1603:
1907: – in set theory, the combinatorial principle that, for every stationary 𝑆⊂ω₁, there exists a sequence of sets 𝐴_𝛿 (𝛿∈𝑆) such that 𝐴_𝛿 is a cofinal subset of 𝛿 and every unbounded subset of ω₁ is contained in some 𝐴_𝛿
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is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any
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1598:{\displaystyle \{S\subseteq \kappa :\exists C\subseteq S{\text{ such that }}C{\text{ is closed unbounded in }}\kappa \}}
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518:; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded. If
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are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).
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1965:
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1922: – Use of filters to describe and characterize all basic topological notions and results.
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which is possible because each is unbounded. Since this is a collection of fewer than
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1396:{\displaystyle \beta _{0}^{\xi },\beta _{1}^{\xi },\beta _{2}^{\xi },\ldots \,,}
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1976:
1964:, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover.
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The limit of this sequence must in fact also be the limit of the sequence
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and therefore this limit is an element of the intersection that is above
58:, and is unbounded (see below) relative to the limit ordinal. The name
1795:
closed under diagonal intersection, containing all sets of the form
1916: – In mathematics, a special subset of a partially ordered set
1651:
1992:
1988:
1944:
Set Theory: The Third
Millennium Edition, Revised and Expanded
1879:
1755:
1718:
is a regular cardinal then club sets are also closed under
1318:{\displaystyle \beta _{0},\beta _{1},\beta _{2},\ldots \,.}
957:{\displaystyle \langle C_{\xi }:\xi <\alpha \rangle \,}
591:
In fact a club set is nothing else but the range of a
1833:{\displaystyle \{\xi <\kappa :\xi \geq \alpha \}\,}
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which shows that the intersection is unbounded. QED.
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Pages displaying wikidata descriptions as a fallback
1149:{\displaystyle \beta _{n+1}^{\xi }>\beta _{n}\,,}
1025:{\displaystyle \bigcap _{\xi <\alpha }C_{\xi }\,}
499:
If a set is both closed and unbounded, then it is a
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215:{\displaystyle \sup(C\cap \alpha )=\alpha \neq 0,}
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514:limit ordinals is a club set with respect to the
1981:Creative Commons Attribution/Share-Alike License
1975:This article incorporates material from Club on
179:
1201:their least upper bound must also be less than
2004:
964:be a sequence of closed unbounded subsets of
8:
1826:
1802:
1592:
1549:
1262:This process generates a countable sequence
950:
925:
62:is a contraction of "closed and unbounded".
1452:is uncountable, this limit must be in each
2662:Positive cone of a partially ordered group
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1687:{\displaystyle (\wp (\kappa ),\subseteq )}
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2645:Positive cone of an ordered vector space
1062:{\displaystyle \beta _{0}<\kappa \,,}
674:{\displaystyle C\subseteq ^{\lambda }}
1863:{\displaystyle \alpha <\kappa \,,}
911:{\displaystyle \alpha <\lambda \,}
542:, then the set of all limit ordinals
7:
435:{\displaystyle \alpha <\kappa ,}
165:{\displaystyle \alpha <\kappa ,}
2172:Properties & Types (
1663:
1586: is closed unbounded in
1564:
1517:From this, it can be seen that if
864:be a limit ordinal of uncountable
595:(i.e. increasing and continuous).
561:{\displaystyle \alpha <\kappa }
490:{\displaystyle \alpha <\beta .}
362:{\displaystyle C\subseteq \kappa }
109:{\displaystyle C\subseteq \kappa }
25:
2628:Positive cone of an ordered field
2482:Ordered topological vector space
1888:{\displaystyle {\mathcal {F}}\,}
1764:{\displaystyle {\mathcal {F}}\,}
805:is contained in some element of
1255:{\displaystyle \beta _{n+1}\,.}
90:is a limit ordinal, then a set
1979:, which is licensed under the
1681:
1672:
1666:
1660:
725:if every union of a subset of
662:
655:
194:
182:
1:
2439:Series-parallel partial order
1928: – Set-theoretic concept
1507:{\displaystyle \beta _{0}\,,}
1073:< ω choose from each
244:{\displaystyle \alpha \in C.}
2118:Cantor's isomorphism theorem
1895:must include all club sets.
510:For example, the set of all
2158:Szpilrajn extension theorem
2133:Hausdorff maximal principle
2108:Boolean prime ideal theorem
1476:{\displaystyle C_{\xi }\,,}
884:{\displaystyle \lambda \,.}
833:The closed unbounded filter
461:{\displaystyle \beta \in C}
2716:
2504:Topological vector lattice
1788:{\displaystyle \kappa \,,}
1445:{\displaystyle \lambda \,}
1424:{\displaystyle C_{\xi }\,}
1218:{\displaystyle \kappa \,,}
1194:{\displaystyle \kappa \,,}
1170:{\displaystyle \lambda \,}
1094:{\displaystyle C_{\xi }\,}
981:{\displaystyle \kappa \,.}
836:
298:then the limit is also in
2026:
1739:{\displaystyle \kappa \,}
1711:{\displaystyle \kappa \,}
1619:{\displaystyle \kappa \,}
1531:{\displaystyle \kappa \,}
857:{\displaystyle \kappa \,}
785:of cardinality less than
516:first uncountable ordinal
2113:Cantor–Bernstein theorem
1177:ordinals, all less than
798:{\displaystyle \lambda }
714:{\displaystyle \lambda }
631:{\displaystyle \lambda }
584:{\displaystyle \kappa .}
291:{\displaystyle \kappa ,}
2657:Partially ordered group
2477:Specialization preorder
1643:{\displaystyle \kappa }
681:(the set of subsets of
568:is closed unbounded in
531:{\displaystyle \kappa }
406:{\displaystyle \kappa }
343:is a limit ordinal and
336:{\displaystyle \kappa }
133:{\displaystyle \kappa }
83:{\displaystyle \kappa }
2143:Kruskal's tree theorem
2138:Knaster–Tarski theorem
2128:Dushnik–Miller theorem
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1864:
1834:
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958:
912:
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858:
819:
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632:
618:is a nonempty set and
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292:
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253:limit of some sequence
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216:
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110:
84:
1890:
1865:
1835:
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1720:diagonal intersection
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1600:
1578: such that
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364:
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293:
270:
246:
217:
167:
135:
111:
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2635:Ordered vector space
1914:Filter (mathematics)
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789:
769:
765:and every subset of
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18:Bounded (set theory)
2473:Alexandrov topology
2419:Lexicographic order
2378:Well-quasi-ordering
1920:Filters in topology
1605:is a non-principal
1382:
1364:
1346:
1128:
598:More generally, if
2454:Transitive closure
2414:Converse/Transpose
2123:Dilworth's theorem
1885:
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1830:
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1421:
1393:
1368:
1350:
1332:
1315:
1252:
1225:so we can call it
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538:is an uncountable
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379:
359:
333:
314:{\displaystyle C.}
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36:mathematical logic
34:, particularly in
27:Set theory concept
2682:
2681:
2640:Partially ordered
2449:Symmetric closure
2434:Reflexive closure
2177:
1771:is any filter on
1650:(that is, on the
1626:-complete proper
1587:
1579:
995:
818:{\displaystyle C}
778:{\displaystyle X}
758:{\displaystyle C}
738:{\displaystyle C}
694:{\displaystyle X}
611:{\displaystyle X}
382:{\displaystyle C}
268:{\displaystyle C}
66:Formal definition
46:is a subset of a
16:(Redirected from
2707:
2424:Linear extension
2173:
2153:Mirsky's theorem
2013:
2006:
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1990:
1962:Basic Set Theory
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1601:
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1588:
1585:
1580:
1577:
1540:regular cardinal
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801:
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784:
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609:
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588:
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582:
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565:
564:
559:
537:
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534:
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496:
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493:
488:
467:
465:
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459:
441:
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412:
410:
409:
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388:
386:
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368:
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342:
340:
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334:
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312:
297:
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289:
274:
272:
271:
266:
250:
248:
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242:
221:
219:
218:
213:
171:
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163:
139:
137:
136:
131:
115:
113:
112:
107:
89:
87:
86:
81:
21:
2715:
2714:
2710:
2709:
2708:
2706:
2705:
2704:
2695:Ordinal numbers
2685:
2684:
2683:
2678:
2674:Young's lattice
2530:
2458:
2397:
2247:Heyting algebra
2195:Boolean algebra
2167:
2148:Laver's theorem
2096:
2062:Boolean algebra
2057:Binary relation
2045:
2022:
2017:
1987:
1935:
1908:
1901:
1872:
1871:
1842:
1841:
1797:
1796:
1773:
1772:
1748:
1747:
1746:is regular and
1727:
1726:
1699:
1698:
1655:
1654:
1632:
1631:
1607:
1606:
1544:
1543:
1519:
1518:
1490:
1485:
1484:
1459:
1454:
1453:
1433:
1432:
1410:
1405:
1404:
1403:and since each
1327:
1326:
1295:
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1269:
1264:
1263:
1232:
1227:
1226:
1203:
1202:
1179:
1178:
1158:
1157:
1132:
1103:
1102:
1080:
1075:
1074:
1039:
1034:
1033:
1011:
990:
989:
966:
965:
928:
920:
919:
893:
892:
869:
868:
845:
844:
841:
835:
807:
806:
787:
786:
767:
766:
747:
746:
727:
726:
703:
702:
701:of cardinality
683:
682:
661:
644:
643:
620:
619:
600:
599:
593:normal function
570:
569:
544:
543:
540:initial ordinal
520:
519:
470:
469:
444:
443:
415:
414:
395:
394:
371:
370:
345:
344:
325:
324:
300:
299:
277:
276:
257:
256:
224:
223:
174:
173:
145:
144:
122:
121:
92:
91:
72:
71:
68:
28:
23:
22:
15:
12:
11:
5:
2713:
2711:
2703:
2702:
2697:
2687:
2686:
2680:
2679:
2677:
2676:
2671:
2666:
2665:
2664:
2654:
2653:
2652:
2647:
2642:
2632:
2631:
2630:
2620:
2615:
2614:
2613:
2608:
2601:Order morphism
2598:
2597:
2596:
2586:
2581:
2576:
2571:
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2565:
2564:
2554:
2549:
2544:
2538:
2536:
2532:
2531:
2529:
2528:
2527:
2526:
2521:
2519:Locally convex
2516:
2511:
2501:
2499:Order topology
2496:
2495:
2494:
2492:Order topology
2489:
2479:
2469:
2467:
2460:
2459:
2457:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2405:
2403:
2399:
2398:
2396:
2395:
2385:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2339:
2338:
2328:
2323:
2322:
2321:
2316:
2311:
2306:
2304:Chain-complete
2296:
2291:
2290:
2289:
2284:
2279:
2274:
2269:
2259:
2254:
2249:
2244:
2239:
2229:
2224:
2219:
2214:
2209:
2204:
2203:
2202:
2192:
2187:
2181:
2179:
2169:
2168:
2166:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2104:
2102:
2098:
2097:
2095:
2094:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2053:
2051:
2047:
2046:
2044:
2043:
2038:
2033:
2027:
2024:
2023:
2018:
2016:
2015:
2008:
2001:
1993:
1986:
1985:
1972:
1955:
1946:. Springer.
1936:
1934:
1931:
1930:
1929:
1926:Stationary set
1923:
1917:
1911:
1900:
1897:
1881:
1859:
1855:
1852:
1849:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1784:
1780:
1757:
1734:
1706:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1639:
1614:
1594:
1591:
1583:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1526:
1503:
1497:
1493:
1472:
1466:
1462:
1440:
1431:is closed and
1417:
1413:
1392:
1388:
1385:
1380:
1375:
1371:
1367:
1362:
1357:
1353:
1349:
1344:
1339:
1335:
1314:
1310:
1307:
1302:
1298:
1294:
1289:
1285:
1281:
1276:
1272:
1251:
1245:
1242:
1239:
1235:
1214:
1210:
1190:
1186:
1165:
1145:
1139:
1135:
1131:
1126:
1121:
1118:
1115:
1111:
1087:
1083:
1058:
1054:
1051:
1046:
1042:
1018:
1014:
1008:
1005:
1002:
998:
977:
973:
952:
949:
946:
943:
940:
935:
931:
927:
906:
903:
900:
880:
876:
852:
837:Main article:
834:
831:
827:stationary set
814:
794:
774:
754:
734:
710:
690:
668:
664:
660:
657:
654:
651:
627:
607:
580:
577:
557:
554:
551:
527:
505:proper classes
486:
483:
480:
477:
457:
454:
451:
442:there is some
431:
428:
425:
422:
402:
378:
358:
355:
352:
332:
310:
307:
287:
284:
264:
240:
237:
234:
231:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
161:
158:
155:
152:
141:if and only if
129:
105:
102:
99:
79:
67:
64:
56:order topology
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2712:
2701:
2698:
2696:
2693:
2692:
2690:
2675:
2672:
2670:
2667:
2663:
2660:
2659:
2658:
2655:
2651:
2648:
2646:
2643:
2641:
2638:
2637:
2636:
2633:
2629:
2626:
2625:
2624:
2623:Ordered field
2621:
2619:
2616:
2612:
2609:
2607:
2604:
2603:
2602:
2599:
2595:
2592:
2591:
2590:
2587:
2585:
2582:
2580:
2579:Hasse diagram
2577:
2575:
2572:
2570:
2567:
2563:
2560:
2559:
2558:
2557:Comparability
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2539:
2537:
2533:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2506:
2505:
2502:
2500:
2497:
2493:
2490:
2488:
2485:
2484:
2483:
2480:
2478:
2474:
2471:
2470:
2468:
2465:
2461:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2429:Product order
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2404:
2402:Constructions
2400:
2394:
2390:
2386:
2383:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2337:
2334:
2333:
2332:
2329:
2327:
2324:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2301:
2300:
2299:Partial order
2297:
2295:
2292:
2288:
2287:Join and meet
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2264:
2263:
2260:
2258:
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2234:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2201:
2198:
2197:
2196:
2193:
2191:
2188:
2186:
2185:Antisymmetric
2183:
2182:
2180:
2176:
2170:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2106:
2105:
2103:
2099:
2093:
2092:Weak ordering
2090:
2088:
2085:
2083:
2080:
2078:
2077:Partial order
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2054:
2052:
2048:
2042:
2039:
2037:
2034:
2032:
2029:
2028:
2025:
2021:
2014:
2009:
2007:
2002:
2000:
1995:
1994:
1991:
1984:
1982:
1978:
1973:
1971:
1970:0-486-42079-5
1967:
1963:
1959:
1956:
1953:
1952:3-540-44085-2
1949:
1945:
1941:
1938:
1937:
1932:
1927:
1924:
1921:
1918:
1915:
1912:
1906:
1903:
1902:
1898:
1896:
1857:
1853:
1850:
1847:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1782:
1778:
1732:
1723:
1721:
1704:
1695:
1678:
1675:
1669:
1653:
1637:
1629:
1612:
1589:
1581:
1573:
1570:
1567:
1561:
1558:
1555:
1552:
1541:
1524:
1515:
1501:
1495:
1491:
1470:
1464:
1460:
1438:
1415:
1411:
1390:
1386:
1383:
1378:
1373:
1369:
1365:
1360:
1355:
1351:
1347:
1342:
1337:
1333:
1312:
1308:
1305:
1300:
1296:
1292:
1287:
1283:
1279:
1274:
1270:
1249:
1243:
1240:
1237:
1233:
1212:
1208:
1188:
1184:
1163:
1143:
1137:
1133:
1129:
1124:
1119:
1116:
1113:
1109:
1085:
1081:
1072:
1069:and for each
1056:
1052:
1049:
1044:
1040:
1016:
1012:
1006:
1003:
1000:
996:
975:
971:
947:
944:
941:
938:
933:
929:
904:
901:
898:
878:
874:
867:
850:
840:
832:
830:
828:
812:
792:
772:
752:
732:
724:
708:
688:
666:
658:
652:
649:
641:
625:
605:
596:
594:
578:
575:
555:
552:
549:
541:
525:
517:
513:
508:
506:
502:
497:
484:
481:
478:
475:
455:
452:
449:
429:
426:
423:
420:
400:
392:
376:
356:
353:
350:
330:
321:
308:
305:
285:
282:
275:is less than
262:
254:
251:Thus, if the
238:
235:
232:
229:
209:
206:
203:
200:
197:
191:
188:
185:
159:
156:
153:
150:
142:
127:
119:
103:
100:
97:
77:
70:Formally, if
65:
63:
61:
57:
53:
49:
48:limit ordinal
45:
41:
37:
33:
19:
2466:& Orders
2444:Star product
2373:Well-founded
2326:Prefix order
2282:Distributive
2272:Complemented
2242:Foundational
2207:Completeness
2163:Zorn's lemma
2067:Cyclic order
2050:Key concepts
2020:Order theory
1974:
1961:
1958:Lévy, Azriel
1943:
1940:Jech, Thomas
1725:In fact, if
1724:
1696:
1516:
1070:
842:
722:
597:
509:
500:
498:
390:
322:
117:
69:
59:
43:
29:
2650:Riesz space
2611:Isomorphism
2487:Normal cone
2409:Composition
2343:Semilattice
2252:Homogeneous
2237:Equivalence
2087:Total order
1630:on the set
1101:an element
839:Club filter
413:if for any
32:mathematics
2700:Set theory
2689:Categories
2618:Order type
2552:Cofinality
2393:Well-order
2368:Transitive
2257:Idempotent
2190:Asymmetric
1977:PlanetMath
1933:References
866:cofinality
468:such that
143:for every
54:under the
40:set theory
2669:Upper set
2606:Embedding
2542:Antichain
2363:Tolerance
2353:Symmetric
2348:Semiorder
2294:Reflexive
2212:Connected
1854:κ
1848:α
1824:α
1821:≥
1818:ξ
1812:κ
1806:ξ
1779:κ
1733:κ
1705:κ
1679:⊆
1670:κ
1664:℘
1638:κ
1613:κ
1590:κ
1571:⊆
1565:∃
1559:κ
1556:⊆
1525:κ
1492:β
1465:ξ
1439:λ
1416:ξ
1387:…
1379:ξ
1370:β
1361:ξ
1352:β
1343:ξ
1334:β
1309:…
1297:β
1284:β
1271:β
1234:β
1209:κ
1185:κ
1164:λ
1134:β
1125:ξ
1110:β
1086:ξ
1053:κ
1041:β
1017:ξ
1007:α
1001:ξ
997:⋂
972:κ
951:⟩
948:α
942:ξ
934:ξ
926:⟨
905:λ
899:α
891:For some
875:λ
851:κ
793:λ
709:λ
667:λ
653:⊆
626:λ
576:κ
556:κ
550:α
526:κ
512:countable
503:. Closed
482:β
476:α
453:∈
450:β
427:κ
421:α
401:κ
391:unbounded
357:κ
354:⊆
331:κ
283:κ
233:∈
230:α
204:≠
201:α
192:α
189:∩
157:κ
151:α
128:κ
104:κ
101:⊆
78:κ
2464:Topology
2331:Preorder
2314:Eulerian
2277:Complete
2227:Directed
2217:Covering
2082:Preorder
2041:Category
2036:Glossary
1942:, 2003.
1905:Clubsuit
1899:See also
640:cardinal
501:club set
50:that is
44:club set
2569:Duality
2547:Cofinal
2535:Related
2514:Fréchet
2391:)
2267:Bounded
2262:Lattice
2235:)
2233:Partial
2101:Results
2072:Lattice
1960:(1979)
1542:, then
642:, then
2594:Subnet
2574:Filter
2524:Normed
2509:Banach
2475:&
2382:Better
2319:Strict
2309:Graded
2200:topics
2031:Topics
1968:
1950:
1628:filter
918:, let
745:is in
118:closed
52:closed
2584:Ideal
2562:Graph
2358:Total
2336:Total
2222:Dense
1870:then
1652:poset
1538:is a
988:Then
825:(see
721:) is
638:is a
369:then
255:from
222:then
2175:list
1966:ISBN
1948:ISBN
1851:<
1840:for
1809:<
1130:>
1050:<
1004:<
945:<
902:<
843:Let
723:club
553:<
479:<
424:<
154:<
60:club
42:, a
38:and
2589:Net
2389:Pre
1697:If
1694:).
829:).
393:in
389:is
323:If
180:sup
172:if
120:in
116:is
30:In
2691::
1722:.
2387:(
2384:)
2380:(
2231:(
2178:)
2012:e
2005:t
1998:v
1983:.
1954:.
1880:F
1858:,
1827:}
1815::
1803:{
1783:,
1756:F
1682:)
1676:,
1673:)
1667:(
1661:(
1593:}
1582:C
1574:S
1568:C
1562::
1553:S
1550:{
1502:,
1496:0
1471:,
1461:C
1412:C
1391:,
1384:,
1374:2
1366:,
1356:1
1348:,
1338:0
1313:.
1306:,
1301:2
1293:,
1288:1
1280:,
1275:0
1250:.
1244:1
1241:+
1238:n
1213:,
1189:,
1144:,
1138:n
1120:1
1117:+
1114:n
1082:C
1071:n
1057:,
1045:0
1013:C
976:.
939::
930:C
879:.
813:C
773:X
753:C
733:C
689:X
663:]
659:X
656:[
650:C
606:X
579:.
485:.
456:C
430:,
377:C
351:C
309:.
306:C
286:,
263:C
239:.
236:C
210:,
207:0
198:=
195:)
186:C
183:(
160:,
98:C
20:)
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