1007:
3065:
2669:
456:
807:
334:
1422:
765:
2920:
544:
1097:
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
2915:
1902:
1737:
1983:
1851:
2831:
2508:
2779:
2355:
1036:
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.
2187:
2142:
230:, 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
667:
627:
587:
3143:
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.
2045:
1502:
150:
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
2309:
461:
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
2263:
1275:
2567:
2097:
2071:
1780:
205:
2702:
1628:
1136:
1648:
2733:
2562:
2535:
2456:
2429:
1529:
1329:
1302:
795:
3125:
3105:
3085:
2402:
2382:
2217:
2003:
1922:
1800:
1589:
1569:
1549:
1462:
1442:
1349:
1229:
1209:
1176:
1156:
1002:{\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)}
339:
239:
3060:{\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,}
1354:
675:
1086:
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
470:
21:
3542:
2836:
3502:
3466:
3395:
3356:
3331:
3306:
1856:
1661:
1805:
2784:
2461:
3180:
Since every irreducible space is connected, the irreducible components will always lie in the connected components.
2738:
2314:
2147:
2102:
3163:
1927:
1094:
182:
46:
that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name
1087:
632:
592:
552:
3184:
1467:
2268:
3485:
2664:{\displaystyle \exists x\in \operatorname {Cl} _{U_{1}}(V_{1})\cap U_{2}=U_{1}\cap U_{2}\neq \emptyset }
2222:
1651:
1234:
1056:
The nonempty open subsets of a hyperconnected space are "large" in the sense that each one is dense in
797:
is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves (by the
3206:
131:. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the
798:
2076:
2050:
1759:
1043:
hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it
188:
2008:
462:
2674:
1606:
1115:
3480:
1080:
1030:
3139:
1633:
3512:
3196:
2711:
2540:
2513:
2434:
2407:
1507:
1307:
1280:
773:
231:
227:
8:
1099:
98:
3490:
3167:
3166:
of a space, the irreducible components need not be disjoint (i.e. they need not form a
3110:
3090:
3070:
2387:
2367:
2202:
2190:
1988:
1907:
1785:
1746:
1574:
1554:
1534:
1447:
1427:
1334:
1214:
1194:
1161:
1141:
451:{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} }{(y^{2}z-x(x-z)(x-2z))}}\right)}
211:
174:
151:
51:
3498:
3476:
3462:
3391:
3352:
3327:
3302:
3268:
1072:
1022:
178:
162:
109:
40:
17:
3263:
2705:
1655:
1039:
For example, the space of real numbers with the standard topology is connected but
329:{\displaystyle {\text{Spec}}\left({\frac {\mathbb {Z} }{x^{4}+y^{3}+z^{2}}}\right)}
87:
3252:"An anti-Hausdorff Fréchet space in which convergent sequences have unique limits"
3508:
3494:
1068:
1061:
1018:
223:
219:
132:
3251:
1184:
More generally, every dense subset of a hyperconnected space is hyperconnected.
1076:
1026:
3536:
3519:
3174:
1417:{\displaystyle X={\overline {S}}={\overline {S_{1}}}\cup {\overline {S_{2}}}}
147:
80:
70:
3439:
3425:
3411:
3372:
3283:
760:{\displaystyle {\text{Proj}}\left({\frac {\mathbb {C} }{(xy,f_{4})}}\right)}
215:
3201:
154:, add an explicit condition that an irreducible space must be nonempty.
1060:
and any pair of them intersects. Thus, a hyperconnected space cannot be
3524:
1597:
A closed subspace of a hyperconnected space need not be hyperconnected.
539:{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} }{(xyz)}}\right)}
3440:"Section 5.9 (0050): Noetherian topological spaces—The Stacks project"
143:
3150:
is contained in a (not necessarily unique) irreducible component of
66:
28:
3233:
3231:
1105:
Every open subspace of a hyperconnected space is hyperconnected.
22:
Connectivity (graph theory) § Super- and hyper-connectivity
3457:
Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004).
3284:"Section 5.8 (004U): Irreducible components—The Stacks project"
1444:
is hyperconnected, one of the two closures is the whole space
123:
A space which satisfies any one of these conditions is called
3228:
3173:
The irreducible components of a
Hausdorff space are just the
549:
since the underlying space is the union of the affine planes
119:
No two points can be separated by disjoint neighbourhoods.
2910:{\displaystyle V=V\cap (U_{1}\cup U_{2})=V_{1}\cup V_{2}}
1047:
be written as a union of two (non-disjoint) closed sets.
3170:). In general, the irreducible components will overlap.
1897:{\displaystyle F,G\subseteq \operatorname {Cl} _{X}(S)}
1732:{\displaystyle V=Z(XY)=Z(X)\cup Z(Y)\subset \Bbbk ^{2}}
1012:
3107:. Since this is true for every non-empty open subset,
3113:
3093:
3073:
2923:
2839:
2787:
2741:
2714:
2677:
2570:
2543:
2516:
2464:
2437:
2410:
2390:
2370:
2317:
2271:
2225:
2205:
2150:
2105:
2079:
2053:
2011:
1991:
1930:
1910:
1859:
1808:
1788:
1762:
1664:
1636:
1609:
1577:
1557:
1537:
1510:
1470:
1450:
1430:
1357:
1337:
1310:
1283:
1237:
1217:
1197:
1164:
1144:
1118:
810:
776:
678:
635:
595:
555:
473:
342:
242:
191:
1138:
be an open subset. Any two disjoint open subsets of
3119:
3099:
3079:
3059:
2909:
2825:
2773:
2727:
2696:
2663:
2556:
2529:
2502:
2450:
2423:
2396:
2376:
2349:
2303:
2257:
2211:
2181:
2136:
2091:
2065:
2039:
1997:
1977:
1916:
1896:
1846:{\displaystyle \operatorname {Cl} _{X}(S)=F\cup G}
1845:
1794:
1774:
1731:
1642:
1622:
1583:
1563:
1543:
1523:
1496:
1456:
1436:
1416:
1343:
1323:
1296:
1269:
1223:
1203:
1170:
1150:
1130:
1001:
789:
759:
661:
621:
581:
538:
450:
328:
199:
20:. For hyper-connectivity in node-link graphs, see
3456:
3237:
2826:{\displaystyle V_{2}:=V\cap U_{2}\neq \emptyset }
2503:{\displaystyle V_{1}:=U_{1}\cap V\neq \emptyset }
222:is an irreducible topological space—applying the
161:is a subset of a topological space for which the
3534:
3158:is contained in some irreducible component of
2774:{\displaystyle V_{1}\cap U_{2}\neq \emptyset }
2350:{\displaystyle U_{1}\cap U_{2}\neq \emptyset }
3493:reprint of 1978 ed.), Berlin, New York:
1158:would themselves be disjoint open subsets of
79:cannot be written as the union of two proper
3475:
3412:"Definition 5.8.1 (004V)—The Stacks project"
2182:{\displaystyle \operatorname {Cl} _{X}(S)=G}
2137:{\displaystyle \operatorname {Cl} _{X}(S)=F}
669:. Another non-example is given by the scheme
173:Two examples of hyperconnected spaces from
3249:
3187:has finitely many irreducible components.
3267:
3132:
1978:{\displaystyle F':=F\cap S,\,G':=G\cap S}
1954:
1720:
1654:(thus infinite) is hyperconnected in the
1637:
1611:
1178:. So at least one of them must be empty.
922:
824:
692:
638:
598:
558:
487:
356:
256:
193:
61:the following conditions are equivalent:
3385:
3321:
3296:
3087:is a non-empty open and dense subset of
135:property, some authors call such spaces
3426:"Lemma 5.8.3 (004W)—The Stacks project"
3373:"Lemma 5.8.3 (004W)—The Stacks project"
1064:unless it contains only a single point.
3535:
3346:
1749:of any irreducible set is irreducible.
662:{\displaystyle \mathbb {A} _{y,z}^{2}}
622:{\displaystyle \mathbb {A} _{x,z}^{2}}
582:{\displaystyle \mathbb {A} _{x,y}^{2}}
16:For the computer networking term, see
1497:{\displaystyle {\overline {S_{1}}}=X}
3146:Every irreducible subset of a space
2304:{\displaystyle U_{1},U_{2}\subset X}
1013:Hyperconnectedness vs. connectedness
3349:Algebraic Geometry. An introduction
1067:Every hyperconnected space is both
1017:Every hyperconnected space is both
13:
2820:
2768:
2658:
2571:
2497:
2344:
210:In algebraic geometry, taking the
14:
3554:
3388:Commutative Algebra: Chapters 1-7
3324:Commutative Algebra: Chapters 1-7
3299:Commutative Algebra: Chapters 1-7
3154:. In particular, every point of
2258:{\displaystyle X=U_{1}\cup U_{2}}
1739:is closed and not hyperconnected.
1270:{\displaystyle S=S_{1}\cup S_{2}}
101:of every proper closed subset of
3543:Properties of topological spaces
3459:Encyclopedia of general topology
3432:
3418:
3404:
3238:Hart, Nagata & Vaughan 2004
2311:open and irreducible such that
3379:
3365:
3340:
3315:
3290:
3276:
3243:
3219:
3019:
3006:
2981:
2968:
2943:
2937:
2878:
2852:
2613:
2600:
2170:
2164:
2125:
2119:
1891:
1885:
1828:
1822:
1713:
1707:
1698:
1692:
1683:
1674:
989:
986:
962:
949:
944:
926:
891:
888:
864:
851:
846:
828:
747:
725:
720:
696:
526:
514:
509:
491:
438:
435:
420:
417:
405:
383:
378:
360:
278:
260:
1:
3450:
3256:Topology and Its Applications
1050:
27:In the mathematical field of
3269:10.1016/0166-8641(93)90147-6
3250:Van Douwen, Eric K. (1993).
3185:Noetherian topological space
2092:{\displaystyle S\subseteq G}
2066:{\displaystyle S\subseteq F}
1775:{\displaystyle S\subseteq X}
1551:, and since it is closed in
1483:
1409:
1389:
1369:
200:{\displaystyle \mathbb {R} }
181:on any infinite set and the
7:
3486:Counterexamples in Topology
3190:
2384:is a non-empty open set in
2364:Firstly, we notice that if
2040:{\displaystyle S=F'\cup G'}
168:
86:Every nonempty open set is
10:
3559:
3461:. Elsevier/North-Holland.
3386:Bourbaki, Nicolas (1989).
3322:Bourbaki, Nicolas (1989).
3297:Bourbaki, Nicolas (1989).
3225:Steen & Seebach, p. 29
2697:{\displaystyle x\in U_{2}}
1652:algebraically closed field
1623:{\displaystyle \Bbbk ^{2}}
1131:{\displaystyle U\subset X}
15:
3207:Geometrically irreducible
1802:is irreducible and write
108:Every subset is dense or
3390:. Springer. p. 95.
3351:. Springer. p. 14.
3326:. Springer. p. 95.
3301:. Springer. p. 95.
3212:
2404:then it intersects both
2219:which can be written as
1075:(though not necessarily
1025:(though not necessarily
57:For a topological space
3347:Perrin, Daniel (2008).
2917:and taking the closure
1853:for two closed subsets
1088:extremally disconnected
463:normal crossing divisor
3520:"Hyperconnected space"
3481:Seebach, J. Arthur Jr.
3133:Irreducible components
3121:
3101:
3081:
3061:
2911:
2827:
2775:
2729:
2698:
2665:
2558:
2531:
2504:
2452:
2425:
2398:
2378:
2351:
2305:
2259:
2213:
2183:
2138:
2093:
2067:
2041:
1999:
1979:
1918:
1898:
1847:
1796:
1776:
1733:
1644:
1643:{\displaystyle \Bbbk }
1624:
1585:
1571:, it must be equal to
1565:
1545:
1525:
1498:
1458:
1438:
1418:
1345:
1325:
1298:
1271:
1225:
1205:
1172:
1152:
1132:
1081:locally path-connected
1031:locally path-connected
1010:
1003:
791:
768:
761:
663:
623:
583:
547:
540:
459:
452:
330:
201:
3140:irreducible component
3122:
3102:
3082:
3062:
2912:
2828:
2776:
2730:
2728:{\displaystyle V_{1}}
2699:
2666:
2559:
2557:{\displaystyle U_{1}}
2532:
2530:{\displaystyle V_{1}}
2505:
2453:
2451:{\displaystyle U_{2}}
2426:
2424:{\displaystyle U_{1}}
2399:
2379:
2352:
2306:
2260:
2214:
2184:
2139:
2094:
2068:
2042:
2000:
1980:
1919:
1899:
1848:
1797:
1777:
1734:
1645:
1625:
1586:
1566:
1546:
1526:
1524:{\displaystyle S_{1}}
1504:. This implies that
1499:
1459:
1439:
1419:
1346:
1326:
1324:{\displaystyle S_{2}}
1299:
1297:{\displaystyle S_{1}}
1272:
1226:
1211:is a dense subset of
1206:
1173:
1153:
1133:
1004:
803:
792:
790:{\displaystyle f_{4}}
762:
671:
664:
624:
584:
541:
466:
453:
331:
235:
202:
3197:Ultraconnected space
3164:connected components
3111:
3091:
3071:
2921:
2837:
2785:
2739:
2712:
2675:
2568:
2541:
2514:
2462:
2435:
2408:
2388:
2368:
2315:
2269:
2223:
2203:
2148:
2103:
2077:
2051:
2009:
1989:
1928:
1908:
1857:
1806:
1786:
1760:
1662:
1634:
1607:
1575:
1555:
1535:
1508:
1468:
1448:
1428:
1355:
1335:
1308:
1281:
1235:
1215:
1195:
1162:
1142:
1116:
808:
799:genus–degree formula
774:
676:
633:
593:
553:
471:
340:
240:
189:
183:right order topology
33:hyperconnected space
658:
618:
578:
3477:Steen, Lynn Arthur
3117:
3097:
3077:
3057:
2907:
2823:
2771:
2725:
2694:
2661:
2554:
2527:
2500:
2458:; indeed, suppose
2448:
2421:
2394:
2374:
2347:
2301:
2255:
2209:
2179:
2134:
2089:
2063:
2037:
1995:
1975:
1914:
1894:
1843:
1792:
1772:
1729:
1640:
1620:
1581:
1561:
1541:
1521:
1494:
1454:
1434:
1414:
1341:
1321:
1294:
1267:
1221:
1201:
1168:
1148:
1128:
999:
787:
757:
659:
636:
619:
596:
579:
556:
536:
448:
326:
212:spectrum of a ring
197:
175:point set topology
152:algebraic geometry
52:algebraic geometry
3504:978-0-486-68735-3
3468:978-0-444-50355-8
3397:978-3-540-64239-8
3358:978-1-84800-055-1
3333:978-3-540-64239-8
3308:978-3-540-64239-8
3120:{\displaystyle X}
3100:{\displaystyle X}
3080:{\displaystyle V}
2397:{\displaystyle X}
2377:{\displaystyle V}
2212:{\displaystyle X}
2189:by definition of
1998:{\displaystyle S}
1917:{\displaystyle X}
1795:{\displaystyle S}
1584:{\displaystyle S}
1564:{\displaystyle S}
1544:{\displaystyle S}
1486:
1457:{\displaystyle X}
1437:{\displaystyle X}
1412:
1392:
1372:
1344:{\displaystyle S}
1224:{\displaystyle X}
1204:{\displaystyle S}
1171:{\displaystyle X}
1151:{\displaystyle U}
1073:locally connected
1023:locally connected
993:
912:
907:
895:
814:
751:
682:
530:
477:
442:
346:
320:
246:
179:cofinite topology
163:subspace topology
48:irreducible space
41:topological space
37:irreducible space
18:Hyperconnectivity
3550:
3529:
3515:
3472:
3444:
3443:
3436:
3430:
3429:
3422:
3416:
3415:
3408:
3402:
3401:
3383:
3377:
3376:
3369:
3363:
3362:
3344:
3338:
3337:
3319:
3313:
3312:
3294:
3288:
3287:
3280:
3274:
3273:
3271:
3247:
3241:
3235:
3226:
3223:
3126:
3124:
3123:
3118:
3106:
3104:
3103:
3098:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3047:
3046:
3034:
3033:
3018:
3017:
3005:
3004:
3003:
3002:
2992:
2980:
2979:
2967:
2966:
2965:
2964:
2954:
2933:
2932:
2916:
2914:
2913:
2908:
2906:
2905:
2893:
2892:
2877:
2876:
2864:
2863:
2832:
2830:
2829:
2824:
2816:
2815:
2797:
2796:
2780:
2778:
2777:
2772:
2764:
2763:
2751:
2750:
2734:
2732:
2731:
2726:
2724:
2723:
2706:point of closure
2703:
2701:
2700:
2695:
2693:
2692:
2670:
2668:
2667:
2662:
2654:
2653:
2641:
2640:
2628:
2627:
2612:
2611:
2596:
2595:
2594:
2593:
2563:
2561:
2560:
2555:
2553:
2552:
2536:
2534:
2533:
2528:
2526:
2525:
2509:
2507:
2506:
2501:
2487:
2486:
2474:
2473:
2457:
2455:
2454:
2449:
2447:
2446:
2430:
2428:
2427:
2422:
2420:
2419:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2356:
2354:
2353:
2348:
2340:
2339:
2327:
2326:
2310:
2308:
2307:
2302:
2294:
2293:
2281:
2280:
2264:
2262:
2261:
2256:
2254:
2253:
2241:
2240:
2218:
2216:
2215:
2210:
2188:
2186:
2185:
2180:
2160:
2159:
2143:
2141:
2140:
2135:
2115:
2114:
2098:
2096:
2095:
2090:
2072:
2070:
2069:
2064:
2046:
2044:
2043:
2038:
2036:
2025:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1962:
1938:
1923:
1921:
1920:
1915:
1903:
1901:
1900:
1895:
1881:
1880:
1852:
1850:
1849:
1844:
1818:
1817:
1801:
1799:
1798:
1793:
1781:
1779:
1778:
1773:
1738:
1736:
1735:
1730:
1728:
1727:
1656:Zariski topology
1649:
1647:
1646:
1641:
1629:
1627:
1626:
1621:
1619:
1618:
1602:Counterexample:
1590:
1588:
1587:
1582:
1570:
1568:
1567:
1562:
1550:
1548:
1547:
1542:
1530:
1528:
1527:
1522:
1520:
1519:
1503:
1501:
1500:
1495:
1487:
1482:
1481:
1472:
1463:
1461:
1460:
1455:
1443:
1441:
1440:
1435:
1423:
1421:
1420:
1415:
1413:
1408:
1407:
1398:
1393:
1388:
1387:
1378:
1373:
1365:
1350:
1348:
1347:
1342:
1330:
1328:
1327:
1322:
1320:
1319:
1303:
1301:
1300:
1295:
1293:
1292:
1276:
1274:
1273:
1268:
1266:
1265:
1253:
1252:
1230:
1228:
1227:
1222:
1210:
1208:
1207:
1202:
1177:
1175:
1174:
1169:
1157:
1155:
1154:
1149:
1137:
1135:
1134:
1129:
1008:
1006:
1005:
1000:
998:
994:
992:
961:
960:
947:
925:
919:
913:
910:
908:
905:
900:
896:
894:
863:
862:
849:
827:
821:
815:
812:
796:
794:
793:
788:
786:
785:
766:
764:
763:
758:
756:
752:
750:
746:
745:
723:
695:
689:
683:
680:
668:
666:
665:
660:
657:
652:
641:
628:
626:
625:
620:
617:
612:
601:
588:
586:
585:
580:
577:
572:
561:
545:
543:
542:
537:
535:
531:
529:
512:
490:
484:
478:
475:
457:
455:
454:
449:
447:
443:
441:
395:
394:
381:
359:
353:
347:
344:
335:
333:
332:
327:
325:
321:
319:
318:
317:
305:
304:
292:
291:
281:
259:
253:
247:
244:
206:
204:
203:
198:
196:
165:is irreducible.
65:No two nonempty
50:is preferred in
3558:
3557:
3553:
3552:
3551:
3549:
3548:
3547:
3533:
3532:
3518:
3505:
3495:Springer-Verlag
3469:
3453:
3448:
3447:
3438:
3437:
3433:
3424:
3423:
3419:
3410:
3409:
3405:
3398:
3384:
3380:
3371:
3370:
3366:
3359:
3345:
3341:
3334:
3320:
3316:
3309:
3295:
3291:
3282:
3281:
3277:
3248:
3244:
3236:
3229:
3224:
3220:
3215:
3193:
3135:
3127:is irreducible.
3112:
3109:
3108:
3092:
3089:
3088:
3072:
3069:
3068:
3042:
3038:
3029:
3025:
3013:
3009:
2998:
2994:
2993:
2988:
2987:
2975:
2971:
2960:
2956:
2955:
2950:
2949:
2928:
2924:
2922:
2919:
2918:
2901:
2897:
2888:
2884:
2872:
2868:
2859:
2855:
2838:
2835:
2834:
2811:
2807:
2792:
2788:
2786:
2783:
2782:
2781:and a fortiori
2759:
2755:
2746:
2742:
2740:
2737:
2736:
2719:
2715:
2713:
2710:
2709:
2688:
2684:
2676:
2673:
2672:
2649:
2645:
2636:
2632:
2623:
2619:
2607:
2603:
2589:
2585:
2584:
2580:
2569:
2566:
2565:
2548:
2544:
2542:
2539:
2538:
2521:
2517:
2515:
2512:
2511:
2482:
2478:
2469:
2465:
2463:
2460:
2459:
2442:
2438:
2436:
2433:
2432:
2415:
2411:
2409:
2406:
2405:
2389:
2386:
2385:
2369:
2366:
2365:
2357:is irreducible.
2335:
2331:
2322:
2318:
2316:
2313:
2312:
2289:
2285:
2276:
2272:
2270:
2267:
2266:
2249:
2245:
2236:
2232:
2224:
2221:
2220:
2204:
2201:
2200:
2155:
2151:
2149:
2146:
2145:
2110:
2106:
2104:
2101:
2100:
2078:
2075:
2074:
2052:
2049:
2048:
2029:
2018:
2010:
2007:
2006:
1990:
1987:
1986:
1955:
1931:
1929:
1926:
1925:
1909:
1906:
1905:
1876:
1872:
1858:
1855:
1854:
1813:
1809:
1807:
1804:
1803:
1787:
1784:
1783:
1761:
1758:
1757:
1723:
1719:
1663:
1660:
1659:
1635:
1632:
1631:
1614:
1610:
1608:
1605:
1604:
1576:
1573:
1572:
1556:
1553:
1552:
1536:
1533:
1532:
1515:
1511:
1509:
1506:
1505:
1477:
1473:
1471:
1469:
1466:
1465:
1449:
1446:
1445:
1429:
1426:
1425:
1403:
1399:
1397:
1383:
1379:
1377:
1364:
1356:
1353:
1352:
1336:
1333:
1332:
1315:
1311:
1309:
1306:
1305:
1288:
1284:
1282:
1279:
1278:
1261:
1257:
1248:
1244:
1236:
1233:
1232:
1216:
1213:
1212:
1196:
1193:
1192:
1163:
1160:
1159:
1143:
1140:
1139:
1117:
1114:
1113:
1053:
1015:
956:
952:
948:
921:
920:
918:
914:
909:
904:
858:
854:
850:
823:
822:
820:
816:
811:
809:
806:
805:
781:
777:
775:
772:
771:
741:
737:
724:
691:
690:
688:
684:
679:
677:
674:
673:
653:
642:
637:
634:
631:
630:
613:
602:
597:
594:
591:
590:
573:
562:
557:
554:
551:
550:
513:
486:
485:
483:
479:
474:
472:
469:
468:
390:
386:
382:
355:
354:
352:
348:
343:
341:
338:
337:
313:
309:
300:
296:
287:
283:
282:
255:
254:
252:
248:
243:
241:
238:
237:
224:lattice theorem
220:integral domain
192:
190:
187:
186:
171:
159:irreducible set
25:
12:
11:
5:
3556:
3546:
3545:
3531:
3530:
3516:
3503:
3473:
3467:
3452:
3449:
3446:
3445:
3431:
3417:
3403:
3396:
3378:
3364:
3357:
3339:
3332:
3314:
3307:
3289:
3275:
3262:(2): 147–158.
3242:
3227:
3217:
3216:
3214:
3211:
3210:
3209:
3204:
3199:
3192:
3189:
3175:singleton sets
3162:. Unlike the
3134:
3131:
3130:
3129:
3116:
3096:
3076:
3056:
3053:
3050:
3045:
3041:
3037:
3032:
3028:
3024:
3021:
3016:
3012:
3008:
3001:
2997:
2991:
2986:
2983:
2978:
2974:
2970:
2963:
2959:
2953:
2948:
2945:
2942:
2939:
2936:
2931:
2927:
2904:
2900:
2896:
2891:
2887:
2883:
2880:
2875:
2871:
2867:
2862:
2858:
2854:
2851:
2848:
2845:
2842:
2822:
2819:
2814:
2810:
2806:
2803:
2800:
2795:
2791:
2770:
2767:
2762:
2758:
2754:
2749:
2745:
2735:which implies
2722:
2718:
2691:
2687:
2683:
2680:
2660:
2657:
2652:
2648:
2644:
2639:
2635:
2631:
2626:
2622:
2618:
2615:
2610:
2606:
2602:
2599:
2592:
2588:
2583:
2579:
2576:
2573:
2551:
2547:
2524:
2520:
2499:
2496:
2493:
2490:
2485:
2481:
2477:
2472:
2468:
2445:
2441:
2418:
2414:
2393:
2373:
2359:
2358:
2346:
2343:
2338:
2334:
2330:
2325:
2321:
2300:
2297:
2292:
2288:
2284:
2279:
2275:
2252:
2248:
2244:
2239:
2235:
2231:
2228:
2208:
2196:
2195:
2178:
2175:
2172:
2169:
2166:
2163:
2158:
2154:
2133:
2130:
2127:
2124:
2121:
2118:
2113:
2109:
2088:
2085:
2082:
2062:
2059:
2056:
2047:which implies
2035:
2032:
2028:
2024:
2021:
2017:
2014:
1994:
1985:are closed in
1974:
1971:
1968:
1965:
1961:
1958:
1953:
1950:
1947:
1944:
1941:
1937:
1934:
1913:
1893:
1890:
1887:
1884:
1879:
1875:
1871:
1868:
1865:
1862:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1816:
1812:
1791:
1771:
1768:
1765:
1751:
1750:
1742:
1741:
1726:
1722:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1639:
1617:
1613:
1599:
1598:
1594:
1593:
1580:
1560:
1540:
1518:
1514:
1493:
1490:
1485:
1480:
1476:
1453:
1433:
1411:
1406:
1402:
1396:
1391:
1386:
1382:
1376:
1371:
1368:
1363:
1360:
1340:
1318:
1314:
1291:
1287:
1264:
1260:
1256:
1251:
1247:
1243:
1240:
1220:
1200:
1186:
1185:
1181:
1180:
1167:
1147:
1127:
1124:
1121:
1107:
1106:
1103:
1091:
1084:
1077:path-connected
1065:
1052:
1049:
1027:path-connected
1014:
1011:
997:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
959:
955:
951:
946:
943:
940:
937:
934:
931:
928:
924:
917:
903:
899:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
861:
857:
853:
848:
845:
842:
839:
836:
833:
830:
826:
819:
784:
780:
755:
749:
744:
740:
736:
733:
730:
727:
722:
719:
716:
713:
710:
707:
704:
701:
698:
694:
687:
656:
651:
648:
645:
640:
616:
611:
608:
605:
600:
576:
571:
568:
565:
560:
534:
528:
525:
522:
519:
516:
511:
508:
505:
502:
499:
496:
493:
489:
482:
446:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
393:
389:
385:
380:
377:
374:
371:
368:
365:
362:
358:
351:
324:
316:
312:
308:
303:
299:
295:
290:
286:
280:
277:
274:
271:
268:
265:
262:
258:
251:
195:
170:
167:
137:anti-Hausdorff
125:hyperconnected
121:
120:
117:
106:
95:
84:
81:closed subsets
74:
9:
6:
4:
3:
2:
3555:
3544:
3541:
3540:
3538:
3527:
3526:
3521:
3517:
3514:
3510:
3506:
3500:
3496:
3492:
3488:
3487:
3482:
3478:
3474:
3470:
3464:
3460:
3455:
3454:
3441:
3435:
3427:
3421:
3413:
3407:
3399:
3393:
3389:
3382:
3374:
3368:
3360:
3354:
3350:
3343:
3335:
3329:
3325:
3318:
3310:
3304:
3300:
3293:
3285:
3279:
3270:
3265:
3261:
3257:
3253:
3246:
3239:
3234:
3232:
3222:
3218:
3208:
3205:
3203:
3200:
3198:
3195:
3194:
3188:
3186:
3181:
3178:
3176:
3171:
3169:
3165:
3161:
3157:
3153:
3149:
3144:
3142:
3141:
3128:
3114:
3094:
3074:
3054:
3051:
3048:
3043:
3039:
3035:
3030:
3026:
3022:
3014:
3010:
2999:
2995:
2989:
2984:
2976:
2972:
2961:
2957:
2951:
2946:
2940:
2934:
2929:
2925:
2902:
2898:
2894:
2889:
2885:
2881:
2873:
2869:
2865:
2860:
2856:
2849:
2846:
2843:
2840:
2817:
2812:
2808:
2804:
2801:
2798:
2793:
2789:
2765:
2760:
2756:
2752:
2747:
2743:
2720:
2716:
2707:
2689:
2685:
2681:
2678:
2655:
2650:
2646:
2642:
2637:
2633:
2629:
2624:
2620:
2616:
2608:
2604:
2597:
2590:
2586:
2581:
2577:
2574:
2549:
2545:
2522:
2518:
2494:
2491:
2488:
2483:
2479:
2475:
2470:
2466:
2443:
2439:
2416:
2412:
2391:
2371:
2361:
2360:
2341:
2336:
2332:
2328:
2323:
2319:
2298:
2295:
2290:
2286:
2282:
2277:
2273:
2250:
2246:
2242:
2237:
2233:
2229:
2226:
2206:
2198:
2197:
2194:
2192:
2176:
2173:
2167:
2161:
2156:
2152:
2131:
2128:
2122:
2116:
2111:
2107:
2086:
2083:
2080:
2060:
2057:
2054:
2033:
2030:
2026:
2022:
2019:
2015:
2012:
1992:
1972:
1969:
1966:
1963:
1959:
1956:
1951:
1948:
1945:
1942:
1939:
1935:
1932:
1911:
1904:(and thus in
1888:
1882:
1877:
1873:
1869:
1866:
1863:
1860:
1840:
1837:
1834:
1831:
1825:
1819:
1814:
1810:
1789:
1769:
1766:
1763:
1753:
1752:
1748:
1744:
1743:
1740:
1724:
1716:
1710:
1704:
1701:
1695:
1689:
1686:
1680:
1677:
1671:
1668:
1665:
1657:
1653:
1615:
1601:
1600:
1596:
1595:
1592:
1578:
1558:
1538:
1516:
1512:
1491:
1488:
1478:
1474:
1451:
1431:
1404:
1400:
1394:
1384:
1380:
1374:
1366:
1361:
1358:
1338:
1316:
1312:
1289:
1285:
1262:
1258:
1254:
1249:
1245:
1241:
1238:
1218:
1198:
1188:
1187:
1183:
1182:
1179:
1165:
1145:
1125:
1122:
1119:
1109:
1108:
1104:
1101:
1100:pseudocompact
1096:
1092:
1089:
1085:
1082:
1078:
1074:
1070:
1066:
1063:
1059:
1055:
1054:
1048:
1046:
1042:
1037:
1034:
1032:
1028:
1024:
1020:
1009:
995:
983:
980:
977:
974:
971:
968:
965:
957:
953:
941:
938:
935:
932:
929:
915:
901:
897:
885:
882:
879:
876:
873:
870:
867:
859:
855:
843:
840:
837:
834:
831:
817:
802:
800:
782:
778:
767:
753:
742:
738:
734:
731:
728:
717:
714:
711:
708:
705:
702:
699:
685:
670:
654:
649:
646:
643:
614:
609:
606:
603:
574:
569:
566:
563:
546:
532:
523:
520:
517:
506:
503:
500:
497:
494:
480:
465:
464:
458:
444:
432:
429:
426:
423:
414:
411:
408:
402:
399:
396:
391:
387:
375:
372:
369:
366:
363:
349:
322:
314:
310:
306:
301:
297:
293:
288:
284:
275:
272:
269:
266:
263:
249:
234:
233:
229:
225:
221:
217:
213:
208:
184:
180:
176:
166:
164:
160:
155:
153:
149:
145:
140:
138:
134:
130:
126:
118:
115:
111:
110:nowhere dense
107:
104:
100:
96:
93:
89:
85:
82:
78:
75:
72:
68:
64:
63:
62:
60:
55:
53:
49:
45:
42:
38:
34:
30:
23:
19:
3523:
3484:
3458:
3434:
3420:
3406:
3387:
3381:
3367:
3348:
3342:
3323:
3317:
3298:
3292:
3278:
3259:
3255:
3245:
3240:, p. 9.
3221:
3182:
3179:
3172:
3159:
3155:
3151:
3147:
3145:
3138:
3136:
2537:is dense in
2363:
1755:
1603:
1531:is dense in
1190:
1111:
1057:
1044:
1040:
1038:
1035:
1016:
804:
769:
672:
548:
467:
460:
236:
216:reduced ring
209:
172:
158:
156:
141:
136:
128:
124:
122:
113:
102:
91:
76:
58:
56:
47:
43:
36:
32:
26:
3202:Sober space
2099:, but then
129:irreducible
3525:PlanetMath
3451:References
3067:therefore
1331:closed in
1095:continuous
1051:Properties
228:nilradical
3483:(1995) ,
3168:partition
3036:∪
2985:∪
2947:⊇
2935:
2895:∪
2866:∪
2850:∩
2821:∅
2818:≠
2805:∩
2769:∅
2766:≠
2753:∩
2682:∈
2659:∅
2656:≠
2643:∩
2617:∩
2598:
2578:∈
2572:∃
2498:∅
2495:≠
2489:∩
2345:∅
2342:≠
2329:∩
2296:⊂
2243:∪
2162:
2117:
2084:⊆
2058:⊆
2027:∪
1970:∩
1946:∩
1883:
1870:⊆
1838:∪
1820:
1767:⊆
1721:k
1717:⊂
1702:∪
1638:k
1612:k
1484:¯
1424:. Since
1410:¯
1395:∪
1390:¯
1370:¯
1255:∪
1123:⊂
1069:connected
1062:Hausdorff
1019:connected
427:−
412:−
400:−
148:vacuously
144:empty set
133:Hausdorff
105:is empty.
67:open sets
3537:Category
3191:See also
2199:A space
2034:′
2023:′
1960:′
1936:′
1756:Suppose
1658:, while
1351:. Then
1191:Suppose
177:are the
169:Examples
99:interior
71:disjoint
29:topology
3513:0507446
2564:, thus
2510:, then
2362:Proof:
2191:closure
1754:Proof:
1747:closure
1189:Proof:
1110:Proof:
232:schemes
226:to the
3511:
3501:
3465:
3394:
3355:
3330:
3305:
3183:Every
2833:. Now
1782:where
1464:, say
906:
770:where
629:, and
218:is an
214:whose
3491:Dover
3213:Notes
2704:is a
2265:with
1630:with
1277:with
88:dense
39:is a
3499:ISBN
3463:ISBN
3392:ISBN
3353:ISBN
3328:ISBN
3303:ISBN
2671:and
2431:and
2005:and
1745:The
1231:and
1112:Let
1093:The
1071:and
1033:).
1021:and
911:Proj
813:Proj
681:Proj
476:Spec
345:Proj
245:Spec
142:The
97:The
69:are
31:, a
3264:doi
3137:An
2708:of
2144:or
2073:or
1924:).
1650:an
1079:or
1045:can
1041:not
1029:or
185:on
157:An
146:is
127:or
112:in
90:in
35:or
3539::
3522:.
3509:MR
3507:,
3497:,
3479:;
3260:51
3258:.
3254:.
3230:^
3177:.
2990:Cl
2952:Cl
2926:Cl
2799::=
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2476::=
2153:Cl
2108:Cl
1964::=
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1083:).
589:,
336:,
207:.
139:.
54:.
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3471:.
3442:.
3428:.
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3375:.
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3336:.
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3286:.
3272:.
3266::
3160:X
3156:X
3152:X
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3115:X
3095:X
3075:V
3055:,
3052:X
3049:=
3044:2
3040:U
3031:1
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3015:2
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3007:(
3000:2
2996:U
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2847:V
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2591:1
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2227:X
2207:X
2193:.
2177:G
2174:=
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2168:S
2165:(
2157:X
2132:F
2129:=
2126:)
2123:S
2120:(
2112:X
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2031:G
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2016:=
2013:S
1993:S
1973:S
1967:G
1957:G
1952:,
1949:S
1943:F
1933:F
1912:X
1892:)
1889:S
1886:(
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1826:S
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996:)
990:)
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