719:
1298:
232:(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.
594:
943:
1338:
458:
412:
537:
1044:
1663:
1628:
1589:
1542:
1498:
1458:
1135:
498:
338:
202:
157:
1204:
1106:
81:
55:
1231:
984:
813:
1010:
879:
563:
305:
772:
1422:
1402:
1382:
1362:
1178:
1155:
1064:
853:
833:
743:
583:
366:
273:
253:
1236:
714:{\displaystyle {\bigl \{}[x,+\infty ){\bigr \}}\cup {\Bigl \{}{\bigl (}-\infty ,x-{\tfrac {1}{n}}{\bigr )}\,{\Big |}\,n\in \mathbb {N} {\Bigr \}}.}
1754:
884:
1307:
1595:, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a
1461:
417:
371:
167:, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in
507:
1780:
1737:
1424:
when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
1015:
1596:
1639:
1604:
1565:
1518:
1474:
1434:
1111:
474:
314:
178:
133:
1183:
1078:
64:
38:
1209:
1732:
1509:
1341:
948:
777:
1501:
1468:
989:
858:
542:
1764:
461:
278:
164:
748:
8:
1742:
1675:
1407:
1387:
1367:
1347:
1163:
1140:
1049:
838:
818:
728:
568:
351:
258:
238:
92:
1553:
1750:
1728:
1712:
Adam Emeryk, Władysław Kulpa. The
Sorgenfrey line has no connected compactification.
1428:
1301:
123:
115:
32:
1545:
1384:(when the codomain carries the standard topology) is the same as the usual limit of
745:
is compact, this cover has a finite subcover, and hence there exists a real number
205:
168:
88:
1695:
87:) and has a number of interesting properties. It is the topology generated by the
1760:
1746:
1557:
1505:
1293:{\displaystyle \forall \alpha \geq \alpha _{0}:L\leq x_{\alpha }<L+\epsilon }
1157:
1072:
229:
1071:
The name "lower limit topology" comes from the following fact: a sequence (or
1774:
501:
468:
172:
84:
58:
1631:
1549:
20:
1592:
345:
308:
160:
341:
204:
with itself is also a useful counterexample, known as the
1696:"general topology - The Sorgenfrey line is a Baire Space"
460:
are also clopen. This shows that the
Sorgenfrey line is
663:
1642:
1607:
1568:
1521:
1477:
1437:
1410:
1390:
1370:
1350:
1310:
1239:
1212:
1186:
1166:
1143:
1114:
1081:
1052:
1018:
992:
951:
887:
861:
841:
821:
780:
751:
731:
597:
571:
545:
510:
477:
420:
374:
354:
317:
281:
261:
241:
181:
136:
67:
41:
504:. To see this, consider a non-empty compact subset
1657:
1622:
1583:
1536:
1492:
1452:
1416:
1396:
1376:
1356:
1332:
1292:
1225:
1198:
1172:
1149:
1129:
1100:
1058:
1038:
1004:
978:
937:
873:
847:
827:
807:
766:
737:
713:
577:
557:
531:
492:
452:
406:
360:
332:
299:
267:
247:
196:
151:
75:
49:
938:{\displaystyle q(x)\in (a(x),x]\cap \mathbb {Q} }
703:
684:
635:
1772:
1300:. The Sorgenfrey line can thus be used to study
61:; it is different from the standard topology on
1665:does not have any connected compactifications.
1333:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1745:reprint of 1978 ed.), Berlin, New York:
676:
642:
625:
600:
453:{\displaystyle \{x\in \mathbb {R} :x\geq a\}}
211:In complete analogy, one can also define the
1727:
447:
421:
407:{\displaystyle \{x\in \mathbb {R} :x<a\}}
401:
375:
532:{\displaystyle C\subseteq \mathbb {R} _{l}}
1344:, then the ordinary right-sided limit of
1645:
1610:
1571:
1524:
1480:
1440:
1326:
1318:
1117:
1032:
931:
697:
689:
681:
519:
480:
431:
385:
320:
184:
139:
69:
43:
565:, consider the following open cover of
1773:
1012:, are pairwise disjoint, the function
1515:In terms of compactness properties,
1180:from the right", meaning for every
1039:{\displaystyle q:C\to \mathbb {Q} }
13:
1240:
650:
617:
14:
1792:
29:right half-open interval topology
1658:{\displaystyle \mathbb {R} _{l}}
1623:{\displaystyle \mathbb {R} _{l}}
1584:{\displaystyle \mathbb {R} _{l}}
1537:{\displaystyle \mathbb {R} _{l}}
1493:{\displaystyle \mathbb {R} _{l}}
1462:perfectly normal Hausdorff space
1453:{\displaystyle \mathbb {R} _{l}}
1130:{\displaystyle \mathbb {R} _{l}}
493:{\displaystyle \mathbb {R} _{l}}
333:{\displaystyle \mathbb {R} _{l}}
217:left half-open interval topology
197:{\displaystyle \mathbb {R} _{l}}
152:{\displaystyle \mathbb {R} _{l}}
881:. Now choose a rational number
1706:
1688:
1322:
1199:{\displaystyle \epsilon >0}
1095:
1082:
1028:
973:
964:
958:
952:
924:
915:
909:
903:
897:
891:
802:
793:
787:
781:
761:
755:
620:
605:
294:
282:
1:
1681:
1101:{\displaystyle (x_{\alpha })}
348:). Furthermore, for all real
222:
228:The lower limit topology is
76:{\displaystyle \mathbb {R} }
50:{\displaystyle \mathbb {R} }
16:Topology on the real numbers
7:
1738:Counterexamples in Topology
1714:Comm. Math. Univ. Carolinae
1669:
1226:{\displaystyle \alpha _{0}}
10:
1797:
1700:Mathematics Stack Exchange
130:and is sometimes written
979:{\displaystyle (a(x),x]}
808:{\displaystyle (a(x),x]}
1137:converges to the limit
855:. This is true for all
774:such that the interval
1733:Seebach, J. Arthur Jr.
1659:
1624:
1585:
1538:
1494:
1454:
1418:
1398:
1378:
1358:
1334:
1294:
1227:
1206:there exists an index
1200:
1174:
1151:
1131:
1102:
1060:
1040:
1006:
1005:{\displaystyle x\in C}
980:
945:. Since the intervals
939:
875:
874:{\displaystyle x\in C}
849:
829:
809:
768:
739:
715:
579:
559:
558:{\displaystyle x\in C}
533:
494:
454:
408:
362:
334:
301:
269:
249:
198:
153:
77:
51:
1660:
1625:
1586:
1539:
1495:
1455:
1419:
1399:
1379:
1359:
1335:
1295:
1228:
1201:
1175:
1152:
1132:
1103:
1066:is at most countable.
1061:
1046:is injective, and so
1041:
1007:
981:
940:
876:
850:
830:
815:contains no point of
810:
769:
740:
716:
580:
560:
534:
495:
455:
409:
363:
335:
302:
300:{\displaystyle [a,b)}
270:
250:
199:
154:
78:
52:
1640:
1605:
1566:
1519:
1475:
1435:
1408:
1388:
1368:
1348:
1308:
1237:
1210:
1184:
1164:
1141:
1112:
1079:
1050:
1016:
990:
949:
885:
859:
839:
819:
778:
767:{\displaystyle a(x)}
749:
729:
595:
569:
543:
508:
475:
462:totally disconnected
418:
372:
352:
315:
279:
259:
239:
213:upper limit topology
179:
134:
65:
39:
25:lower limit topology
1469:countability axioms
500:must be an at most
93:half-open intervals
1781:Topological spaces
1729:Steen, Lynn Arthur
1676:List of topologies
1655:
1620:
1581:
1534:
1490:
1450:
1414:
1394:
1374:
1354:
1330:
1302:right-sided limits
1290:
1223:
1196:
1170:
1147:
1127:
1098:
1056:
1036:
1002:
986:, parametrized by
976:
935:
871:
845:
825:
805:
764:
735:
711:
672:
575:
555:
529:
490:
450:
404:
358:
330:
297:
265:
245:
194:
149:
111:are real numbers.
83:(generated by the
73:
47:
1756:978-0-486-68735-3
1429:separation axioms
1417:{\displaystyle x}
1397:{\displaystyle f}
1377:{\displaystyle x}
1357:{\displaystyle f}
1173:{\displaystyle L}
1150:{\displaystyle L}
1059:{\displaystyle C}
848:{\displaystyle x}
828:{\displaystyle C}
738:{\displaystyle C}
671:
578:{\displaystyle C}
361:{\displaystyle a}
268:{\displaystyle b}
248:{\displaystyle a}
124:Robert Sorgenfrey
116:topological space
1788:
1767:
1720:
1719:(1977), 483–487.
1710:
1704:
1703:
1692:
1664:
1662:
1661:
1656:
1654:
1653:
1648:
1629:
1627:
1626:
1621:
1619:
1618:
1613:
1590:
1588:
1587:
1582:
1580:
1579:
1574:
1543:
1541:
1540:
1535:
1533:
1532:
1527:
1510:second-countable
1499:
1497:
1496:
1491:
1489:
1488:
1483:
1459:
1457:
1456:
1451:
1449:
1448:
1443:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1339:
1337:
1336:
1331:
1329:
1321:
1299:
1297:
1296:
1291:
1277:
1276:
1258:
1257:
1232:
1230:
1229:
1224:
1222:
1221:
1205:
1203:
1202:
1197:
1179:
1177:
1176:
1171:
1156:
1154:
1153:
1148:
1136:
1134:
1133:
1128:
1126:
1125:
1120:
1107:
1105:
1104:
1099:
1094:
1093:
1065:
1063:
1062:
1057:
1045:
1043:
1042:
1037:
1035:
1011:
1009:
1008:
1003:
985:
983:
982:
977:
944:
942:
941:
936:
934:
880:
878:
877:
872:
854:
852:
851:
846:
834:
832:
831:
826:
814:
812:
811:
806:
773:
771:
770:
765:
744:
742:
741:
736:
720:
718:
717:
712:
707:
706:
700:
688:
687:
680:
679:
673:
664:
646:
645:
639:
638:
629:
628:
604:
603:
584:
582:
581:
576:
564:
562:
561:
556:
538:
536:
535:
530:
528:
527:
522:
499:
497:
496:
491:
489:
488:
483:
459:
457:
456:
451:
434:
413:
411:
410:
405:
388:
367:
365:
364:
359:
339:
337:
336:
331:
329:
328:
323:
306:
304:
303:
298:
274:
272:
271:
266:
254:
252:
251:
246:
206:Sorgenfrey plane
203:
201:
200:
195:
193:
192:
187:
169:general topology
158:
156:
155:
150:
148:
147:
142:
82:
80:
79:
74:
72:
56:
54:
53:
48:
46:
1796:
1795:
1791:
1790:
1789:
1787:
1786:
1785:
1771:
1770:
1757:
1747:Springer-Verlag
1724:
1723:
1711:
1707:
1694:
1693:
1689:
1684:
1672:
1649:
1644:
1643:
1641:
1638:
1637:
1614:
1609:
1608:
1606:
1603:
1602:
1575:
1570:
1569:
1567:
1564:
1563:
1558:locally compact
1528:
1523:
1522:
1520:
1517:
1516:
1502:first-countable
1484:
1479:
1478:
1476:
1473:
1472:
1444:
1439:
1438:
1436:
1433:
1432:
1409:
1406:
1405:
1389:
1386:
1385:
1369:
1366:
1365:
1349:
1346:
1345:
1325:
1317:
1309:
1306:
1305:
1272:
1268:
1253:
1249:
1238:
1235:
1234:
1217:
1213:
1211:
1208:
1207:
1185:
1182:
1181:
1165:
1162:
1161:
1160:it "approaches
1142:
1139:
1138:
1121:
1116:
1115:
1113:
1110:
1109:
1089:
1085:
1080:
1077:
1076:
1051:
1048:
1047:
1031:
1017:
1014:
1013:
991:
988:
987:
950:
947:
946:
930:
886:
883:
882:
860:
857:
856:
840:
837:
836:
820:
817:
816:
779:
776:
775:
750:
747:
746:
730:
727:
726:
702:
701:
696:
683:
682:
675:
674:
662:
641:
640:
634:
633:
624:
623:
599:
598:
596:
593:
592:
570:
567:
566:
544:
541:
540:
523:
518:
517:
509:
506:
505:
484:
479:
478:
476:
473:
472:
430:
419:
416:
415:
384:
373:
370:
369:
353:
350:
349:
324:
319:
318:
316:
313:
312:
280:
277:
276:
275:, the interval
260:
257:
256:
240:
237:
236:
225:
188:
183:
182:
180:
177:
176:
143:
138:
137:
135:
132:
131:
120:Sorgenfrey line
68:
66:
63:
62:
42:
40:
37:
36:
17:
12:
11:
5:
1794:
1784:
1783:
1769:
1768:
1755:
1722:
1721:
1705:
1686:
1685:
1683:
1680:
1679:
1678:
1671:
1668:
1667:
1666:
1652:
1647:
1635:
1617:
1612:
1600:
1578:
1573:
1561:
1531:
1526:
1513:
1487:
1482:
1465:
1447:
1442:
1425:
1413:
1393:
1373:
1353:
1328:
1324:
1320:
1316:
1313:
1289:
1286:
1283:
1280:
1275:
1271:
1267:
1264:
1261:
1256:
1252:
1248:
1245:
1242:
1220:
1216:
1195:
1192:
1189:
1169:
1158:if and only if
1146:
1124:
1119:
1097:
1092:
1088:
1084:
1068:
1067:
1055:
1034:
1030:
1027:
1024:
1021:
1001:
998:
995:
975:
972:
969:
966:
963:
960:
957:
954:
933:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
870:
867:
864:
844:
824:
804:
801:
798:
795:
792:
789:
786:
783:
763:
760:
757:
754:
734:
723:
722:
721:
710:
705:
699:
695:
692:
686:
678:
670:
667:
661:
658:
655:
652:
649:
644:
637:
632:
627:
622:
619:
616:
613:
610:
607:
602:
587:
586:
574:
554:
551:
548:
526:
521:
516:
513:
487:
482:
469:compact subset
465:
449:
446:
443:
440:
437:
433:
429:
426:
423:
403:
400:
397:
394:
391:
387:
383:
380:
377:
357:
327:
322:
296:
293:
290:
287:
284:
264:
244:
233:
224:
221:
191:
186:
146:
141:
118:is called the
114:The resulting
85:open intervals
71:
45:
15:
9:
6:
4:
3:
2:
1793:
1782:
1779:
1778:
1776:
1766:
1762:
1758:
1752:
1748:
1744:
1740:
1739:
1734:
1730:
1726:
1725:
1718:
1715:
1709:
1701:
1697:
1691:
1687:
1677:
1674:
1673:
1650:
1636:
1633:
1615:
1601:
1598:
1594:
1576:
1562:
1559:
1555:
1551:
1547:
1529:
1514:
1511:
1507:
1503:
1485:
1470:
1466:
1463:
1445:
1430:
1426:
1411:
1391:
1371:
1351:
1343:
1314:
1311:
1303:
1287:
1284:
1281:
1278:
1273:
1269:
1265:
1262:
1259:
1254:
1250:
1246:
1243:
1218:
1214:
1193:
1190:
1187:
1167:
1159:
1144:
1122:
1090:
1086:
1074:
1070:
1069:
1053:
1025:
1022:
1019:
999:
996:
993:
970:
967:
961:
955:
927:
921:
918:
912:
906:
900:
894:
888:
868:
865:
862:
842:
822:
799:
796:
790:
784:
758:
752:
732:
724:
708:
693:
690:
668:
665:
659:
656:
653:
647:
630:
614:
611:
608:
591:
590:
589:
588:
572:
552:
549:
546:
524:
514:
511:
503:
502:countable set
485:
470:
466:
463:
444:
441:
438:
435:
427:
424:
398:
395:
392:
389:
381:
378:
355:
347:
343:
325:
310:
291:
288:
285:
262:
242:
235:For any real
234:
231:
227:
226:
220:
218:
214:
209:
207:
189:
174:
170:
166:
162:
144:
129:
125:
121:
117:
112:
110:
106:
102:
98:
94:
90:
86:
60:
57:, the set of
34:
30:
26:
22:
1736:
1716:
1713:
1708:
1699:
1690:
1467:In terms of
1427:In terms of
340:(i.e., both
216:
212:
210:
127:
119:
113:
108:
104:
100:
96:
59:real numbers
35:defined on
28:
24:
18:
1632:Baire space
1597:quasimetric
1550:paracompact
835:apart from
368:, the sets
159:. Like the
21:mathematics
1682:References
1593:metrizable
1552:, but not
1508:, but not
1233:such that
539:. Fix an
223:Properties
161:Cantor set
1735:(1995) ,
1554:σ-compact
1506:separable
1323:→
1288:ϵ
1274:α
1266:≤
1251:α
1247:≥
1244:α
1241:∀
1215:α
1188:ϵ
1091:α
1029:→
997:∈
928:∩
901:∈
866:∈
694:∈
660:−
651:∞
648:−
631:∪
618:∞
550:∈
515:⊆
442:≥
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