1018:
3076:
2680:
467:
818:
345:
1433:
776:
2931:
555:
1108:
image of a hyperconnected space is hyperconnected. In particular, any continuous function from a hyperconnected space to a
Hausdorff space must be constant. It follows that every hyperconnected space is
2926:
1913:
1748:
1994:
1862:
2842:
2519:
2790:
2366:
1047:
Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint.
2198:
2153:
241:, which is within every prime, to show the spectrum of the quotient map is a homeomorphism, this reduces to the irreducibility of the spectrum of an integral domain. For example, the
678:
638:
598:
3154:
in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.
2056:
1513:
161:
a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors, especially those interested in applications to
2320:
472:
are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization). A non-example is given by the
2274:
1286:
2578:
2108:
2082:
1791:
216:
2713:
1639:
1147:
1659:
2744:
2573:
2546:
2467:
2440:
1540:
1340:
1313:
806:
3136:
3116:
3096:
2413:
2393:
2228:
2014:
1933:
1811:
1600:
1580:
1560:
1473:
1453:
1360:
1240:
1220:
1187:
1167:
1013:{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} }{(f_{4}(0,y,z,w))}}\right),{\text{ }}{\text{Proj}}\left({\frac {\mathbb {C} }{(f_{4}(x,0,z,w))}}\right)}
350:
250:
3553:
3071:{\displaystyle \operatorname {Cl} _{X}(V)\supseteq {\operatorname {Cl} }_{U_{1}}(V_{1})\cup {\operatorname {Cl} }_{U_{2}}(V_{2})=U_{1}\cup U_{2}=X,}
1365:
686:
1097:
Since the closure of every non-empty open set in a hyperconnected space is the whole space, which is an open set, every hyperconnected space is
3513:
3477:
3406:
3367:
3342:
3317:
481:
2847:
3174:
1867:
1672:
1105:
1098:
3195:
1816:
2795:
2472:
3496:
32:
3191:
Since every irreducible space is connected, the irreducible components will always lie in the connected components.
2749:
2325:
1662:
2158:
2113:
3217:
809:
1938:
57:
that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name
643:
603:
563:
473:
1478:
3491:
2279:
1091:
1041:
3150:
2675:{\displaystyle \exists x\in \operatorname {Cl} _{U_{1}}(V_{1})\cap U_{2}=U_{1}\cap U_{2}\neq \emptyset }
2233:
1245:
1067:
The nonempty open subsets of a hyperconnected space are "large" in the sense that each one is dense in
808:
is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves (by the
3207:
242:
238:
142:. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the
1110:
109:
2087:
2061:
1770:
1054:
hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it
199:
3501:
3178:
2201:
2019:
1757:
222:
185:
162:
62:
2685:
1617:
1126:
3509:
3487:
3473:
3402:
3363:
3338:
3313:
1083:
1033:
189:
173:
120:
51:
28:
3274:
2716:
1666:
1644:
98:
3523:
2722:
2551:
2524:
2445:
2418:
1518:
1318:
1291:
784:
3519:
3505:
1079:
1072:
1029:
234:
230:
143:
3177:
of a space, the irreducible components need not be disjoint (i.e. they need not form a
3121:
3101:
3081:
2398:
2378:
2213:
1999:
1918:
1796:
1585:
1565:
1545:
1458:
1438:
1345:
1225:
1205:
1172:
1152:
1087:
1037:
462:{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} }{(y^{2}z-x(x-z)(x-2z))}}\right)}
193:
3547:
3279:
3185:
158:
91:
81:
1050:
For example, the space of real numbers with the standard topology is connected but
340:{\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} }{x^{4}+y^{3}+z^{2}}}\right)}
226:
3263:"An anti-Hausdorff Fréchet space in which convergent sequences have unique limits"
3212:
3262:
1195:
More generally, every dense subset of a hyperconnected space is hyperconnected.
17:
3535:
3530:
1428:{\displaystyle X={\overline {S}}={\overline {S_{1}}}\cup {\overline {S_{2}}}}
3450:
3436:
3422:
3383:
3294:
771:{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} }{(xy,f_{4})}}\right)}
154:
165:, add an explicit condition that an irreducible space must be nonempty.
77:
39:
1071:
and any pair of them intersects. Thus, a hyperconnected space cannot be
1608:
A closed subspace of a hyperconnected space need not be hyperconnected.
550:{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} }{(xyz)}}\right)}
3451:"Section 5.9 (0050): Noetherian topological spaces—The Stacks project"
3161:
is contained in a (not necessarily unique) irreducible component of
3244:
3242:
1116:
Every open subspace of a hyperconnected space is hyperconnected.
33:
Connectivity (graph theory) § Super- and hyper-connectivity
3468:
Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004).
3295:"Section 5.8 (004U): Irreducible components—The Stacks project"
1455:
is hyperconnected, one of the two closures is the whole space
134:
A space which satisfies any one of these conditions is called
3184:
The irreducible components of a
Hausdorff space are just the
560:
since the underlying space is the union of the affine planes
130:
No two points can be separated by disjoint neighbourhoods.
2921:{\displaystyle V=V\cap (U_{1}\cup U_{2})=V_{1}\cup V_{2}}
1058:
be written as a union of two (non-disjoint) closed sets.
3181:). In general, the irreducible components will overlap.
1908:{\displaystyle F,G\subseteq \operatorname {Cl} _{X}(S)}
1743:{\displaystyle V=Z(XY)=Z(X)\cup Z(Y)\subset \Bbbk ^{2}}
3118:. Since this is true for every non-empty open subset,
3124:
3104:
3084:
2934:
2850:
2798:
2752:
2725:
2688:
2581:
2554:
2527:
2475:
2448:
2421:
2401:
2381:
2328:
2282:
2236:
2216:
2161:
2116:
2090:
2064:
2022:
2002:
1941:
1921:
1870:
1819:
1799:
1773:
1675:
1647:
1620:
1588:
1568:
1548:
1521:
1481:
1461:
1441:
1368:
1348:
1321:
1294:
1248:
1228:
1208:
1175:
1155:
1129:
821:
787:
689:
646:
606:
566:
484:
353:
253:
202:
1149:
be an open subset. Any two disjoint open subsets of
3130:
3110:
3090:
3070:
2920:
2836:
2784:
2738:
2707:
2674:
2567:
2540:
2513:
2461:
2434:
2407:
2387:
2360:
2314:
2268:
2222:
2192:
2147:
2102:
2076:
2050:
2008:
1988:
1927:
1907:
1857:{\displaystyle \operatorname {Cl} _{X}(S)=F\cup G}
1856:
1805:
1785:
1742:
1653:
1633:
1594:
1574:
1554:
1534:
1507:
1467:
1447:
1427:
1354:
1334:
1307:
1280:
1234:
1214:
1181:
1161:
1141:
1012:
800:
770:
672:
632:
592:
549:
461:
339:
210:
31:. For hyper-connectivity in node-link graphs, see
3248:
2837:{\displaystyle V_{2}:=V\cap U_{2}\neq \emptyset }
2514:{\displaystyle V_{1}:=U_{1}\cap V\neq \emptyset }
233:is an irreducible topological space—applying the
172:is a subset of a topological space for which the
3169:is contained in some irreducible component of
2785:{\displaystyle V_{1}\cap U_{2}\neq \emptyset }
2361:{\displaystyle U_{1}\cap U_{2}\neq \emptyset }
3504:reprint of 1978 ed.), Berlin, New York:
1169:would themselves be disjoint open subsets of
90:cannot be written as the union of two proper
8:
3423:"Definition 5.8.1 (004V)—The Stacks project"
2193:{\displaystyle \operatorname {Cl} _{X}(S)=G}
2148:{\displaystyle \operatorname {Cl} _{X}(S)=F}
680:. Another non-example is given by the scheme
184:Two examples of hyperconnected spaces from
3198:has finitely many irreducible components.
3278:
3123:
3103:
3083:
3053:
3040:
3024:
3009:
3004:
2999:
2986:
2971:
2966:
2961:
2939:
2933:
2912:
2899:
2883:
2870:
2849:
2822:
2803:
2797:
2770:
2757:
2751:
2730:
2724:
2699:
2687:
2660:
2647:
2634:
2618:
2600:
2595:
2580:
2559:
2553:
2532:
2526:
2493:
2480:
2474:
2453:
2447:
2426:
2420:
2400:
2380:
2346:
2333:
2327:
2300:
2287:
2281:
2260:
2247:
2235:
2215:
2166:
2160:
2121:
2115:
2089:
2063:
2021:
2001:
1989:{\displaystyle F':=F\cap S,\,G':=G\cap S}
1965:
1940:
1920:
1887:
1869:
1824:
1818:
1798:
1772:
1734:
1731:
1674:
1665:(thus infinite) is hyperconnected in the
1648:
1646:
1625:
1622:
1619:
1587:
1567:
1547:
1526:
1520:
1488:
1482:
1480:
1460:
1440:
1414:
1408:
1394:
1388:
1375:
1367:
1347:
1326:
1320:
1299:
1293:
1272:
1259:
1247:
1227:
1207:
1189:. So at least one of them must be empty.
1174:
1154:
1128:
967:
933:
932:
929:
920:
915:
869:
835:
834:
831:
822:
820:
792:
786:
752:
703:
702:
699:
690:
688:
664:
653:
649:
648:
645:
624:
613:
609:
608:
605:
584:
573:
569:
568:
565:
498:
497:
494:
485:
483:
401:
367:
366:
363:
354:
352:
324:
311:
298:
267:
266:
263:
254:
252:
204:
203:
201:
72:the following conditions are equivalent:
3098:is a non-empty open and dense subset of
146:property, some authors call such spaces
3437:"Lemma 5.8.3 (004W)—The Stacks project"
3384:"Lemma 5.8.3 (004W)—The Stacks project"
3229:
1075:unless it contains only a single point.
1760:of any irreducible set is irreducible.
673:{\displaystyle \mathbb {A} _{y,z}^{2}}
633:{\displaystyle \mathbb {A} _{x,z}^{2}}
593:{\displaystyle \mathbb {A} _{x,y}^{2}}
27:For the computer networking term, see
1508:{\displaystyle {\overline {S_{1}}}=X}
7:
3157:Every irreducible subset of a space
2315:{\displaystyle U_{1},U_{2}\subset X}
1024:Hyperconnectedness vs. connectedness
3360:Algebraic Geometry. An introduction
1078:Every hyperconnected space is both
1028:Every hyperconnected space is both
2831:
2779:
2669:
2582:
2508:
2355:
221:In algebraic geometry, taking the
25:
3399:Commutative Algebra: Chapters 1-7
3335:Commutative Algebra: Chapters 1-7
3310:Commutative Algebra: Chapters 1-7
3165:. In particular, every point of
2269:{\displaystyle X=U_{1}\cup U_{2}}
1750:is closed and not hyperconnected.
1281:{\displaystyle S=S_{1}\cup S_{2}}
112:of every proper closed subset of
3554:Properties of topological spaces
3470:Encyclopedia of general topology
3249:Hart, Nagata & Vaughan 2004
2322:open and irreducible such that
3030:
3017:
2992:
2979:
2954:
2948:
2889:
2863:
2624:
2611:
2181:
2175:
2136:
2130:
1902:
1896:
1839:
1833:
1724:
1718:
1709:
1703:
1694:
1685:
1000:
997:
973:
960:
955:
937:
902:
899:
875:
862:
857:
839:
758:
736:
731:
707:
537:
525:
520:
502:
449:
446:
431:
428:
416:
394:
389:
371:
289:
271:
1:
3267:Topology and Its Applications
38:In the mathematical field of
3280:10.1016/0166-8641(93)90147-6
3261:Van Douwen, Eric K. (1993).
3196:Noetherian topological space
2103:{\displaystyle S\subseteq G}
2077:{\displaystyle S\subseteq F}
1786:{\displaystyle S\subseteq X}
1562:, and since it is closed in
1494:
1420:
1400:
1380:
211:{\displaystyle \mathbb {R} }
192:on any infinite set and the
3497:Counterexamples in Topology
2395:is a non-empty open set in
2375:Firstly, we notice that if
2051:{\displaystyle S=F'\cup G'}
97:Every nonempty open set is
3570:
3472:. Elsevier/North-Holland.
3397:Bourbaki, Nicolas (1989).
3333:Bourbaki, Nicolas (1989).
3308:Bourbaki, Nicolas (1989).
3236:Steen & Seebach, p. 29
2708:{\displaystyle x\in U_{2}}
1663:algebraically closed field
1634:{\displaystyle \Bbbk ^{2}}
1142:{\displaystyle U\subset X}
26:
3218:Geometrically irreducible
1813:is irreducible and write
119:Every subset is dense or
3401:. Springer. p. 95.
3362:. Springer. p. 14.
3337:. Springer. p. 95.
3312:. Springer. p. 95.
2415:then it intersects both
2230:which can be written as
1086:(though not necessarily
1036:(though not necessarily
68:For a topological space
3358:Perrin, Daniel (2008).
2928:and taking the closure
1864:for two closed subsets
1099:extremally disconnected
474:normal crossing divisor
3531:"Hyperconnected space"
3492:Seebach, J. Arthur Jr.
3144:Irreducible components
3132:
3112:
3092:
3072:
2922:
2838:
2786:
2740:
2709:
2676:
2569:
2542:
2515:
2463:
2436:
2409:
2389:
2362:
2316:
2270:
2224:
2194:
2149:
2104:
2078:
2052:
2010:
1990:
1929:
1909:
1858:
1807:
1787:
1744:
1655:
1654:{\displaystyle \Bbbk }
1635:
1596:
1582:, it must be equal to
1576:
1556:
1536:
1509:
1469:
1449:
1429:
1356:
1336:
1309:
1282:
1236:
1216:
1183:
1163:
1143:
1092:locally path-connected
1042:locally path-connected
1021:
1014:
802:
779:
772:
674:
634:
594:
558:
551:
470:
463:
341:
212:
3151:irreducible component
3133:
3113:
3093:
3073:
2923:
2839:
2787:
2741:
2739:{\displaystyle V_{1}}
2710:
2677:
2570:
2568:{\displaystyle U_{1}}
2543:
2541:{\displaystyle V_{1}}
2516:
2464:
2462:{\displaystyle U_{2}}
2437:
2435:{\displaystyle U_{1}}
2410:
2390:
2363:
2317:
2271:
2225:
2195:
2150:
2105:
2079:
2053:
2011:
1991:
1930:
1910:
1859:
1808:
1788:
1745:
1656:
1636:
1597:
1577:
1557:
1537:
1535:{\displaystyle S_{1}}
1515:. This implies that
1510:
1470:
1450:
1430:
1357:
1337:
1335:{\displaystyle S_{2}}
1310:
1308:{\displaystyle S_{1}}
1283:
1237:
1222:is a dense subset of
1217:
1184:
1164:
1144:
1015:
814:
803:
801:{\displaystyle f_{4}}
773:
682:
675:
635:
595:
552:
477:
464:
342:
246:
213:
3208:Ultraconnected space
3175:connected components
3122:
3102:
3082:
2932:
2848:
2796:
2750:
2723:
2686:
2579:
2552:
2525:
2473:
2446:
2419:
2399:
2379:
2326:
2280:
2234:
2214:
2159:
2114:
2088:
2062:
2020:
2000:
1939:
1919:
1868:
1817:
1797:
1771:
1673:
1645:
1618:
1586:
1566:
1546:
1519:
1479:
1459:
1439:
1366:
1346:
1319:
1292:
1246:
1226:
1206:
1173:
1153:
1127:
819:
810:genus–degree formula
785:
687:
644:
604:
564:
482:
351:
251:
200:
194:right order topology
44:hyperconnected space
669:
629:
589:
3488:Steen, Lynn Arthur
3128:
3108:
3088:
3068:
2918:
2834:
2782:
2736:
2705:
2672:
2565:
2538:
2511:
2469:; indeed, suppose
2459:
2432:
2405:
2385:
2358:
2312:
2266:
2220:
2190:
2145:
2100:
2074:
2048:
2006:
1986:
1925:
1905:
1854:
1803:
1783:
1740:
1651:
1631:
1592:
1572:
1552:
1532:
1505:
1465:
1445:
1425:
1352:
1332:
1305:
1278:
1232:
1212:
1179:
1159:
1139:
1010:
798:
768:
670:
647:
630:
607:
590:
567:
547:
459:
337:
223:spectrum of a ring
208:
186:point set topology
163:algebraic geometry
63:algebraic geometry
3515:978-0-486-68735-3
3479:978-0-444-50355-8
3408:978-3-540-64239-8
3369:978-1-84800-055-1
3344:978-3-540-64239-8
3319:978-3-540-64239-8
3131:{\displaystyle X}
3111:{\displaystyle X}
3091:{\displaystyle V}
2408:{\displaystyle X}
2388:{\displaystyle V}
2223:{\displaystyle X}
2200:by definition of
2009:{\displaystyle S}
1928:{\displaystyle X}
1806:{\displaystyle S}
1595:{\displaystyle S}
1575:{\displaystyle S}
1555:{\displaystyle S}
1497:
1468:{\displaystyle X}
1448:{\displaystyle X}
1423:
1403:
1383:
1355:{\displaystyle S}
1235:{\displaystyle X}
1215:{\displaystyle S}
1182:{\displaystyle X}
1162:{\displaystyle U}
1084:locally connected
1034:locally connected
1004:
923:
918:
906:
825:
762:
693:
541:
488:
453:
357:
331:
257:
190:cofinite topology
174:subspace topology
59:irreducible space
52:topological space
48:irreducible space
29:Hyperconnectivity
16:(Redirected from
3561:
3540:
3526:
3483:
3455:
3454:
3447:
3441:
3440:
3433:
3427:
3426:
3419:
3413:
3412:
3394:
3388:
3387:
3380:
3374:
3373:
3355:
3349:
3348:
3330:
3324:
3323:
3305:
3299:
3298:
3291:
3285:
3284:
3282:
3258:
3252:
3246:
3237:
3234:
3137:
3135:
3134:
3129:
3117:
3115:
3114:
3109:
3097:
3095:
3094:
3089:
3077:
3075:
3074:
3069:
3058:
3057:
3045:
3044:
3029:
3028:
3016:
3015:
3014:
3013:
3003:
2991:
2990:
2978:
2977:
2976:
2975:
2965:
2944:
2943:
2927:
2925:
2924:
2919:
2917:
2916:
2904:
2903:
2888:
2887:
2875:
2874:
2843:
2841:
2840:
2835:
2827:
2826:
2808:
2807:
2791:
2789:
2788:
2783:
2775:
2774:
2762:
2761:
2745:
2743:
2742:
2737:
2735:
2734:
2717:point of closure
2714:
2712:
2711:
2706:
2704:
2703:
2681:
2679:
2678:
2673:
2665:
2664:
2652:
2651:
2639:
2638:
2623:
2622:
2607:
2606:
2605:
2604:
2574:
2572:
2571:
2566:
2564:
2563:
2547:
2545:
2544:
2539:
2537:
2536:
2520:
2518:
2517:
2512:
2498:
2497:
2485:
2484:
2468:
2466:
2465:
2460:
2458:
2457:
2441:
2439:
2438:
2433:
2431:
2430:
2414:
2412:
2411:
2406:
2394:
2392:
2391:
2386:
2367:
2365:
2364:
2359:
2351:
2350:
2338:
2337:
2321:
2319:
2318:
2313:
2305:
2304:
2292:
2291:
2275:
2273:
2272:
2267:
2265:
2264:
2252:
2251:
2229:
2227:
2226:
2221:
2199:
2197:
2196:
2191:
2171:
2170:
2154:
2152:
2151:
2146:
2126:
2125:
2109:
2107:
2106:
2101:
2083:
2081:
2080:
2075:
2057:
2055:
2054:
2049:
2047:
2036:
2015:
2013:
2012:
2007:
1995:
1993:
1992:
1987:
1973:
1949:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1892:
1891:
1863:
1861:
1860:
1855:
1829:
1828:
1812:
1810:
1809:
1804:
1792:
1790:
1789:
1784:
1749:
1747:
1746:
1741:
1739:
1738:
1667:Zariski topology
1660:
1658:
1657:
1652:
1640:
1638:
1637:
1632:
1630:
1629:
1613:Counterexample:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1561:
1559:
1558:
1553:
1541:
1539:
1538:
1533:
1531:
1530:
1514:
1512:
1511:
1506:
1498:
1493:
1492:
1483:
1474:
1472:
1471:
1466:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1424:
1419:
1418:
1409:
1404:
1399:
1398:
1389:
1384:
1376:
1361:
1359:
1358:
1353:
1341:
1339:
1338:
1333:
1331:
1330:
1314:
1312:
1311:
1306:
1304:
1303:
1287:
1285:
1284:
1279:
1277:
1276:
1264:
1263:
1241:
1239:
1238:
1233:
1221:
1219:
1218:
1213:
1188:
1186:
1185:
1180:
1168:
1166:
1165:
1160:
1148:
1146:
1145:
1140:
1019:
1017:
1016:
1011:
1009:
1005:
1003:
972:
971:
958:
936:
930:
924:
921:
919:
916:
911:
907:
905:
874:
873:
860:
838:
832:
826:
823:
807:
805:
804:
799:
797:
796:
777:
775:
774:
769:
767:
763:
761:
757:
756:
734:
706:
700:
694:
691:
679:
677:
676:
671:
668:
663:
652:
639:
637:
636:
631:
628:
623:
612:
599:
597:
596:
591:
588:
583:
572:
556:
554:
553:
548:
546:
542:
540:
523:
501:
495:
489:
486:
468:
466:
465:
460:
458:
454:
452:
406:
405:
392:
370:
364:
358:
355:
346:
344:
343:
338:
336:
332:
330:
329:
328:
316:
315:
303:
302:
292:
270:
264:
258:
255:
217:
215:
214:
209:
207:
176:is irreducible.
76:No two nonempty
61:is preferred in
21:
3569:
3568:
3564:
3563:
3562:
3560:
3559:
3558:
3544:
3543:
3529:
3516:
3506:Springer-Verlag
3486:
3480:
3467:
3464:
3459:
3458:
3449:
3448:
3444:
3435:
3434:
3430:
3421:
3420:
3416:
3409:
3396:
3395:
3391:
3382:
3381:
3377:
3370:
3357:
3356:
3352:
3345:
3332:
3331:
3327:
3320:
3307:
3306:
3302:
3293:
3292:
3288:
3260:
3259:
3255:
3247:
3240:
3235:
3231:
3226:
3204:
3146:
3138:is irreducible.
3120:
3119:
3100:
3099:
3080:
3079:
3049:
3036:
3020:
3005:
2998:
2982:
2967:
2960:
2935:
2930:
2929:
2908:
2895:
2879:
2866:
2846:
2845:
2818:
2799:
2794:
2793:
2792:and a fortiori
2766:
2753:
2748:
2747:
2726:
2721:
2720:
2695:
2684:
2683:
2656:
2643:
2630:
2614:
2596:
2591:
2577:
2576:
2555:
2550:
2549:
2528:
2523:
2522:
2489:
2476:
2471:
2470:
2449:
2444:
2443:
2422:
2417:
2416:
2397:
2396:
2377:
2376:
2368:is irreducible.
2342:
2329:
2324:
2323:
2296:
2283:
2278:
2277:
2256:
2243:
2232:
2231:
2212:
2211:
2162:
2157:
2156:
2117:
2112:
2111:
2086:
2085:
2060:
2059:
2040:
2029:
2018:
2017:
1998:
1997:
1966:
1942:
1937:
1936:
1917:
1916:
1883:
1866:
1865:
1820:
1815:
1814:
1795:
1794:
1769:
1768:
1730:
1671:
1670:
1643:
1642:
1621:
1616:
1615:
1584:
1583:
1564:
1563:
1544:
1543:
1522:
1517:
1516:
1484:
1477:
1476:
1457:
1456:
1437:
1436:
1410:
1390:
1364:
1363:
1344:
1343:
1322:
1317:
1316:
1295:
1290:
1289:
1268:
1255:
1244:
1243:
1224:
1223:
1204:
1203:
1171:
1170:
1151:
1150:
1125:
1124:
1064:
1026:
963:
959:
931:
925:
865:
861:
833:
827:
817:
816:
788:
783:
782:
748:
735:
701:
695:
685:
684:
642:
641:
602:
601:
562:
561:
524:
496:
490:
480:
479:
397:
393:
365:
359:
349:
348:
320:
307:
294:
293:
265:
259:
249:
248:
235:lattice theorem
231:integral domain
198:
197:
182:
170:irreducible set
36:
23:
22:
18:Irreducible set
15:
12:
11:
5:
3567:
3565:
3557:
3556:
3546:
3545:
3542:
3541:
3527:
3514:
3484:
3478:
3463:
3460:
3457:
3456:
3442:
3428:
3414:
3407:
3389:
3375:
3368:
3350:
3343:
3325:
3318:
3300:
3286:
3273:(2): 147–158.
3253:
3238:
3228:
3227:
3225:
3222:
3221:
3220:
3215:
3210:
3203:
3200:
3186:singleton sets
3173:. Unlike the
3145:
3142:
3141:
3140:
3127:
3107:
3087:
3067:
3064:
3061:
3056:
3052:
3048:
3043:
3039:
3035:
3032:
3027:
3023:
3019:
3012:
3008:
3002:
2997:
2994:
2989:
2985:
2981:
2974:
2970:
2964:
2959:
2956:
2953:
2950:
2947:
2942:
2938:
2915:
2911:
2907:
2902:
2898:
2894:
2891:
2886:
2882:
2878:
2873:
2869:
2865:
2862:
2859:
2856:
2853:
2833:
2830:
2825:
2821:
2817:
2814:
2811:
2806:
2802:
2781:
2778:
2773:
2769:
2765:
2760:
2756:
2746:which implies
2733:
2729:
2702:
2698:
2694:
2691:
2671:
2668:
2663:
2659:
2655:
2650:
2646:
2642:
2637:
2633:
2629:
2626:
2621:
2617:
2613:
2610:
2603:
2599:
2594:
2590:
2587:
2584:
2562:
2558:
2535:
2531:
2510:
2507:
2504:
2501:
2496:
2492:
2488:
2483:
2479:
2456:
2452:
2429:
2425:
2404:
2384:
2370:
2369:
2357:
2354:
2349:
2345:
2341:
2336:
2332:
2311:
2308:
2303:
2299:
2295:
2290:
2286:
2263:
2259:
2255:
2250:
2246:
2242:
2239:
2219:
2207:
2206:
2189:
2186:
2183:
2180:
2177:
2174:
2169:
2165:
2144:
2141:
2138:
2135:
2132:
2129:
2124:
2120:
2099:
2096:
2093:
2073:
2070:
2067:
2058:which implies
2046:
2043:
2039:
2035:
2032:
2028:
2025:
2005:
1996:are closed in
1985:
1982:
1979:
1976:
1972:
1969:
1964:
1961:
1958:
1955:
1952:
1948:
1945:
1924:
1904:
1901:
1898:
1895:
1890:
1886:
1882:
1879:
1876:
1873:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1827:
1823:
1802:
1782:
1779:
1776:
1762:
1761:
1753:
1752:
1737:
1733:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1650:
1628:
1624:
1610:
1609:
1605:
1604:
1591:
1571:
1551:
1529:
1525:
1504:
1501:
1496:
1491:
1487:
1464:
1444:
1422:
1417:
1413:
1407:
1402:
1397:
1393:
1387:
1382:
1379:
1374:
1371:
1351:
1329:
1325:
1302:
1298:
1275:
1271:
1267:
1262:
1258:
1254:
1251:
1231:
1211:
1197:
1196:
1192:
1191:
1178:
1158:
1138:
1135:
1132:
1118:
1117:
1114:
1102:
1095:
1088:path-connected
1076:
1063:
1060:
1038:path-connected
1025:
1022:
1008:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
970:
966:
962:
957:
954:
951:
948:
945:
942:
939:
935:
928:
914:
910:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
872:
868:
864:
859:
856:
853:
850:
847:
844:
841:
837:
830:
795:
791:
766:
760:
755:
751:
747:
744:
741:
738:
733:
730:
727:
724:
721:
718:
715:
712:
709:
705:
698:
667:
662:
659:
656:
651:
627:
622:
619:
616:
611:
587:
582:
579:
576:
571:
545:
539:
536:
533:
530:
527:
522:
519:
516:
513:
510:
507:
504:
500:
493:
457:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
404:
400:
396:
391:
388:
385:
382:
379:
376:
373:
369:
362:
335:
327:
323:
319:
314:
310:
306:
301:
297:
291:
288:
285:
282:
279:
276:
273:
269:
262:
206:
181:
178:
148:anti-Hausdorff
136:hyperconnected
132:
131:
128:
117:
106:
95:
92:closed subsets
85:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3566:
3555:
3552:
3551:
3549:
3538:
3537:
3532:
3528:
3525:
3521:
3517:
3511:
3507:
3503:
3499:
3498:
3493:
3489:
3485:
3481:
3475:
3471:
3466:
3465:
3461:
3452:
3446:
3443:
3438:
3432:
3429:
3424:
3418:
3415:
3410:
3404:
3400:
3393:
3390:
3385:
3379:
3376:
3371:
3365:
3361:
3354:
3351:
3346:
3340:
3336:
3329:
3326:
3321:
3315:
3311:
3304:
3301:
3296:
3290:
3287:
3281:
3276:
3272:
3268:
3264:
3257:
3254:
3250:
3245:
3243:
3239:
3233:
3230:
3223:
3219:
3216:
3214:
3211:
3209:
3206:
3205:
3201:
3199:
3197:
3192:
3189:
3187:
3182:
3180:
3176:
3172:
3168:
3164:
3160:
3155:
3153:
3152:
3143:
3139:
3125:
3105:
3085:
3065:
3062:
3059:
3054:
3050:
3046:
3041:
3037:
3033:
3025:
3021:
3010:
3006:
3000:
2995:
2987:
2983:
2972:
2968:
2962:
2957:
2951:
2945:
2940:
2936:
2913:
2909:
2905:
2900:
2896:
2892:
2884:
2880:
2876:
2871:
2867:
2860:
2857:
2854:
2851:
2828:
2823:
2819:
2815:
2812:
2809:
2804:
2800:
2776:
2771:
2767:
2763:
2758:
2754:
2731:
2727:
2718:
2700:
2696:
2692:
2689:
2666:
2661:
2657:
2653:
2648:
2644:
2640:
2635:
2631:
2627:
2619:
2615:
2608:
2601:
2597:
2592:
2588:
2585:
2560:
2556:
2533:
2529:
2505:
2502:
2499:
2494:
2490:
2486:
2481:
2477:
2454:
2450:
2427:
2423:
2402:
2382:
2372:
2371:
2352:
2347:
2343:
2339:
2334:
2330:
2309:
2306:
2301:
2297:
2293:
2288:
2284:
2261:
2257:
2253:
2248:
2244:
2240:
2237:
2217:
2209:
2208:
2205:
2203:
2187:
2184:
2178:
2172:
2167:
2163:
2142:
2139:
2133:
2127:
2122:
2118:
2097:
2094:
2091:
2071:
2068:
2065:
2044:
2041:
2037:
2033:
2030:
2026:
2023:
2003:
1983:
1980:
1977:
1974:
1970:
1967:
1962:
1959:
1956:
1953:
1950:
1946:
1943:
1922:
1915:(and thus in
1899:
1893:
1888:
1884:
1880:
1877:
1874:
1871:
1851:
1848:
1845:
1842:
1836:
1830:
1825:
1821:
1800:
1780:
1777:
1774:
1764:
1763:
1759:
1755:
1754:
1751:
1735:
1727:
1721:
1715:
1712:
1706:
1700:
1697:
1691:
1688:
1682:
1679:
1676:
1668:
1664:
1626:
1612:
1611:
1607:
1606:
1603:
1589:
1569:
1549:
1527:
1523:
1502:
1499:
1489:
1485:
1462:
1442:
1415:
1411:
1405:
1395:
1391:
1385:
1377:
1372:
1369:
1349:
1327:
1323:
1300:
1296:
1273:
1269:
1265:
1260:
1256:
1252:
1249:
1229:
1209:
1199:
1198:
1194:
1193:
1190:
1176:
1156:
1136:
1133:
1130:
1120:
1119:
1115:
1112:
1111:pseudocompact
1107:
1103:
1100:
1096:
1093:
1089:
1085:
1081:
1077:
1074:
1070:
1066:
1065:
1061:
1059:
1057:
1053:
1048:
1045:
1043:
1039:
1035:
1031:
1023:
1020:
1006:
994:
991:
988:
985:
982:
979:
976:
968:
964:
952:
949:
946:
943:
940:
926:
912:
908:
896:
893:
890:
887:
884:
881:
878:
870:
866:
854:
851:
848:
845:
842:
828:
813:
811:
793:
789:
778:
764:
753:
749:
745:
742:
739:
728:
725:
722:
719:
716:
713:
710:
696:
681:
665:
660:
657:
654:
625:
620:
617:
614:
585:
580:
577:
574:
557:
543:
534:
531:
528:
517:
514:
511:
508:
505:
491:
476:
475:
469:
455:
443:
440:
437:
434:
425:
422:
419:
413:
410:
407:
402:
398:
386:
383:
380:
377:
374:
360:
333:
325:
321:
317:
312:
308:
304:
299:
295:
286:
283:
280:
277:
274:
260:
245:
244:
240:
236:
232:
228:
224:
219:
195:
191:
187:
179:
177:
175:
171:
166:
164:
160:
156:
151:
149:
145:
141:
137:
129:
126:
122:
121:nowhere dense
118:
115:
111:
107:
104:
100:
96:
93:
89:
86:
83:
79:
75:
74:
73:
71:
66:
64:
60:
56:
53:
49:
45:
41:
34:
30:
19:
3534:
3495:
3469:
3445:
3431:
3417:
3398:
3392:
3378:
3359:
3353:
3334:
3328:
3309:
3303:
3289:
3270:
3266:
3256:
3251:, p. 9.
3232:
3193:
3190:
3183:
3170:
3166:
3162:
3158:
3156:
3149:
3147:
2548:is dense in
2374:
1766:
1614:
1542:is dense in
1201:
1122:
1068:
1055:
1051:
1049:
1046:
1027:
815:
780:
683:
559:
478:
471:
247:
227:reduced ring
220:
183:
169:
167:
152:
147:
139:
135:
133:
124:
113:
102:
87:
69:
67:
58:
54:
47:
43:
37:
3213:Sober space
2110:, but then
140:irreducible
3536:PlanetMath
3462:References
3078:therefore
1342:closed in
1106:continuous
1062:Properties
239:nilradical
3494:(1995) ,
3179:partition
3047:∪
2996:∪
2958:⊇
2946:
2906:∪
2877:∪
2861:∩
2832:∅
2829:≠
2816:∩
2780:∅
2777:≠
2764:∩
2693:∈
2670:∅
2667:≠
2654:∩
2628:∩
2609:
2589:∈
2583:∃
2509:∅
2506:≠
2500:∩
2356:∅
2353:≠
2340:∩
2307:⊂
2254:∪
2173:
2128:
2095:⊆
2069:⊆
2038:∪
1981:∩
1957:∩
1894:
1881:⊆
1849:∪
1831:
1778:⊆
1732:k
1728:⊂
1713:∪
1649:k
1623:k
1495:¯
1435:. Since
1421:¯
1406:∪
1401:¯
1381:¯
1266:∪
1134:⊂
1080:connected
1073:Hausdorff
1030:connected
438:−
423:−
411:−
159:vacuously
155:empty set
144:Hausdorff
116:is empty.
78:open sets
3548:Category
3202:See also
2210:A space
2045:′
2034:′
1971:′
1947:′
1767:Suppose
1669:, while
1362:. Then
1202:Suppose
188:are the
180:Examples
110:interior
82:disjoint
40:topology
3524:0507446
2575:, thus
2521:, then
2373:Proof:
2202:closure
1765:Proof:
1758:closure
1200:Proof:
1121:Proof:
243:schemes
237:to the
3522:
3512:
3476:
3405:
3366:
3341:
3316:
3194:Every
2844:. Now
1793:where
1475:, say
917:
781:where
640:, and
229:is an
225:whose
3502:Dover
3224:Notes
2715:is a
2276:with
1641:with
1288:with
99:dense
50:is a
3510:ISBN
3474:ISBN
3403:ISBN
3364:ISBN
3339:ISBN
3314:ISBN
2682:and
2442:and
2016:and
1756:The
1242:and
1123:Let
1104:The
1082:and
1044:).
1032:and
922:Proj
824:Proj
692:Proj
487:Spec
356:Proj
256:Spec
153:The
108:The
80:are
42:, a
3275:doi
3148:An
2719:of
2155:or
2084:or
1935:).
1661:an
1090:or
1056:can
1052:not
1040:or
196:on
168:An
157:is
138:or
123:in
101:in
46:or
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3533:.
3520:MR
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3508:,
3490:;
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3269:.
3265:.
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2937:Cl
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1975::=
1951::=
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218:.
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3277::
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