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
328:
337:
Eine offene
Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.
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:
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential
734:
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.
1501:
517:
406:
413:
uses the term "region" to identify an open connected set, and reserves the term "domain" to identify an internally connected,
254:
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.
778:
150:
of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (
1422:
476:
459:
1386:
1265:
1103:
730:
Vorher war, für diese
Punktmengen die Bezeichnung "Gebiet" in Gebrauch, die wir (§ 5, S. 85) anders verwenden werden.
1496:
657:
1407:
17:
1397:
1418:
1402:
801:
1176:
170:(generalized functions defined on the boundary). Commonly considered types of domains are domains with
1001:
360:
343:
1207:
915:: in the second edition of the book, Zane C. Motteler appropriately translates this term as "region".
356:
1477:
1455:
1441:
79:
861:
845:
821:
725:
587:
1451:
709:
550:
210:
1343:
680:
562:
1145:
688:
642:
496:
465:
417:, each point of which is an accumulation point of interior points, following his former master
147:
797:
754:
648:
633:
595:
542:
1473:
534:
287:
163:
70:
38:
1374:
1253:
1164:
1117:
1091:
1079:
1063:
1035:
954:
1323:
1296:
1075:
1025:
829:
793:
250:
246:
92:
54:
1465:
1362:
1331:
1304:
1225:
1211:
1193:
1155:
1017:
612:
8:
912:
728:), commenting the just given definition of open set ("offene Menge"), precisely states:-"
447:
291:
171:
1375:
1165:
1036:
1414:
989:
428:
151:
128:
1254:
1189:
1118:
1080:
1064:
984:
967:
955:
1506:
1382:
1261:
1099:
1055:
279:
62:
1113:
398:
were often used informally (sometimes interchangeably) without explicit definition.
1461:
1358:
1327:
1300:
1233:
1221:
1151:
1013:
979:
471:
242:
217:
175:
88:
1092:
1319:
1292:
1237:
1021:
453:
332:
275:
271:
267:
257:, the definition of a domain is extended to include any connected open subset of
179:
31:
1249:
1141:
872:
to be a connected portion of the complex plane consisting only of inner points.
167:
155:
1490:
1311:
1284:
410:
390:. The rough concept is older. In the 19th and early 20th century, the terms
371:
229:
159:
1370:
1339:
950:
418:
1129:
653:
414:
213:
is bounded; sometimes smoothness conditions are imposed on its boundary.
194:
132:
993:
666:) called the region an open set and the domain a concatenated open set.
296:
1167:
Functions of a complex variable and some of their applications, vol. 1
1059:
402:
368:
359:, the concept of a domain as an open connected set was introduced by
238:
50:
237:. For example, the entire complex plane is a domain, as is the open
387:
58:
816:
informally and apparently interchangeably. By the second edition (
924:
An internally connected set is a set whose interior is connected.
383:
450: – Subset of complex n-space bounded by analytic functions
308:
66:
30:"Region (mathematics)" redirects here. Not to be confused with
283:
1318:. Translated by Motteler, Zane C. (2nd ed.). Springer.
302:
1178:Функции комплексного переменного и некоторые их приложения
367:). In this definition, Carathéodory considers obviously
499:
may be defined on sets that are not topological spaces.
245:, and so forth. Often, a complex domain serves as the
1277:
Applied
Complex Variables for Scientists and Engineers
840:
to be the open region along with its boundary curve. (
143:
is the union of a domain and all of its limit points.
1066:
Functions of a
Complex Variable: Theory and Technique
65:. In particular, it is any non-empty connected open
1357:] (in Italian). Circolo matematico di Catania.
1054:
591:
1413:
1289:Equazioni alle derivate parziali di tipo ellittico
1082:Introduction to Complex Variables and Applications
903:Precisely, in the first edition of his monograph,
513:
1430:
1038:Theory of Functions of a Complex Variable, vol. I
972:Transactions of the American Mathematical Society
911:", meaning literally "field" in a way similar to
1488:
1471:
1316:Partial Differential Equations of Elliptic Type
817:
1431:Свешников, Алексей; Ти́хонов, Андре́й (1967).
456: – Region with boundary of finite measure
1424:The Theory Of Functions Of A Complex Variable
468: – All numbers between two given numbers
1136:. Prindle, Weber & Schmidt. p. 105.
1045:
1033:
1000:
882:
880:
878:
558:
364:
347:
27:Connected open subset of a topological space
1395:
1232:
1213:Theorie der reellen Funktionen. Erster Band
1162:
676:
460:Hermitian symmetric space#Classical domains
131:of a domain with none, some, or all of its
965:
663:
1450:
1351:Lezioni di analisi infinitesimale, vol. I
1200:A course in mathematical analysis, vol. 2
1174:
1147:Theory of Functions of a Complex Variable
1112:
1089:
1074:
983:
875:
805:
629:
608:
530:
479: – Geometric theory based on regions
421:: according to this convention, if a set
382:") was occasionally previously used as a
294:, whose extent are called, respectively,
1439:
1248:
857:
684:
583:
462: – Manifold with inversion symmetry
1310:
1283:
1188:
1140:
949:
904:
891:
887:
841:
746:
538:
407:elliptic partial differential equations
374:sets. Hahn also remarks that the word "
14:
1489:
1338:
934:
856:as a connected portion of the plane. (
569:for a connected open set and the term
228:) is any connected open subset of the
1433:Теория функций комплексной переменной
1369:
546:
146:Various degrees of smoothness of the
119:One common convention is to define a
1274:
1206:
1202:] (in French). Gauthier-Villars.
1195:Cours d'analyse mathématique, tome 2
1163:Fuchs, Boris; Shabat, Boris (1964).
1128:
907:, p. 1) uses the Italian term "
757:) alongside the informal expression
721:
705:
692:
658:"246A, Notes 2: complex integration"
652:
315:
24:
1175:Фукс, Борис; Шабат, Борис (1949).
1094:Complex Variables and Applications
1034:Carathéodory, Constantin (1964) .
1012:] (in German). B. G. Teubner.
1006:Vorlesungen über reelle Funktionen
25:
1518:
1355:Lessons in infinitesimal analysis
1046:Carathéodory, Constantin (1950).
985:10.1090/S0002-9947-1956-0079100-2
830:interior of a simple closed curve
592:Carrier, Krook & Pearson 1966
197:, i.e., contained in some ball.
1256:Advanced Engineering Mathematics
1242:The Geometry of Domains in Space
1218:Theory of Real Functions, vol. I
1443:Functions of a Complex Variable
1134:Functions of a Complex Variable
927:
918:
897:
739:
615:), who does not require that a
557:for the domain of a function; (
477:Whitehead's point-free geometry
109:non-empty connected open subset
1502:Partial differential equations
1344:"Parte Prima – La Derivazione"
1120:Foundations of Modern Analysis
732:" (Free English translation:-"
715:
698:
669:
622:
601:
576:
523:
514:Sveshnikov & Tikhonov 1978
506:
489:
166:on the boundary and spaces of
123:as a connected open set but a
114:
13:
1:
1396:Solomentsev, Evgeny (2001) ,
1381:(2nd ed.). McGraw-Hill.
1220:] (in German). Springer.
1098:(2nd ed.). McGraw-Hill.
943:
87:. A connected open subset of
1086:(1st ed.). McGraw-Hill.
800:). The first edition of the
753:informally throughout (e.g.
636:) generally uses the phrase
7:
1479:A Course Of Modern Analysis
1460:(1st ed.). Cambridge.
1457:A Course Of Modern Analysis
1403:Encyclopedia of Mathematics
818:Whittaker & Watson 1915
573:for a connected closed set.
441:
91:is frequently used for the
10:
1523:
1482:(2nd ed.). Cambridge.
1010:Lectures on real functions
966:Bremermann, H. J. (1956).
913:its meaning in agriculture
201:is defined similarly. An
29:
1377:Real and Complex Analysis
1030:Reprinted 1968 (Chelsea).
255:several complex variables
1440:Townsend, Edgar (1915).
1291:(in Italian). Springer.
1184:(in Russian). Физматгиз.
1090:Churchill, Ruel (1960).
1062:; Pearson, Carl (1966).
1050:(in German). Birkhäuser.
1042:(2nd ed.). Chelsea.
1002:Carathéodory, Constantin
483:
186:boundary, and so forth.
80:complex coordinate space
1429:English translation of
1275:Kwok, Yue-Kuen (2002).
1260:(3rd ed.). Wiley.
1173:English translation of
1044:English translation of
677:Fuchs & Shabat 1964
361:Constantin Carathéodory
344:Constantin Carathéodory
466:Interval (mathematics)
353:
336:
1497:Mathematical analysis
619:be connected or open.
427:is a region then its
319:
99:authors use the term
71:real coordinate space
39:mathematical analysis
1435:(in Russian). Наука.
802:influential textbook
640:, but later defines
553:) reserves the term
363: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
1472:Whittaker, Edmund;
1415:Sveshnikov, Aleksei
1048:Functionentheorie I
968:"Complex Convexity"
448:Analytic polyhedron
886:See (Miranda
777:to be the largest
765:, and defines the
638:open connected set
253:. In the study of
209:is a domain whose
176:Lipschitz boundary
1452:Whittaker, Edmund
1124:. Academic Press.
808:) uses the terms
559:Carathéodory 1964
365:Carathéodory 1918
348:Carathéodory 1918
280:three-dimensional
158:), properties of
63:topological space
16:(Redirected from
1514:
1483:
1469:
1447:
1436:
1428:
1419:Tikhonov, Andrey
1410:
1392:
1380:
1366:
1348:
1335:
1308:
1280:
1271:
1259:
1245:
1229:
1203:
1190:Goursat, Édouard
1185:
1183:
1172:
1170:
1159:
1137:
1125:
1123:
1109:
1097:
1087:
1085:
1071:
1069:
1051:
1043:
1041:
1029:
997:
987:
962:
960:
957:Complex Analysis
938:
931:
925:
922:
916:
901:
895:
884:
873:
791:
787:
781:
776:
772:
762:
749:) uses the term
743:
737:
726:p. 61 footnote 3
719:
713:
710:p. 85 footnote 1
702:
696:
673:
667:
661:
643:simply connected
626:
620:
605:
599:
580:
574:
565:) uses the term
527:
521:
510:
500:
493:
472:Lipschitz domain
437:
435:
426:
351:
331:
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:
1522:
1521:
1517:
1516:
1515:
1513:
1512:
1511:
1487:
1486:
1470:
1389:
1346:
1268:
1250:Kreyszig, Erwin
1181:
1142:Forsyth, Andrew
1114:Dieudonné, Jean
1106:
1088:
1076:Churchill, Ruel
1056:Carrier, George
946:
941:
932:
928:
923:
919:
902:
898:
885:
876:
789:
785:
779:
774:
773:for a function
770:
760:
744:
740:
720:
716:
703:
699:
695:, §1.4, p. 23.)
674:
670:
664:Bremermann 1956
634:§3.19 pp. 64–67
627:
623:
606:
602:
581:
577:
528:
524:
511:
507:
503:
494:
490:
486:
454:Caccioppoli set
444:
433:
431:
422:
352:
342:
327:
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:
1520:
1510:
1509:
1504:
1499:
1485:
1484:
1474:Watson, George
1448:
1437:
1411:
1393:
1387:
1367:
1336:
1312:Miranda, Carlo
1309:Translated as
1285:Miranda, Carlo
1281:
1272:
1266:
1246:
1234:Krantz, Steven
1230:
1204:
1186:
1160:
1138:
1126:
1110:
1104:
1072:
1070:. McGraw-Hill.
1052:
1031:
998:
963:
961:. McGraw-Hill.
945:
942:
940:
939:
937:, p. 66).
926:
917:
896:
894:, p. 2).
890:, p. 1,
874:
806:Whittaker 1902
738:
714:
697:
675:For instance (
668:
630:Dieudonné 1960
628:For instance (
621:
609:Churchill 1960
607:For instance (
600:
582:For instance (
575:
535:§1.9 pp. 16–17
531:Churchill 1948
529:For instance (
522:
518:§1.3 pp. 21–22
512:For instance (
504:
502:
501:
487:
485:
482:
481:
480:
474:
469:
463:
457:
451:
443:
440:
350:, p. 222)
340:
317:
314:
222:complex domain
199:Bounded region
191:bounded domain
160:Sobolev spaces
156:Stokes theorem
116:
113:
26:
9:
6:
4:
3:
2:
1519:
1508:
1505:
1503:
1500:
1498:
1495:
1494:
1492:
1481:
1480:
1475:
1467:
1463:
1459:
1458:
1453:
1449:
1445:
1444:
1438:
1434:
1426:
1425:
1420:
1416:
1412:
1409:
1405:
1404:
1399:
1394:
1390:
1388:9780070542334
1384:
1379:
1378:
1372:
1371:Rudin, Walter
1368:
1364:
1360:
1356:
1352:
1345:
1341:
1340:Picone, Mauro
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1273:
1269:
1267:9780471507284
1263:
1258:
1257:
1251:
1247:
1244:. Birkhäuser.
1243:
1239:
1238:Parks, Harold
1235:
1231:
1227:
1223:
1219:
1215:
1214:
1209:
1205:
1201:
1197:
1196:
1191:
1187:
1180:
1179:
1169:
1168:
1161:
1157:
1153:
1150:. Cambridge.
1149:
1148:
1143:
1139:
1135:
1131:
1127:
1122:
1121:
1115:
1111:
1107:
1105:9780070108530
1101:
1096:
1095:
1084:
1083:
1077:
1073:
1068:
1067:
1061:
1057:
1053:
1049:
1040:
1039:
1032:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
986:
981:
977:
973:
969:
964:
959:
958:
952:
951:Ahlfors, Lars
948:
947:
936:
930:
921:
914:
910:
906:
905:Miranda (1955
900:
893:
889:
883:
881:
879:
871:
867:
863:
859:
858:Townsend 1915
855:
851:
847:
843:
839:
835:
834:closed region
831:
827:
823:
819:
815:
811:
807:
803:
799:
795:
783:
782:-neighborhood
768:
764:
756:
752:
748:
745:For example (
742:
735:
731:
727:
723:
718:
711:
707:
701:
694:
690:
686:
685:Kreyszig 1972
682:
678:
672:
665:
659:
655:
650:
646:
644:
639:
635:
631:
625:
618:
614:
610:
604:
597:
593:
589:
585:
584:Townsend 1915
579:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
532:
526:
519:
515:
509:
505:
498:
492:
488:
478:
475:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
445:
439:
438:is a domain.
436:
430:
425:
420:
416:
412:
411:Carlo Miranda
408:
404:
399:
397:
393:
389:
385:
381:
377:
373:
370:
366:
362:
358:
355:According to
349:
345:
339:
338:
334:
330:
325:
323:
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:
112:
110:
106:
102:
96:
94:
90:
85:
81:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
18:Closed region
1478:
1456:
1442:
1432:
1423:
1401:
1376:
1354:
1350:
1315:
1288:
1279:. Cambridge.
1276:
1255:
1241:
1217:
1212:
1199:
1194:
1177:
1166:
1146:
1133:
1130:Eves, Howard
1119:
1093:
1081:
1065:
1047:
1037:
1009:
1005:
978:(1): 17–51.
975:
971:
956:
929:
920:
908:
899:
869:
865:
864:) defines a
853:
849:
842:Goursat 1905
837:
833:
825:
824:) define an
822:§3.21, p. 44
813:
809:
766:
759:part of the
758:
750:
747:Forsyth 1893
741:
733:
729:
717:
700:
689:§11.1 p. 469
681:§6 pp. 22–23
671:
654:Tao, Terence
641:
637:
624:
616:
603:
578:
570:
566:
554:
551:§10.1 p. 213
539:Ahlfors 1953
525:
508:
491:
432:
423:
419:Mauro Picone
400:
395:
391:
379:
375:
354:
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:
1171:. Pergamon.
935:Picone 1923
846:§262, p. 10
826:open region
794:holomorphic
769:of a point
649:§9.7 p. 215
415:perfect set
241:, the open
224:(or simply
115:Conventions
1491:Categories
1466:33.0390.01
1363:49.0172.07
1332:0198.14101
1305:0065.08503
1226:48.0261.09
1208:Hahn, Hans
1156:25.0652.01
1060:Krook, Max
1018:46.0376.12
944:References
862:§10, p. 20
848:) defines
828:to be the
798:§32, p. 52
755:§16, p. 21
722:Hahn (1921
613:§1.9 p. 17
596:§2.2 p. 32
588:§10, p. 20
547:Rudin 1974
543:§2.2 p. 58
403:monographs
322:Definition
211:complement
174:boundary,
172:continuous
1408:EMS Press
1373:(1974) .
1252:(1972) .
788:in which
706:Hahn 1921
693:Kwok 2002
662:, also, (
571:continuum
497:functions
495:However,
369:non-empty
357:Hans Hahn
329:‹See Tfd›
239:unit disk
55:connected
51:non-empty
1507:Topology
1476:(1915).
1454:(1902).
1421:(1978).
1398:"Domain"
1342:(1923).
1314:(1970).
1287:(1955).
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