730:
1309:
243:(has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open intervals.
605:
954:
1349:
469:
423:
548:
1055:
1674:
1639:
1600:
1553:
1509:
1469:
1146:
509:
349:
213:
168:
1215:
1117:
92:
66:
1242:
995:
824:
1021:
890:
574:
316:
783:
1433:
1413:
1393:
1373:
1189:
1166:
1075:
864:
844:
754:
594:
377:
284:
264:
1247:
725:{\displaystyle {\bigl \{}[x,+\infty ){\bigr \}}\cup {\Bigl \{}{\bigl (}-\infty ,x-{\tfrac {1}{n}}{\bigr )}\,{\Big |}\,n\in \mathbb {N} {\Bigr \}}.}
17:
1765:
895:
1318:
1606:, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a
1472:
428:
382:
178:, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in
518:
1791:
1748:
1435:
when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
1026:
1607:
1650:
1615:
1576:
1529:
1485:
1445:
1122:
485:
325:
189:
144:
1194:
1089:
75:
49:
1220:
1743:
1520:
1352:
959:
788:
1512:
1479:
1000:
869:
553:
1775:
472:
289:
175:
759:
8:
1753:
1686:
1418:
1398:
1378:
1358:
1174:
1151:
1060:
849:
829:
739:
579:
362:
269:
249:
103:
1564:
1761:
1739:
1723:
Adam Emeryk, Władysław Kulpa. The
Sorgenfrey line has no connected compactification.
1439:
1312:
134:
126:
43:
1556:
1395:(when the codomain carries the standard topology) is the same as the usual limit of
756:
is compact, this cover has a finite subcover, and hence there exists a real number
216:
179:
99:
1706:
98:) and has a number of interesting properties. It is the topology generated by the
1771:
1757:
1568:
1516:
1304:{\displaystyle \forall \alpha \geq \alpha _{0}:L\leq x_{\alpha }<L+\epsilon }
1168:
1083:
240:
1082:
The name "lower limit topology" comes from the following fact: a sequence (or
1785:
512:
479:
183:
95:
69:
1642:
1560:
31:
1603:
356:
319:
171:
352:
215:
with itself is also a useful counterexample, known as the
1707:"general topology - The Sorgenfrey line is a Baire Space"
471:
are also clopen. This shows that the
Sorgenfrey line is
674:
1653:
1618:
1579:
1532:
1488:
1448:
1421:
1401:
1381:
1361:
1321:
1250:
1223:
1197:
1177:
1154:
1125:
1092:
1063:
1029:
1003:
962:
898:
872:
852:
832:
791:
762:
742:
608:
582:
556:
521:
488:
431:
385:
365:
328:
292:
272:
252:
192:
147:
78:
52:
515:. To see this, consider a non-empty compact subset
1668:
1633:
1594:
1547:
1503:
1463:
1427:
1407:
1387:
1367:
1343:
1303:
1236:
1209:
1183:
1160:
1140:
1111:
1069:
1049:
1015:
989:
948:
884:
858:
838:
818:
777:
748:
724:
588:
568:
542:
503:
463:
417:
371:
343:
310:
278:
258:
207:
162:
86:
60:
949:{\displaystyle q(x)\in (a(x),x]\cap \mathbb {Q} }
714:
695:
646:
1783:
1311:. The Sorgenfrey line can thus be used to study
72:; it is different from the standard topology on
1676:does not have any connected compactifications.
1344:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1756:reprint of 1978 ed.), Berlin, New York:
687:
653:
636:
611:
464:{\displaystyle \{x\in \mathbb {R} :x\geq a\}}
222:In complete analogy, one can also define the
1738:
458:
432:
418:{\displaystyle \{x\in \mathbb {R} :x<a\}}
412:
386:
543:{\displaystyle C\subseteq \mathbb {R} _{l}}
1355:, then the ordinary right-sided limit of
1656:
1621:
1582:
1535:
1491:
1451:
1337:
1329:
1128:
1043:
942:
708:
700:
692:
530:
491:
442:
396:
331:
195:
150:
80:
54:
576:, consider the following open cover of
14:
1784:
1023:, are pairwise disjoint, the function
1526:In terms of compactness properties,
1191:from the right", meaning for every
1050:{\displaystyle q:C\to \mathbb {Q} }
24:
1251:
661:
628:
25:
1803:
40:right half-open interval topology
1669:{\displaystyle \mathbb {R} _{l}}
1634:{\displaystyle \mathbb {R} _{l}}
1595:{\displaystyle \mathbb {R} _{l}}
1548:{\displaystyle \mathbb {R} _{l}}
1504:{\displaystyle \mathbb {R} _{l}}
1473:perfectly normal Hausdorff space
1464:{\displaystyle \mathbb {R} _{l}}
1141:{\displaystyle \mathbb {R} _{l}}
504:{\displaystyle \mathbb {R} _{l}}
344:{\displaystyle \mathbb {R} _{l}}
228:left half-open interval topology
208:{\displaystyle \mathbb {R} _{l}}
163:{\displaystyle \mathbb {R} _{l}}
892:. Now choose a rational number
1717:
1699:
1333:
1210:{\displaystyle \epsilon >0}
1106:
1093:
1039:
984:
975:
969:
963:
935:
926:
920:
914:
908:
902:
813:
804:
798:
792:
772:
766:
631:
616:
305:
293:
13:
1:
1692:
1112:{\displaystyle (x_{\alpha })}
359:). Furthermore, for all real
233:
239:The lower limit topology is
87:{\displaystyle \mathbb {R} }
61:{\displaystyle \mathbb {R} }
27:Topology on the real numbers
7:
1749:Counterexamples in Topology
1725:Comm. Math. Univ. Carolinae
1680:
1237:{\displaystyle \alpha _{0}}
18:Half-open interval topology
10:
1808:
1711:Mathematics Stack Exchange
141:and is sometimes written
990:{\displaystyle (a(x),x]}
819:{\displaystyle (a(x),x]}
1148:converges to the limit
866:. This is true for all
785:such that the interval
1744:Seebach, J. Arthur Jr.
1670:
1635:
1596:
1549:
1505:
1465:
1429:
1409:
1389:
1369:
1345:
1305:
1238:
1217:there exists an index
1211:
1185:
1162:
1142:
1113:
1071:
1051:
1017:
1016:{\displaystyle x\in C}
991:
956:. Since the intervals
950:
886:
885:{\displaystyle x\in C}
860:
840:
820:
779:
750:
726:
590:
570:
569:{\displaystyle x\in C}
544:
505:
465:
419:
373:
345:
312:
280:
260:
209:
164:
88:
62:
1671:
1636:
1597:
1550:
1506:
1466:
1430:
1410:
1390:
1370:
1346:
1306:
1239:
1212:
1186:
1163:
1143:
1114:
1077:is at most countable.
1072:
1057:is injective, and so
1052:
1018:
992:
951:
887:
861:
841:
826:contains no point of
821:
780:
751:
727:
591:
571:
545:
506:
466:
420:
374:
346:
313:
311:{\displaystyle [a,b)}
281:
261:
210:
165:
89:
63:
1651:
1616:
1577:
1530:
1486:
1446:
1419:
1399:
1379:
1359:
1319:
1248:
1221:
1195:
1175:
1152:
1123:
1090:
1061:
1027:
1001:
960:
896:
870:
850:
830:
789:
778:{\displaystyle a(x)}
760:
740:
606:
580:
554:
519:
486:
473:totally disconnected
429:
383:
363:
326:
290:
270:
250:
224:upper limit topology
190:
145:
76:
50:
36:lower limit topology
1480:countability axioms
511:must be an at most
104:half-open intervals
1792:Topological spaces
1740:Steen, Lynn Arthur
1687:List of topologies
1666:
1631:
1592:
1545:
1501:
1461:
1425:
1405:
1385:
1365:
1341:
1313:right-sided limits
1301:
1234:
1207:
1181:
1158:
1138:
1109:
1067:
1047:
1013:
997:, parametrized by
987:
946:
882:
856:
836:
816:
775:
746:
722:
683:
586:
566:
540:
501:
461:
415:
369:
341:
308:
276:
256:
205:
160:
122:are real numbers.
94:(generated by the
84:
58:
1767:978-0-486-68735-3
1440:separation axioms
1428:{\displaystyle x}
1408:{\displaystyle f}
1388:{\displaystyle x}
1368:{\displaystyle f}
1184:{\displaystyle L}
1161:{\displaystyle L}
1070:{\displaystyle C}
859:{\displaystyle x}
839:{\displaystyle C}
749:{\displaystyle C}
682:
589:{\displaystyle C}
372:{\displaystyle a}
279:{\displaystyle b}
259:{\displaystyle a}
135:Robert Sorgenfrey
127:topological space
16:(Redirected from
1799:
1778:
1731:
1730:(1977), 483–487.
1721:
1715:
1714:
1703:
1675:
1673:
1672:
1667:
1665:
1664:
1659:
1640:
1638:
1637:
1632:
1630:
1629:
1624:
1601:
1599:
1598:
1593:
1591:
1590:
1585:
1554:
1552:
1551:
1546:
1544:
1543:
1538:
1521:second-countable
1510:
1508:
1507:
1502:
1500:
1499:
1494:
1470:
1468:
1467:
1462:
1460:
1459:
1454:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1350:
1348:
1347:
1342:
1340:
1332:
1310:
1308:
1307:
1302:
1288:
1287:
1269:
1268:
1243:
1241:
1240:
1235:
1233:
1232:
1216:
1214:
1213:
1208:
1190:
1188:
1187:
1182:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1137:
1136:
1131:
1118:
1116:
1115:
1110:
1105:
1104:
1076:
1074:
1073:
1068:
1056:
1054:
1053:
1048:
1046:
1022:
1020:
1019:
1014:
996:
994:
993:
988:
955:
953:
952:
947:
945:
891:
889:
888:
883:
865:
863:
862:
857:
845:
843:
842:
837:
825:
823:
822:
817:
784:
782:
781:
776:
755:
753:
752:
747:
731:
729:
728:
723:
718:
717:
711:
699:
698:
691:
690:
684:
675:
657:
656:
650:
649:
640:
639:
615:
614:
595:
593:
592:
587:
575:
573:
572:
567:
549:
547:
546:
541:
539:
538:
533:
510:
508:
507:
502:
500:
499:
494:
470:
468:
467:
462:
445:
424:
422:
421:
416:
399:
378:
376:
375:
370:
350:
348:
347:
342:
340:
339:
334:
317:
315:
314:
309:
285:
283:
282:
277:
265:
263:
262:
257:
217:Sorgenfrey plane
214:
212:
211:
206:
204:
203:
198:
180:general topology
169:
167:
166:
161:
159:
158:
153:
93:
91:
90:
85:
83:
67:
65:
64:
59:
57:
21:
1807:
1806:
1802:
1801:
1800:
1798:
1797:
1796:
1782:
1781:
1768:
1758:Springer-Verlag
1735:
1734:
1722:
1718:
1705:
1704:
1700:
1695:
1683:
1660:
1655:
1654:
1652:
1649:
1648:
1625:
1620:
1619:
1617:
1614:
1613:
1586:
1581:
1580:
1578:
1575:
1574:
1569:locally compact
1539:
1534:
1533:
1531:
1528:
1527:
1513:first-countable
1495:
1490:
1489:
1487:
1484:
1483:
1455:
1450:
1449:
1447:
1444:
1443:
1420:
1417:
1416:
1400:
1397:
1396:
1380:
1377:
1376:
1360:
1357:
1356:
1336:
1328:
1320:
1317:
1316:
1283:
1279:
1264:
1260:
1249:
1246:
1245:
1228:
1224:
1222:
1219:
1218:
1196:
1193:
1192:
1176:
1173:
1172:
1171:it "approaches
1153:
1150:
1149:
1132:
1127:
1126:
1124:
1121:
1120:
1100:
1096:
1091:
1088:
1087:
1062:
1059:
1058:
1042:
1028:
1025:
1024:
1002:
999:
998:
961:
958:
957:
941:
897:
894:
893:
871:
868:
867:
851:
848:
847:
831:
828:
827:
790:
787:
786:
761:
758:
757:
741:
738:
737:
713:
712:
707:
694:
693:
686:
685:
673:
652:
651:
645:
644:
635:
634:
610:
609:
607:
604:
603:
581:
578:
577:
555:
552:
551:
534:
529:
528:
520:
517:
516:
495:
490:
489:
487:
484:
483:
441:
430:
427:
426:
395:
384:
381:
380:
364:
361:
360:
335:
330:
329:
327:
324:
323:
291:
288:
287:
286:, the interval
271:
268:
267:
251:
248:
247:
236:
199:
194:
193:
191:
188:
187:
154:
149:
148:
146:
143:
142:
131:Sorgenfrey line
79:
77:
74:
73:
53:
51:
48:
47:
28:
23:
22:
15:
12:
11:
5:
1805:
1795:
1794:
1780:
1779:
1766:
1733:
1732:
1716:
1697:
1696:
1694:
1691:
1690:
1689:
1682:
1679:
1678:
1677:
1663:
1658:
1646:
1628:
1623:
1611:
1589:
1584:
1572:
1542:
1537:
1524:
1498:
1493:
1476:
1458:
1453:
1436:
1424:
1404:
1384:
1364:
1339:
1335:
1331:
1327:
1324:
1300:
1297:
1294:
1291:
1286:
1282:
1278:
1275:
1272:
1267:
1263:
1259:
1256:
1253:
1231:
1227:
1206:
1203:
1200:
1180:
1169:if and only if
1157:
1135:
1130:
1108:
1103:
1099:
1095:
1079:
1078:
1066:
1045:
1041:
1038:
1035:
1032:
1012:
1009:
1006:
986:
983:
980:
977:
974:
971:
968:
965:
944:
940:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
881:
878:
875:
855:
835:
815:
812:
809:
806:
803:
800:
797:
794:
774:
771:
768:
765:
745:
734:
733:
732:
721:
716:
710:
706:
703:
697:
689:
681:
678:
672:
669:
666:
663:
660:
655:
648:
643:
638:
633:
630:
627:
624:
621:
618:
613:
598:
597:
585:
565:
562:
559:
537:
532:
527:
524:
498:
493:
480:compact subset
476:
460:
457:
454:
451:
448:
444:
440:
437:
434:
414:
411:
408:
405:
402:
398:
394:
391:
388:
368:
338:
333:
307:
304:
301:
298:
295:
275:
255:
244:
235:
232:
202:
197:
157:
152:
129:is called the
125:The resulting
96:open intervals
82:
56:
26:
9:
6:
4:
3:
2:
1804:
1793:
1790:
1789:
1787:
1777:
1773:
1769:
1763:
1759:
1755:
1751:
1750:
1745:
1741:
1737:
1736:
1729:
1726:
1720:
1712:
1708:
1702:
1698:
1688:
1685:
1684:
1661:
1647:
1644:
1626:
1612:
1609:
1605:
1587:
1573:
1570:
1566:
1562:
1558:
1540:
1525:
1522:
1518:
1514:
1496:
1481:
1477:
1474:
1456:
1441:
1437:
1422:
1402:
1382:
1362:
1354:
1325:
1322:
1314:
1298:
1295:
1292:
1289:
1284:
1280:
1276:
1273:
1270:
1265:
1261:
1257:
1254:
1229:
1225:
1204:
1201:
1198:
1178:
1170:
1155:
1133:
1101:
1097:
1085:
1081:
1080:
1064:
1036:
1033:
1030:
1010:
1007:
1004:
981:
978:
972:
966:
938:
932:
929:
923:
917:
911:
905:
899:
879:
876:
873:
853:
833:
810:
807:
801:
795:
769:
763:
743:
735:
719:
704:
701:
679:
676:
670:
667:
664:
658:
641:
625:
622:
619:
602:
601:
600:
599:
583:
563:
560:
557:
535:
525:
522:
514:
513:countable set
496:
481:
477:
474:
455:
452:
449:
446:
438:
435:
409:
406:
403:
400:
392:
389:
366:
358:
354:
336:
321:
302:
299:
296:
273:
253:
246:For any real
245:
242:
238:
237:
231:
229:
225:
220:
218:
200:
185:
181:
177:
173:
155:
140:
136:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
71:
68:, the set of
45:
41:
37:
33:
19:
1747:
1727:
1724:
1719:
1710:
1701:
1478:In terms of
1438:In terms of
351:(i.e., both
227:
223:
221:
138:
130:
124:
119:
115:
111:
107:
70:real numbers
46:defined on
39:
35:
29:
1643:Baire space
1608:quasimetric
1561:paracompact
846:apart from
379:, the sets
170:. Like the
32:mathematics
1693:References
1604:metrizable
1563:, but not
1519:, but not
1244:such that
550:. Fix an
234:Properties
172:Cantor set
1746:(1995) ,
1565:σ-compact
1517:separable
1334:→
1299:ϵ
1285:α
1277:≤
1262:α
1258:≥
1255:α
1252:∀
1226:α
1199:ϵ
1102:α
1040:→
1008:∈
939:∩
912:∈
877:∈
705:∈
671:−
662:∞
659:−
642:∪
629:∞
561:∈
526:⊆
453:≥
439:∈
393:∈
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1681:See also
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1353:function
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