98:
The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and
English works. In English, some
332:
Eine offene
Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.
1496:
107:, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as
401:
396:
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential
254:
729:
Previously, the term "Gebiet" was occasionally used for such point sets, and it will be used by us in (§ 5, p. 85) with a different meaning.
471:
1381:
1260:
1098:
512:
408:
uses the term "region" to identify an open connected set, and reserves the term "domain" to identify an internally connected,
1491:
324:. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain.
1402:
150:
of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (
1417:
1413:
1397:
796:
773:
725:
Vorher war, für diese
Punktmengen die Bezeichnung "Gebiet" in Gebrauch, die wir (§ 5, S. 85) anders verwenden werden.
996:
355:
338:
1202:
454:
351:
79:
652:
1446:
1392:
210:
637:
491:
460:
147:
1468:
287:
163:
70:
38:
1171:
170:(generalized functions defined on the boundary). Commonly considered types of domains are domains with
1070:
910:: in the second edition of the book, Zane C. Motteler appropriately translates this term as "region".
824:
788:
250:
246:
92:
54:
1472:
1450:
1436:
907:
442:
291:
171:
856:
840:
816:
720:
582:
1409:
984:
704:
545:
423:
128:
1338:
1184:
675:
557:
151:
1140:
683:
412:, each point of which is an accumulation point of interior points, following his former master
1501:
1377:
1256:
1108:
1094:
1050:
792:
749:
643:
628:
590:
537:
279:
62:
529:
1456:
1369:
1353:
1322:
1295:
1248:
1228:
1216:
1159:
1146:
1112:
1086:
1074:
1058:
1030:
1008:
974:
949:
466:
242:
217:
175:
88:
1318:
1291:
1020:
1460:
1357:
1326:
1314:
1299:
1287:
1232:
1220:
1206:
1188:
1150:
1016:
1012:
607:
448:
327:
275:
271:
267:
179:
31:
723:), commenting the just given definition of open set ("offene Menge"), precisely states:-"
1370:
1160:
1031:
1244:
1136:
167:
155:
1249:
1113:
1075:
1059:
979:
962:
950:
1485:
1306:
1279:
405:
366:
229:
159:
393:
were often used informally (sometimes interchangeably) without explicit definition.
1365:
1334:
945:
413:
1087:
257:, the definition of a domain is extended to include any connected open subset of
1124:
648:
409:
194:
132:
867:
to be a connected portion of the complex plane consisting only of inner points.
296:
385:. The rough concept is older. In the 19th and early 20th century, the terms
17:
1054:
397:
363:
238:
50:
382:
213:
is bounded; sometimes smoothness conditions are imposed on its boundary.
58:
988:
661:) called the region an open set and the domain a concatenated open set.
378:
1162:
Functions of a complex variable and some of their applications, vol. 1
354:, the concept of a domain as an open connected set was introduced by
308:
66:
237:. For example, the entire complex plane is a domain, as is the open
811:
informally and apparently interchangeably. By the second edition (
283:
919:
An internally connected set is a set whose interior is connected.
302:
445: – Subset of complex n-space bounded by analytic functions
30:"Region (mathematics)" redirects here. Not to be confused with
1313:. Translated by Motteler, Zane C. (2nd ed.). Springer.
1173:Функции комплексного переменного и некоторые их приложения
362:). In this definition, Carathéodory considers obviously
494:
may be defined on sets that are not topological spaces.
245:, and so forth. Often, a complex domain serves as the
1272:
Applied
Complex Variables for Scientists and Engineers
835:
to be the open region along with its boundary curve. (
143:
is the union of a domain and all of its limit points.
1061:
Functions of a
Complex Variable: Theory and Technique
65:. In particular, it is any non-empty connected open
1352:] (in Italian). Circolo matematico di Catania.
586:
1284:Equazioni alle derivate parziali di tipo ellittico
1077:Introduction to Complex Variables and Applications
898:Precisely, in the first edition of his monograph,
508:
1033:Theory of Functions of a Complex Variable, vol. I
967:Transactions of the American Mathematical Society
906:", meaning literally "field" in a way similar to
1311:Partial Differential Equations of Elliptic Type
812:
319:
1426:Свешников, Алексей; Ти́хонов, Андре́й (1967).
451: – Region with boundary of finite measure
1419:The Theory Of Functions Of A Complex Variable
463: – All numbers between two given numbers
8:
1131:. Prindle, Weber & Schmidt. p. 105.
877:
875:
873:
553:
359:
342:
27:Connected open subset of a topological space
1208:Theorie der reellen Funktionen. Erster Band
671:
455:Hermitian symmetric space#Classical domains
131:of a domain with none, some, or all of its
658:
1346:Lezioni di analisi infinitesimale, vol. I
1195:A course in mathematical analysis, vol. 2
1142:Theory of Functions of a Complex Variable
978:
800:
624:
603:
525:
474: – Geometric theory based on regions
416:: according to this convention, if a set
377:") was occasionally previously used as a
294:, whose extent are called, respectively,
852:
679:
578:
457: – Manifold with inversion symmetry
899:
886:
882:
836:
741:
533:
500:
483:
402:elliptic partial differential equations
369:sets. Hahn also remarks that the word "
929:
851:as a connected portion of the plane. (
564:for a connected open set and the term
228:) is any connected open subset of the
1428:Теория функций комплексной переменной
541:
146:Various degrees of smoothness of the
119:One common convention is to define a
7:
1197:] (in French). Gauthier-Villars.
1190:Cours d'analyse mathématique, tome 2
1158:Fuchs, Boris; Shabat, Boris (1964).
902:, p. 1) uses the Italian term "
752:) alongside the informal expression
716:
700:
687:
653:"246A, Notes 2: complex integration"
1170:Фукс, Борис; Шабат, Борис (1949).
1089:Complex Variables and Applications
1029:Carathéodory, Constantin (1964) .
1007:] (in German). B. G. Teubner.
1001:Vorlesungen über reelle Funktionen
25:
1350:Lessons in infinitesimal analysis
1041:Carathéodory, Constantin (1950).
980:10.1090/S0002-9947-1956-0079100-2
825:interior of a simple closed curve
587:Carrier, Krook & Pearson 1966
197:, i.e., contained in some ball.
1251:Advanced Engineering Mathematics
1237:The Geometry of Domains in Space
1213:Theory of Real Functions, vol. I
1438:Functions of a Complex Variable
1129:Functions of a Complex Variable
610:), who does not require that a
552:for the domain of a function; (
472:Whitehead's point-free geometry
109:non-empty connected open subset
1497:Partial differential equations
1339:"Parte Prima – La Derivazione"
1115:Foundations of Modern Analysis
727:" (Free English translation:-"
509:Sveshnikov & Tikhonov 1978
166:on the boundary and spaces of
123:as a connected open set but a
1:
1391:Solomentsev, Evgeny (2001) ,
1376:(2nd ed.). McGraw-Hill.
1215:] (in German). Springer.
1093:(2nd ed.). McGraw-Hill.
87:. A connected open subset of
1081:(1st ed.). McGraw-Hill.
795:). The first edition of the
748:informally throughout (e.g.
631:) generally uses the phrase
1474:A Course Of Modern Analysis
1455:(1st ed.). Cambridge.
1452:A Course Of Modern Analysis
1398:Encyclopedia of Mathematics
813:Whittaker & Watson 1915
568:for a connected closed set.
91:is frequently used for the
1518:
1477:(2nd ed.). Cambridge.
1005:Lectures on real functions
961:Bremermann, H. J. (1956).
908:its meaning in agriculture
201:is defined similarly. An
29:
1372:Real and Complex Analysis
1025:Reprinted 1968 (Chelsea).
255:several complex variables
1435:Townsend, Edgar (1915).
1286:(in Italian). Springer.
1179:(in Russian). Физматгиз.
1085:Churchill, Ruel (1960).
1057:; Pearson, Carl (1966).
1045:(in German). Birkhäuser.
1037:(2nd ed.). Chelsea.
997:Carathéodory, Constantin
186:boundary, and so forth.
80:complex coordinate space
1424:English translation of
1270:Kwok, Yue-Kuen (2002).
1255:(3rd ed.). Wiley.
1168:English translation of
1039:English translation of
672:Fuchs & Shabat 1964
356:Constantin Carathéodory
339:Constantin Carathéodory
461:Interval (mathematics)
348:
331:
1492:Mathematical analysis
614:be connected or open.
422:is a region then its
99:authors use the term
71:real coordinate space
39:mathematical analysis
1430:(in Russian). Наука.
797:influential textbook
635:, but later defines
548:) reserves the term
358:in his famous book (
251:holomorphic function
247:domain of definition
193:is a domain that is
103:, some use the term
93:domain of a function
1467:Whittaker, Edmund;
1410:Sveshnikov, Aleksei
1043:Functionentheorie I
963:"Complex Convexity"
443:Analytic polyhedron
881:See (Miranda
772:to be the largest
760:, and defines the
633:open connected set
253:. In the study of
209:is a domain whose
176:Lipschitz boundary
1447:Whittaker, Edmund
1119:. Academic Press.
803:) uses the terms
554:Carathéodory 1964
360:Carathéodory 1918
343:Carathéodory 1918
280:three-dimensional
158:), properties of
63:topological space
16:(Redirected from
1509:
1478:
1464:
1442:
1431:
1423:
1414:Tikhonov, Andrey
1405:
1387:
1375:
1361:
1343:
1330:
1303:
1275:
1266:
1254:
1240:
1224:
1198:
1185:Goursat, Édouard
1180:
1178:
1167:
1165:
1154:
1132:
1120:
1118:
1104:
1092:
1082:
1080:
1066:
1064:
1046:
1038:
1036:
1024:
992:
982:
957:
955:
952:Complex Analysis
933:
926:
920:
917:
911:
896:
890:
879:
868:
786:
782:
776:
771:
767:
757:
744:) uses the term
738:
732:
721:p. 61 footnote 3
714:
708:
705:p. 85 footnote 1
697:
691:
668:
662:
656:
638:simply connected
621:
615:
600:
594:
575:
569:
560:) uses the term
522:
516:
505:
495:
488:
467:Lipschitz domain
432:
430:
421:
346:
316:Historical notes
268:Euclidean spaces
262:
243:upper half-plane
236:
218:complex analysis
184:
162:, and to define
89:coordinate space
86:
77:
21:
1517:
1516:
1512:
1511:
1510:
1508:
1507:
1506:
1482:
1481:
1466:
1465:
1445:
1434:
1425:
1408:
1390:
1384:
1364:
1341:
1333:
1305:
1278:
1269:
1263:
1245:Kreyszig, Erwin
1243:
1227:
1201:
1183:
1176:
1169:
1157:
1137:Forsyth, Andrew
1135:
1123:
1109:Dieudonné, Jean
1107:
1101:
1084:
1083:
1071:Churchill, Ruel
1069:
1051:Carrier, George
1049:
1040:
1028:
995:
960:
944:
941:
936:
927:
923:
918:
914:
897:
893:
880:
871:
784:
780:
774:
769:
768:for a function
765:
755:
739:
735:
715:
711:
698:
694:
690:, §1.4, p. 23.)
669:
665:
659:Bremermann 1956
647:
629:§3.19 pp. 64–67
622:
618:
601:
597:
576:
572:
523:
519:
506:
502:
498:
489:
485:
481:
449:Caccioppoli set
439:
428:
426:
417:
347:
337:
318:
258:
232:
207:external domain
203:exterior domain
180:
152:Green's theorem
117:
82:
73:
35:
32:Macbeath region
28:
23:
22:
15:
12:
11:
5:
1515:
1513:
1505:
1504:
1499:
1494:
1484:
1483:
1480:
1479:
1469:Watson, George
1443:
1432:
1406:
1388:
1382:
1362:
1331:
1307:Miranda, Carlo
1304:Translated as
1280:Miranda, Carlo
1276:
1267:
1261:
1241:
1229:Krantz, Steven
1225:
1199:
1181:
1155:
1133:
1121:
1105:
1099:
1067:
1065:. McGraw-Hill.
1047:
1026:
993:
958:
956:. McGraw-Hill.
940:
937:
935:
934:
932:, p. 66).
921:
912:
891:
889:, p. 2).
885:, p. 1,
869:
801:Whittaker 1902
733:
709:
692:
670:For instance (
663:
625:Dieudonné 1960
623:For instance (
616:
604:Churchill 1960
602:For instance (
595:
577:For instance (
570:
530:§1.9 pp. 16–17
526:Churchill 1948
524:For instance (
517:
513:§1.3 pp. 21–22
507:For instance (
499:
497:
496:
482:
480:
477:
476:
475:
469:
464:
458:
452:
446:
438:
435:
345:, p. 222)
335:
317:
314:
222:complex domain
199:Bounded region
191:bounded domain
160:Sobolev spaces
156:Stokes theorem
116:
113:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1514:
1503:
1500:
1498:
1495:
1493:
1490:
1489:
1487:
1476:
1475:
1470:
1462:
1458:
1454:
1453:
1448:
1444:
1440:
1439:
1433:
1429:
1421:
1420:
1415:
1411:
1407:
1404:
1400:
1399:
1394:
1389:
1385:
1383:9780070542334
1379:
1374:
1373:
1367:
1366:Rudin, Walter
1363:
1359:
1355:
1351:
1347:
1340:
1336:
1335:Picone, Mauro
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1268:
1264:
1262:9780471507284
1258:
1253:
1252:
1246:
1242:
1239:. Birkhäuser.
1238:
1234:
1233:Parks, Harold
1230:
1226:
1222:
1218:
1214:
1210:
1209:
1204:
1200:
1196:
1192:
1191:
1186:
1182:
1175:
1174:
1164:
1163:
1156:
1152:
1148:
1145:. Cambridge.
1144:
1143:
1138:
1134:
1130:
1126:
1122:
1117:
1116:
1110:
1106:
1102:
1100:9780070108530
1096:
1091:
1090:
1079:
1078:
1072:
1068:
1063:
1062:
1056:
1052:
1048:
1044:
1035:
1034:
1027:
1022:
1018:
1014:
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1006:
1002:
998:
994:
990:
986:
981:
976:
972:
968:
964:
959:
954:
953:
947:
946:Ahlfors, Lars
943:
942:
938:
931:
925:
922:
916:
913:
909:
905:
901:
900:Miranda (1955
895:
892:
888:
884:
878:
876:
874:
870:
866:
862:
858:
854:
853:Townsend 1915
850:
846:
842:
838:
834:
830:
829:closed region
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
778:
777:-neighborhood
763:
759:
751:
747:
743:
740:For example (
737:
734:
730:
726:
722:
718:
713:
710:
706:
702:
696:
693:
689:
685:
681:
680:Kreyszig 1972
677:
673:
667:
664:
660:
654:
650:
645:
641:
639:
634:
630:
626:
620:
617:
613:
609:
605:
599:
596:
592:
588:
584:
580:
579:Townsend 1915
574:
571:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
521:
518:
514:
510:
504:
501:
493:
487:
484:
478:
473:
470:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
440:
436:
434:
433:is a domain.
431:
425:
420:
415:
411:
407:
406:Carlo Miranda
403:
399:
394:
392:
388:
384:
380:
376:
372:
368:
365:
361:
357:
353:
350:According to
344:
340:
334:
333:
329:
325:
323:
315:
313:
311:
310:
305:
304:
299:
298:
293:
289:
285:
281:
277:
273:
269:
264:
261:
256:
252:
248:
244:
240:
235:
231:
230:complex plane
227:
223:
219:
214:
212:
208:
204:
200:
196:
192:
187:
185:
183:
177:
173:
169:
165:
161:
157:
153:
149:
144:
142:
141:closed domain
138:
137:closed region
134:
130:
126:
122:
114:
112:
110:
106:
102:
96:
94:
90:
85:
81:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
18:Closed region
1473:
1451:
1437:
1427:
1418:
1396:
1371:
1349:
1345:
1310:
1283:
1274:. Cambridge.
1271:
1250:
1236:
1212:
1207:
1194:
1189:
1172:
1161:
1141:
1128:
1125:Eves, Howard
1114:
1088:
1076:
1060:
1042:
1032:
1004:
1000:
973:(1): 17–51.
970:
966:
951:
924:
915:
903:
894:
864:
860:
859:) defines a
848:
844:
837:Goursat 1905
832:
828:
820:
819:) define an
817:§3.21, p. 44
808:
804:
761:
754:part of the
753:
745:
742:Forsyth 1893
736:
728:
724:
712:
695:
684:§11.1 p. 469
676:§6 pp. 22–23
666:
649:Tao, Terence
636:
632:
619:
611:
598:
573:
565:
561:
549:
546:§10.1 p. 213
534:Ahlfors 1953
520:
503:
486:
427:
418:
414:Mauro Picone
395:
390:
386:
374:
370:
349:
326:
321:
320:
307:
301:
295:
282:regions are
265:
259:
233:
225:
221:
215:
206:
202:
198:
190:
188:
181:
145:
140:
136:
133:limit points
124:
120:
118:
108:
104:
100:
97:
83:
74:
46:
42:
36:
1166:. Pergamon.
930:Picone 1923
841:§262, p. 10
821:open region
789:holomorphic
764:of a point
644:§9.7 p. 215
410:perfect set
241:, the open
224:(or simply
115:Conventions
1486:Categories
1461:33.0390.01
1358:49.0172.07
1327:0198.14101
1300:0065.08503
1221:48.0261.09
1203:Hahn, Hans
1151:25.0652.01
1055:Krook, Max
1013:46.0376.12
939:References
857:§10, p. 20
843:) defines
823:to be the
793:§32, p. 52
750:§16, p. 21
717:Hahn (1921
608:§1.9 p. 17
591:§2.2 p. 32
583:§10, p. 20
542:Rudin 1974
538:§2.2 p. 58
398:monographs
322:Definition
211:complement
174:boundary,
172:continuous
1403:EMS Press
1368:(1974) .
1247:(1972) .
783:in which
701:Hahn 1921
688:Kwok 2002
657:, also, (
566:continuum
492:functions
490:However,
364:non-empty
352:Hans Hahn
239:unit disk
55:connected
51:non-empty
1502:Topology
1471:(1915).
1449:(1902).
1416:(1978).
1393:"Domain"
1337:(1923).
1309:(1970).
1282:(1955).
1235:(1999).
1205:(1921).
1187:(1905).
1139:(1893).
1127:(1966).
1111:(1960).
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