4571:
4565:
4559:
4553:
4547:
4541:
4527:
4525:
4524:
4519:
4507:
4505:
4504:
4499:
4491:
4490:
4482:
4472:
4470:
4469:
4464:
4456:
4455:
4447:
4437:
4435:
4434:
4429:
4427:
4426:
4411:is the union of
4410:
4408:
4407:
4402:
4390:
4388:
4387:
4382:
4380:
4379:
4363:
4361:
4360:
4355:
4353:
4352:
4344:
4335:and then by 3.,
4334:
4332:
4331:
4326:
4324:
4323:
4307:
4305:
4304:
4299:
4287:
4285:
4284:
4279:
4277:
4276:
4261:be the union of
4260:
4258:
4257:
4252:
4237:
4235:
4234:
4229:
4227:
4226:
4210:
4208:
4207:
4202:
4194:
4193:
4177:
4175:
4174:
4169:
4167:
4166:
4150:
4148:
4147:
4142:
4127:
4125:
4124:
4119:
4107:
4105:
4104:
4099:
4087:
4085:
4084:
4079:
4077:
4076:
4060:
4058:
4057:
4052:
4050:
4049:
4041:
4032:, which implies
4031:
4029:
4028:
4023:
4021:
4020:
3998:
3996:
3995:
3990:
3988:
3987:
3971:
3969:
3968:
3963:
3961:
3960:
3938:
3936:
3935:
3930:
3918:
3916:
3915:
3910:
3898:
3896:
3895:
3890:
3888:
3887:
3879:
3869:
3867:
3866:
3861:
3844:. Similarly, if
3843:
3841:
3840:
3835:
3823:
3821:
3820:
3815:
3813:
3812:
3804:
3794:
3792:
3791:
3786:
3784:
3783:
3775:
3765:
3763:
3762:
3757:
3755:
3754:
3746:
3730:
3728:
3727:
3722:
3704:
3702:
3701:
3696:
3694:
3693:
3685:
3663:
3661:
3660:
3655:
3643:
3641:
3640:
3635:
3633:
3632:
3624:
3614:
3612:
3611:
3606:
3604:
3603:
3595:
3579:
3577:
3576:
3571:
3563:
3562:
3554:
3544:
3542:
3541:
3536:
3534:
3533:
3517:
3515:
3514:
3509:
3497:
3495:
3494:
3489:
3471:
3469:
3468:
3463:
3451:
3449:
3448:
3443:
3431:
3429:
3428:
3423:
3408:
3406:
3405:
3400:
3392:
3391:
3383:
3373:
3371:
3370:
3365:
3347:
3345:
3344:
3339:
3337:
3336:
3320:
3318:
3317:
3312:
3300:
3298:
3297:
3292:
3280:
3278:
3277:
3272:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3220:
3218:
3217:
3212:
3210:
3209:
3193:
3191:
3190:
3185:
3183:
3182:
3166:
3164:
3163:
3158:
3146:
3144:
3143:
3138:
3120:
3118:
3117:
3112:
3094:
3092:
3091:
3086:
3084:
3083:
3067:
3065:
3064:
3059:
3046:
3044:
3043:
3038:
3036:
3035:
3017:
3015:
3014:
3009:
2997:
2995:
2994:
2989:
2987:
2986:
2967:
2965:
2964:
2959:
2957:
2956:
2941:is a tower. Let
2940:
2938:
2937:
2932:
2930:
2929:
2913:
2911:
2910:
2905:
2889:
2887:
2886:
2881:
2869:
2867:
2866:
2861:
2859:
2858:
2850:
2840:
2838:
2837:
2832:
2820:
2818:
2817:
2812:
2797:
2795:
2794:
2789:
2777:
2775:
2774:
2769:
2761:
2743:
2741:
2740:
2735:
2723:
2721:
2720:
2715:
2713:
2712:
2691:
2689:
2688:
2683:
2681:
2680:
2662:
2660:
2659:
2654:
2636:
2634:
2633:
2628:
2616:
2614:
2613:
2608:
2606:
2605:
2586:
2584:
2583:
2578:
2570:
2569:
2561:
2551:
2549:
2548:
2543:
2531:
2529:
2528:
2523:
2515:
2514:
2506:
2496:
2494:
2493:
2488:
2473:
2471:
2470:
2465:
2454:
2453:
2426:
2425:
2417:
2404:
2402:
2401:
2396:
2388:
2387:
2379:
2369:
2367:
2366:
2361:
2353:
2352:
2336:
2334:
2333:
2328:
2316:
2314:
2313:
2308:
2284:
2282:
2281:
2276:
2252:
2250:
2249:
2244:
2242:
2241:
2225:
2223:
2222:
2217:
2196:
2194:
2193:
2188:
2176:
2174:
2173:
2168:
2157:
2156:
2143:
2141:
2140:
2135:
2108:
2106:
2105:
2100:
2071:
2070:
2048:
2046:
2045:
2040:
2024:
2022:
2021:
2016:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1964:
1962:
1961:
1956:
1944:
1942:
1941:
1936:
1924:
1922:
1921:
1916:
1884:
1882:
1881:
1876:
1864:
1862:
1861:
1856:
1844:
1842:
1841:
1836:
1824:
1822:
1821:
1816:
1804:
1802:
1801:
1796:
1784:
1782:
1781:
1776:
1764:
1762:
1761:
1756:
1744:
1742:
1741:
1736:
1722:
1720:
1719:
1714:
1702:
1700:
1699:
1694:
1682:
1680:
1679:
1674:
1650:
1648:
1647:
1642:
1630:
1628:
1627:
1622:
1610:
1608:
1607:
1602:
1590:
1588:
1587:
1582:
1570:
1568:
1567:
1562:
1544:
1542:
1541:
1536:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1484:
1482:
1481:
1476:
1464:
1462:
1461:
1456:
1444:
1442:
1441:
1436:
1418:
1416:
1415:
1410:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1346:
1344:
1343:
1338:
1327:be the union of
1326:
1324:
1323:
1318:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1266:
1264:
1263:
1258:
1243:
1241:
1240:
1235:
1224:
1223:
1201:
1199:
1198:
1193:
1181:
1179:
1178:
1173:
1150:
1148:
1147:
1142:
1140:
1139:
1134:
1121:
1119:
1118:
1113:
1111:
1110:
1105:
1060:
1058:
1057:
1052:
1050:
1049:
1044:
984:
982:
981:
976:
964:
962:
961:
956:
938:
936:
935:
930:
912:
910:
909:
904:
896:
895:
887:
877:
875:
874:
869:
857:
855:
854:
849:
829:
828:
820:
810:
808:
807:
802:
784:
782:
781:
776:
764:
762:
761:
756:
744:
742:
741:
736:
724:
722:
721:
716:
704:
702:
701:
696:
681:
679:
678:
673:
661:
659:
658:
653:
651:
636:
634:
633:
628:
626:
607:
605:
604:
599:
587:
585:
584:
579:
577:
562:
560:
559:
554:
542:
540:
539:
534:
522:
520:
519:
514:
503:is contained in
502:
500:
499:
494:
482:
480:
479:
474:
472:
457:
455:
454:
449:
447:
446:
430:
428:
427:
422:
410:
408:
407:
402:
390:
388:
387:
382:
380:
366:be the union of
365:
363:
362:
357:
345:
343:
342:
337:
335:
320:
318:
317:
312:
310:
295:
293:
292:
287:
285:
270:
268:
267:
262:
250:
248:
247:
242:
240:
239:
223:
221:
220:
215:
203:
201:
200:
195:
193:
172:
170:
169:
164:
152:
150:
149:
144:
125:
123:
122:
117:
115:
114:
98:
96:
95:
90:
66:Kuratowski lemma
5386:
5385:
5381:
5380:
5379:
5377:
5376:
5375:
5361:Axiom of choice
5351:
5350:
5349:
5344:
5340:Young's lattice
5196:
5124:
5063:
4913:Heyting algebra
4861:Boolean algebra
4833:
4814:Laver's theorem
4762:
4728:Boolean algebra
4723:Binary relation
4711:
4688:
4683:
4648:
4640:. McGraw-Hill.
4632:
4611:, Von Nostrand.
4583:
4580:
4575:
4574:
4566:
4562:
4554:
4550:
4546:, Theorem 4.22.
4542:
4538:
4533:
4510:
4509:
4475:
4474:
4440:
4439:
4418:
4413:
4412:
4393:
4392:
4371:
4366:
4365:
4337:
4336:
4315:
4310:
4309:
4290:
4289:
4268:
4263:
4262:
4243:
4242:
4218:
4213:
4212:
4185:
4180:
4179:
4158:
4153:
4152:
4133:
4132:
4131:Finally, since
4110:
4109:
4090:
4089:
4068:
4063:
4062:
4034:
4033:
4012:
4001:
4000:
3979:
3974:
3973:
3952:
3941:
3940:
3921:
3920:
3901:
3900:
3872:
3871:
3846:
3845:
3826:
3825:
3797:
3796:
3768:
3767:
3733:
3732:
3707:
3706:
3666:
3665:
3646:
3645:
3617:
3616:
3582:
3581:
3547:
3546:
3525:
3520:
3519:
3500:
3499:
3474:
3473:
3454:
3453:
3434:
3433:
3414:
3413:
3376:
3375:
3350:
3349:
3328:
3323:
3322:
3303:
3302:
3283:
3282:
3263:
3262:
3243:
3242:
3223:
3222:
3201:
3196:
3195:
3174:
3169:
3168:
3149:
3148:
3123:
3122:
3097:
3096:
3075:
3070:
3069:
3050:
3049:
3027:
3022:
3021:
3000:
2999:
2978:
2973:
2972:
2948:
2943:
2942:
2921:
2916:
2915:
2896:
2895:
2872:
2871:
2843:
2842:
2823:
2822:
2803:
2802:
2780:
2779:
2754:
2749:
2748:
2726:
2725:
2704:
2699:
2698:
2672:
2667:
2666:
2639:
2638:
2619:
2618:
2597:
2592:
2591:
2554:
2553:
2534:
2533:
2499:
2498:
2479:
2478:
2445:
2410:
2409:
2372:
2371:
2344:
2339:
2338:
2319:
2318:
2287:
2286:
2255:
2254:
2233:
2228:
2227:
2202:
2201:
2179:
2178:
2146:
2145:
2111:
2110:
2054:
2053:
2031:
2030:
2007:
2006:
1987:
1986:
1967:
1966:
1947:
1946:
1927:
1926:
1907:
1906:
1899:axiom of choice
1891:
1867:
1866:
1847:
1846:
1827:
1826:
1807:
1806:
1787:
1786:
1767:
1766:
1747:
1746:
1727:
1726:
1705:
1704:
1685:
1684:
1653:
1652:
1633:
1632:
1613:
1612:
1593:
1592:
1573:
1572:
1547:
1546:
1527:
1526:
1507:
1506:
1487:
1486:
1467:
1466:
1447:
1446:
1421:
1420:
1395:
1394:
1375:
1374:
1349:
1348:
1329:
1328:
1309:
1308:
1289:
1288:
1269:
1268:
1249:
1248:
1215:
1204:
1203:
1184:
1183:
1164:
1163:
1157:
1129:
1124:
1123:
1100:
1095:
1094:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1039:
1034:
1033:
1031:
1027:
1023:
1019:
1004:. Suppose that
990:
967:
966:
941:
940:
915:
914:
880:
879:
860:
859:
813:
812:
787:
786:
767:
766:
747:
746:
727:
726:
707:
706:
687:
686:
664:
663:
644:
639:
638:
619:
614:
613:
590:
589:
570:
565:
564:
545:
544:
525:
524:
505:
504:
485:
484:
465:
460:
459:
438:
433:
432:
413:
412:
393:
392:
373:
368:
367:
348:
347:
328:
323:
322:
303:
298:
297:
278:
273:
272:
253:
252:
231:
226:
225:
206:
205:
186:
181:
180:
155:
154:
135:
134:
128:totally ordered
106:
101:
100:
81:
80:
74:
54:axiom of choice
41:totally ordered
33:Felix Hausdorff
17:
12:
11:
5:
5384:
5382:
5374:
5373:
5368:
5363:
5353:
5352:
5346:
5345:
5343:
5342:
5337:
5332:
5331:
5330:
5320:
5319:
5318:
5313:
5308:
5298:
5297:
5296:
5286:
5281:
5280:
5279:
5274:
5267:Order morphism
5264:
5263:
5262:
5252:
5247:
5242:
5237:
5232:
5231:
5230:
5220:
5215:
5210:
5204:
5202:
5198:
5197:
5195:
5194:
5193:
5192:
5187:
5185:Locally convex
5182:
5177:
5167:
5165:Order topology
5162:
5161:
5160:
5158:Order topology
5155:
5145:
5135:
5133:
5126:
5125:
5123:
5122:
5117:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5071:
5069:
5065:
5064:
5062:
5061:
5051:
5041:
5036:
5031:
5026:
5021:
5016:
5011:
5006:
5005:
5004:
4994:
4989:
4988:
4987:
4982:
4977:
4972:
4970:Chain-complete
4962:
4957:
4956:
4955:
4950:
4945:
4940:
4935:
4925:
4920:
4915:
4910:
4905:
4895:
4890:
4885:
4880:
4875:
4870:
4869:
4868:
4858:
4853:
4847:
4845:
4835:
4834:
4832:
4831:
4826:
4821:
4816:
4811:
4806:
4801:
4796:
4791:
4786:
4781:
4776:
4770:
4768:
4764:
4763:
4761:
4760:
4755:
4750:
4745:
4740:
4735:
4730:
4725:
4719:
4717:
4713:
4712:
4710:
4709:
4704:
4699:
4693:
4690:
4689:
4684:
4682:
4681:
4674:
4667:
4659:
4653:
4652:
4646:
4629:
4619:
4612:
4602:
4579:
4576:
4573:
4572:
4560:
4548:
4535:
4534:
4532:
4529:
4517:
4497:
4494:
4488:
4485:
4462:
4459:
4453:
4450:
4425:
4421:
4400:
4378:
4374:
4350:
4347:
4322:
4318:
4297:
4275:
4271:
4250:
4225:
4221:
4211:, which means
4200:
4197:
4192:
4188:
4165:
4161:
4140:
4117:
4097:
4088:; i.e., is in
4075:
4071:
4047:
4044:
4019:
4015:
4011:
4008:
3986:
3982:
3959:
3955:
3951:
3948:
3928:
3908:
3885:
3882:
3859:
3856:
3853:
3833:
3810:
3807:
3781:
3778:
3752:
3749:
3743:
3740:
3720:
3717:
3714:
3691:
3688:
3682:
3679:
3676:
3673:
3653:
3630:
3627:
3601:
3598:
3592:
3589:
3569:
3566:
3560:
3557:
3532:
3528:
3507:
3487:
3484:
3481:
3461:
3441:
3421:
3398:
3395:
3389:
3386:
3363:
3360:
3357:
3335:
3331:
3310:
3290:
3270:
3250:
3230:
3208:
3204:
3181:
3177:
3156:
3136:
3133:
3130:
3110:
3107:
3104:
3082:
3078:
3057:
3034:
3030:
3020:comparable in
3007:
2985:
2981:
2955:
2951:
2928:
2924:
2903:
2892:
2891:
2879:
2856:
2853:
2830:
2810:
2799:
2787:
2767:
2764:
2760:
2757:
2745:
2733:
2711:
2707:
2679:
2675:
2652:
2649:
2646:
2626:
2604:
2600:
2576:
2573:
2567:
2564:
2541:
2521:
2518:
2512:
2509:
2486:
2475:
2474:
2463:
2460:
2457:
2452:
2448:
2444:
2441:
2438:
2435:
2432:
2429:
2423:
2420:
2394:
2391:
2385:
2382:
2359:
2356:
2351:
2347:
2326:
2306:
2303:
2300:
2297:
2294:
2274:
2271:
2268:
2265:
2262:
2240:
2236:
2215:
2212:
2209:
2186:
2166:
2163:
2160:
2155:
2133:
2130:
2127:
2124:
2121:
2118:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2069:
2064:
2061:
2038:
2027:
2026:
2014:
1994:
1974:
1954:
1934:
1914:
1890:
1887:
1874:
1854:
1834:
1814:
1794:
1785:is a chain in
1774:
1754:
1734:
1712:
1692:
1672:
1669:
1666:
1663:
1660:
1640:
1620:
1600:
1580:
1560:
1557:
1554:
1545:). Second, if
1534:
1514:
1494:
1474:
1454:
1434:
1431:
1428:
1408:
1405:
1402:
1393:, then either
1382:
1362:
1359:
1356:
1336:
1316:
1296:
1276:
1256:
1233:
1230:
1227:
1222:
1218:
1214:
1211:
1191:
1171:
1156:
1153:
1138:
1133:
1109:
1104:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1048:
1043:
1029:
1025:
1021:
1017:
989:
986:
974:
954:
951:
948:
928:
925:
922:
902:
899:
893:
890:
867:
847:
844:
841:
838:
835:
832:
826:
823:
800:
797:
794:
774:
754:
734:
714:
694:
671:
650:
647:
625:
622:
597:
576:
573:
552:
532:
512:
492:
471:
468:
445:
441:
420:
400:
379:
376:
355:
334:
331:
309:
306:
284:
281:
260:
238:
234:
213:
192:
189:
162:
142:
113:
109:
99:, every chain
88:
73:
70:
15:
13:
10:
9:
6:
4:
3:
2:
5383:
5372:
5369:
5367:
5364:
5362:
5359:
5358:
5356:
5341:
5338:
5336:
5333:
5329:
5326:
5325:
5324:
5321:
5317:
5314:
5312:
5309:
5307:
5304:
5303:
5302:
5299:
5295:
5292:
5291:
5290:
5289:Ordered field
5287:
5285:
5282:
5278:
5275:
5273:
5270:
5269:
5268:
5265:
5261:
5258:
5257:
5256:
5253:
5251:
5248:
5246:
5245:Hasse diagram
5243:
5241:
5238:
5236:
5233:
5229:
5226:
5225:
5224:
5223:Comparability
5221:
5219:
5216:
5214:
5211:
5209:
5206:
5205:
5203:
5199:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5172:
5171:
5168:
5166:
5163:
5159:
5156:
5154:
5151:
5150:
5149:
5146:
5144:
5140:
5137:
5136:
5134:
5131:
5127:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5095:Product order
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5072:
5070:
5068:Constructions
5066:
5060:
5056:
5052:
5049:
5045:
5042:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5010:
5007:
5003:
5000:
4999:
4998:
4995:
4993:
4990:
4986:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4967:
4966:
4965:Partial order
4963:
4961:
4958:
4954:
4953:Join and meet
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4934:
4931:
4930:
4929:
4926:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4900:
4896:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4867:
4864:
4863:
4862:
4859:
4857:
4854:
4852:
4851:Antisymmetric
4849:
4848:
4846:
4842:
4836:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4771:
4769:
4765:
4759:
4758:Weak ordering
4756:
4754:
4751:
4749:
4746:
4744:
4743:Partial order
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4720:
4718:
4714:
4708:
4705:
4703:
4700:
4698:
4695:
4694:
4691:
4687:
4680:
4675:
4673:
4668:
4666:
4661:
4660:
4657:
4649:
4643:
4639:
4635:
4634:Rudin, Walter
4630:
4627:
4623:
4622:James Munkres
4620:
4617:
4613:
4610:
4606:
4603:
4600:
4599:0-387-90092-6
4596:
4590:
4586:
4582:
4581:
4577:
4569:
4564:
4561:
4557:
4552:
4549:
4545:
4540:
4537:
4530:
4528:
4515:
4495:
4492:
4486:
4483:
4460:
4457:
4451:
4448:
4423:
4419:
4398:
4376:
4372:
4348:
4345:
4320:
4316:
4295:
4273:
4269:
4248:
4239:
4223:
4219:
4195:
4190:
4186:
4163:
4159:
4129:
4073:
4069:
4045:
4042:
4017:
4013:
4009:
4006:
3984:
3980:
3957:
3953:
3949:
3946:
3926:
3906:
3883:
3880:
3857:
3854:
3851:
3831:
3808:
3805:
3779:
3776:
3750:
3747:
3741:
3738:
3718:
3715:
3712:
3689:
3686:
3680:
3677:
3674:
3671:
3651:
3628:
3625:
3599:
3596:
3590:
3587:
3567:
3564:
3558:
3555:
3530:
3526:
3505:
3498:, then since
3485:
3482:
3479:
3459:
3439:
3419:
3410:
3396:
3393:
3387:
3384:
3361:
3358:
3355:
3333:
3329:
3308:
3288:
3248:
3206:
3202:
3179:
3175:
3134:
3131:
3128:
3108:
3105:
3102:
3080:
3076:
3055:
3047:
3032:
3028:
3005:
2983:
2979:
2969:
2953:
2949:
2926:
2922:
2901:
2877:
2854:
2851:
2828:
2808:
2800:
2785:
2765:
2762:
2758:
2755:
2746:
2731:
2709:
2705:
2697:
2696:
2695:
2693:
2677:
2673:
2650:
2647:
2644:
2624:
2602:
2598:
2588:
2574:
2571:
2565:
2562:
2539:
2519:
2516:
2510:
2507:
2484:
2461:
2450:
2446:
2439:
2433:
2430:
2427:
2421:
2418:
2408:
2407:
2406:
2392:
2389:
2383:
2380:
2354:
2349:
2345:
2324:
2301:
2295:
2292:
2272:
2269:
2266:
2263:
2260:
2238:
2234:
2213:
2210:
2207:
2198:
2184:
2161:
2131:
2128:
2122:
2116:
2096:
2081:
2075:
2062:
2059:
2050:
2036:
2012:
1992:
1972:
1952:
1932:
1912:
1904:
1903:
1902:
1900:
1896:
1888:
1886:
1872:
1852:
1832:
1812:
1792:
1772:
1752:
1732:
1723:
1710:
1690:
1667:
1661:
1658:
1638:
1631:cannot be in
1618:
1598:
1578:
1558:
1555:
1552:
1532:
1512:
1492:
1472:
1452:
1432:
1429:
1426:
1406:
1403:
1400:
1380:
1360:
1357:
1354:
1334:
1314:
1294:
1274:
1254:
1245:
1228:
1220:
1216:
1212:
1209:
1189:
1169:
1162:
1161:Hilbert space
1154:
1152:
1136:
1107:
1046:
1014:
1011:
1007:
1003:
999:
995:
987:
985:
972:
952:
949:
946:
926:
923:
920:
900:
897:
891:
888:
865:
842:
836:
833:
830:
824:
821:
798:
795:
792:
772:
752:
732:
712:
692:
683:
669:
648:
645:
623:
620:
609:
595:
574:
571:
550:
530:
510:
490:
469:
466:
443:
439:
418:
398:
377:
374:
353:
332:
329:
307:
304:
282:
279:
258:
236:
232:
211:
190:
187:
177:
174:
160:
140:
132:
129:
111:
107:
86:
79:
71:
69:
67:
63:
59:
55:
50:
47:
45:
42:
38:
34:
30:
26:
22:
5371:Order theory
5132:& Orders
5110:Star product
5039:Well-founded
4992:Prefix order
4948:Distributive
4938:Complemented
4908:Foundational
4873:Completeness
4829:Zorn's lemma
4798:
4733:Cyclic order
4716:Key concepts
4686:Order theory
4637:
4631:Appendix of
4625:
4615:
4608:
4588:
4585:Halmos, Paul
4563:
4551:
4539:
4240:
4130:
4128:is a tower.
3411:
3048:if for each
3019:
2970:
2893:
2665:tower (over
2664:
2589:
2476:
2199:
2051:
2028:
1895:Zorn's lemma
1892:
1724:
1246:
1158:
1015:
1009:
1005:
1001:
993:
991:
684:
610:
543:is a union,
178:
175:
75:
65:
61:
51:
48:
29:Zorn's lemma
24:
18:
5316:Riesz space
5277:Isomorphism
5153:Normal cone
5075:Composition
5009:Semilattice
4918:Homogeneous
4903:Equivalence
4753:Total order
4618:, Springer.
4605:John Kelley
4556:Halmos 1960
3221:. We claim
2914:containing
2370:, then let
1897:, from the
1155:Application
1061:, define (x
21:mathematics
5355:Categories
5284:Order type
5218:Cofinality
5059:Well-order
5034:Transitive
4923:Idempotent
4856:Asymmetric
4628:, Pearson.
4578:References
4570:, Appendix
4568:Rudin 1986
4544:Rudin 1986
4178:, we have
2552:such that
2285:such that
2109:such that
31:proved by
5335:Upper set
5272:Embedding
5208:Antichain
5029:Tolerance
5019:Symmetric
5014:Semiorder
4960:Reflexive
4878:Connected
4516:◻
4487:~
4473:and thus
4458:⊂
4452:~
4349:~
4288:. By 2.,
4199:Γ
4139:Γ
4116:Γ
4096:Γ
4046:~
3950:⊂
3919:. Hence,
3884:~
3870:, we see
3855:⊂
3809:~
3780:~
3751:~
3690:~
3681:⊂
3675:⊂
3629:~
3600:~
3591:⊂
3565:⊂
3559:~
3545:, either
3483:⊂
3412:We claim
3394:⊂
3388:~
3359:⊂
3269:Γ
3229:Γ
3155:Γ
3132:⊂
3106:⊂
3095:, either
2855:~
2801:For each
2763:⊂
2648:⊂
2566:~
2511:~
2451:∗
2434:∪
2422:~
2384:~
2358:∅
2350:∗
2296:∪
2270:−
2264:∈
2239:∗
2211:∈
2200:For each
2129:∈
2094:→
2088:∅
2082:−
1711:◻
1662:∪
1571:for some
1556:⊋
1430:⊂
1404:⊂
1217:ℓ
1213:≃
1069:) < (x
973:◻
924:∈
892:~
837:∪
825:~
796:≥
637:and thus
431:contains
126:(i.e., a
72:Statement
56:over ZF (
5130:Topology
4997:Preorder
4980:Eulerian
4943:Complete
4893:Directed
4883:Covering
4748:Preorder
4707:Category
4702:Glossary
4636:(1986).
4626:Topology
4624:(2000),
4607:(1955),
4587:(1960).
4391:. Since
2759:′
1024:) and (x
988:Examples
913:; i.e.,
649:′
624:′
575:′
470:′
378:′
333:′
308:′
283:′
191:′
39:, every
5235:Duality
5213:Cofinal
5201:Related
5180:Fréchet
5057:)
4933:Bounded
4928:Lattice
4901:)
4899:Partial
4767:Results
4738:Lattice
4558:, § 16.
2005:. Then
1651:and so
1611:, then
1373:are in
939:and so
811:, then
271:. Then
64:or the
5260:Subnet
5240:Filter
5190:Normed
5175:Banach
5141:&
5048:Better
4985:Strict
4975:Graded
4866:topics
4697:Topics
4644:
4597:
4364:is in
4308:is in
3899:is in
3824:is in
3644:is in
3452:be in
3147:. Let
2870:is in
2778:is in
2724:is in
2590:Fix a
2317:is in
2226:, let
1985:is in
1945:is in
1307:. Let
1089:< x
1077:) if y
346:. Let
131:subset
44:subset
23:, the
5250:Ideal
5228:Graph
5024:Total
5002:Total
4888:Dense
4531:Notes
3472:. If
2477:Note
2337:. If
1889:Proof
1765:. If
1085:and x
1016:If (x
785:. If
4841:list
4642:ISBN
4595:ISBN
4241:Let
3972:and
1905:Let
1247:Let
5255:Net
5055:Pre
3731:or
3580:or
3374:or
3321:in
3261:in
3121:or
3068:in
3018:is
2821:in
2694:if
2617:in
2177:of
1591:in
1485:in
1419:or
1081:= y
1073:, y
1065:, y
1028:, y
1020:, y
1000:on
992:If
765:in
523:as
251:in
19:In
5357::
4508:.
4438:,
3409:.
2841:,
2663:a
2587:.
2197:.
1901:.
1151:.
965:.
682:.
608:.
5053:(
5050:)
5046:(
4897:(
4844:)
4678:e
4671:t
4664:v
4650:.
4496:C
4493:=
4484:C
4461:C
4449:C
4424:0
4420:T
4399:C
4377:0
4373:T
4346:C
4321:0
4317:T
4296:C
4274:0
4270:T
4249:C
4224:0
4220:T
4196:=
4191:0
4187:T
4164:0
4160:T
4074:0
4070:T
4043:C
4018:0
4014:T
4010:=
4007:U
3985:0
3981:T
3958:0
3954:T
3947:U
3927:U
3907:U
3881:A
3858:A
3852:C
3832:U
3806:A
3777:A
3748:A
3742:=
3739:C
3719:C
3716:=
3713:A
3687:A
3678:C
3672:A
3652:U
3626:A
3597:A
3588:C
3568:C
3556:A
3531:0
3527:T
3506:C
3486:C
3480:A
3460:U
3440:A
3420:U
3397:A
3385:C
3362:C
3356:A
3334:0
3330:T
3309:A
3289:U
3249:C
3207:0
3203:T
3180:0
3176:T
3135:A
3129:C
3109:C
3103:A
3081:0
3077:T
3056:A
3033:0
3029:T
3006:C
2984:0
2980:T
2954:0
2950:T
2927:0
2923:C
2902:F
2890:.
2878:T
2852:C
2829:T
2809:C
2786:T
2766:T
2756:T
2744:.
2732:T
2710:0
2706:C
2692:)
2678:0
2674:C
2651:F
2645:T
2625:F
2603:0
2599:C
2575:C
2572:=
2563:C
2540:C
2520:C
2517:=
2508:C
2485:C
2462:.
2459:}
2456:)
2447:C
2443:(
2440:f
2437:{
2431:C
2428:=
2419:C
2393:C
2390:=
2381:C
2355:=
2346:C
2325:F
2305:}
2302:x
2299:{
2293:C
2273:C
2267:P
2261:x
2235:C
2214:F
2208:C
2185:P
2165:)
2162:P
2159:(
2154:P
2132:S
2126:)
2123:S
2120:(
2117:f
2097:P
2091:}
2085:{
2079:)
2076:P
2073:(
2068:P
2063::
2060:f
2037:P
2013:F
1993:F
1973:F
1953:F
1933:F
1913:F
1873:A
1853:P
1833:Q
1813:Q
1793:P
1773:Q
1753:H
1733:P
1691:Q
1671:}
1668:B
1665:{
1659:Q
1639:Q
1619:B
1599:P
1579:B
1559:A
1553:B
1533:H
1513:A
1493:Q
1473:S
1453:A
1433:S
1427:T
1407:T
1401:S
1381:Q
1361:T
1358:,
1355:S
1335:Q
1315:A
1295:Q
1275:H
1255:P
1232:)
1229:A
1226:(
1221:2
1210:H
1190:A
1170:H
1137:2
1132:R
1108:2
1103:R
1091:1
1087:0
1083:1
1079:0
1075:1
1071:1
1067:0
1063:0
1047:2
1042:R
1030:1
1026:1
1022:0
1018:0
1010:A
1006:A
1002:A
994:A
953:x
950:=
947:y
927:C
921:y
901:C
898:=
889:C
866:C
846:}
843:y
840:{
834:C
831:=
822:C
799:x
793:y
773:P
753:x
733:C
713:C
693:P
670:P
646:P
621:P
596:P
572:P
551:C
531:C
511:C
491:C
467:P
444:0
440:C
419:C
399:P
375:C
354:C
330:C
305:P
280:P
259:P
237:0
233:C
212:P
188:P
161:P
141:C
112:0
108:C
87:P
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