2733:
643:
2459:
930:
1802:
1889:
1553:
507:
1699:
1100:
699:
464:
231:
2246:
1977:
306:
1047:
826:
91:
1614:
162:
130:
2331:
2191:
2112:
1922:
40:"), then the region can be divided into a finite number of subregions such that each subregion is interior to a circle of a given set having its center in the subregion.
2497:
2285:
1266:
1233:
1448:
1390:
1121:
792:
1640:
965:
2004:
2138:
2558:
2517:
2336:
2158:
2076:
2024:
1719:
1576:
1468:
1370:
1350:
1290:
1163:
1009:
989:
771:
719:
2056:
1422:
1322:
1195:
751:
498:
411:
2774:
831:
2690:
In J. C. Abbott (Ed.), The
Chauvenet Papers: A collection of Prize-Winning Expository Papers in Mathematics. Mathematical Association of America.
2649:
2608:
1724:
2793:
1269:
1139:
2767:
2713:
364:
1807:
1473:
638:{\displaystyle P=\langle a=x_{0}<t_{1}<x_{1}<t_{2}<\cdots <x_{\ell -1}<t_{\ell }<x_{\ell }=b\rangle }
37:
355:
where it is one of the first third-order theorems that is hard to prove in terms of the comprehension axioms needed.
1645:
2760:
1052:
651:
416:
175:
2196:
1927:
236:
49:
1018:
2798:
2595:. Graduate Studies in Mathematics. Vol. 4. Providence, Rhode Island: American Mathematical Society.
2740:
799:
132:
be a full cover of , that is, a collection of closed subintervals of with the property that for every
67:
1135:
33:
1581:
1200:
143:
111:
2290:
352:
93:. However, Pierre Cousin did not receive any credit. Cousin's theorem was generally attributed to
2698:
968:
2163:
2084:
1894:
2464:
2251:
2655:
2645:
2604:
45:
2744:
1427:
1375:
1106:
777:
2637:
2596:
1619:
935:
57:
1982:
2454:{\displaystyle x_{n+1}=\mathrm {min} (b,t_{n+1}+{\tfrac {1}{2}}\delta (t_{n+1}))>x_{n}}
2117:
2525:
2502:
2143:
2061:
2009:
1704:
1561:
1453:
1355:
1335:
1275:
1148:
994:
974:
756:
704:
94:
2029:
1470:
is open, since it is downwards closed and any point in it is included in the open ray
1395:
1295:
1168:
724:
471:
384:
2787:
101:. Lebesgue was aware of this result in 1898, and proved it in his 1903 dissertation.
61:
17:
53:
925:{\displaystyle (x_{j-1},x_{j})\subseteq B{\big (}t_{j},\delta (t_{j}){\big )}}
29:
2659:
2732:
2600:
1797:{\displaystyle x_{n}>\mathrm {max} (a,r-{\tfrac {1}{2}}\delta (r))}
2590:
2702:
2625:
2641:
44:
This result was originally proved by Pierre Cousin, a student of
2697:, Master of Arts Thesis. University of California, Berkeley.
293:
149:
117:
36:") there is a circle of finite radius (in modern term, a "
28:
If for every point of a closed region (in modern terms, "
2748:
2592:
The
Integrals of Lebesgue, Denjoy, Perron, and Henstock
1884:{\displaystyle x_{n}>b-(t_{n}+\delta (t_{n})-x_{n})}
2399:
1768:
1548:{\displaystyle \cap [a,t_{n}+\delta (t_{n}))\subset S}
2528:
2505:
2467:
2339:
2293:
2254:
2199:
2166:
2146:
2120:
2087:
2064:
2032:
2012:
1985:
1930:
1897:
1810:
1727:
1707:
1648:
1622:
1584:
1564:
1476:
1456:
1430:
1398:
1378:
1358:
1338:
1298:
1278:
1236:
1203:
1171:
1151:
1109:
1055:
1021:
997:
977:
938:
834:
802:
780:
759:
727:
707:
654:
510:
474:
419:
387:
239:
178:
146:
114:
70:
2624:
Kurtz, Douglas S; Swartz, Charles W (October 2011).
1701:
to handle edge cases) we have a partition of length
2695:
2552:
2511:
2491:
2453:
2325:
2279:
2240:
2185:
2152:
2132:
2106:
2070:
2050:
2018:
1998:
1971:
1916:
1883:
1796:
1713:
1693:
1634:
1608:
1570:
1547:
1462:
1442:
1416:
1384:
1364:
1344:
1316:
1284:
1260:
1227:
1189:
1157:
1115:
1094:
1041:
1003:
983:
959:
924:
820:
786:
765:
745:
713:
693:
637:
492:
458:
405:
300:
225:
156:
124:
85:
363:Cousin's theorem is instrumental in the study of
1694:{\displaystyle r\in [a,a+\delta (a))\subset S}
2768:
1197:is said to be inductive if it satisfies that
917:
878:
164:contains all subintervals of which contains
8:
1616:. By that assumption (and using that either
1095:{\displaystyle \delta :\to \mathbb {R} ^{+}}
694:{\displaystyle \delta :\to \mathbb {R} ^{+}}
632:
517:
459:{\displaystyle \delta :\to \mathbb {R} ^{+}}
413:is a strictly positive real-valued function
226:{\displaystyle {I_{1},I_{2},\cdots ,I_{n}}}
2775:
2761:
2688:The Borel Theorem and its Generalizations
2527:
2504:
2466:
2445:
2420:
2398:
2383:
2359:
2344:
2338:
2317:
2298:
2292:
2259:
2253:
2241:{\displaystyle r<x_{n}+\delta (x_{n})}
2229:
2210:
2198:
2177:
2165:
2145:
2119:
2092:
2086:
2063:
2031:
2011:
1990:
1984:
1972:{\displaystyle b<t_{n}+\delta (t_{n})}
1960:
1941:
1929:
1902:
1896:
1872:
1856:
1837:
1815:
1809:
1767:
1741:
1732:
1726:
1706:
1647:
1621:
1583:
1563:
1527:
1508:
1475:
1455:
1429:
1397:
1377:
1357:
1337:
1297:
1277:
1235:
1202:
1170:
1150:
1108:
1086:
1082:
1081:
1054:
1035:
1034:
1020:
996:
976:
937:
916:
915:
906:
887:
877:
876:
861:
842:
833:
801:
779:
758:
726:
706:
685:
681:
680:
653:
620:
607:
588:
569:
556:
543:
530:
509:
473:
450:
446:
445:
418:
386:
292:
291:
279:
260:
244:
238:
233:of non-overlapping intervals for , where
216:
197:
184:
179:
177:
148:
147:
145:
116:
115:
113:
77:
73:
72:
69:
2160:. To show this, we split into the cases
301:{\displaystyle I_{i}=\in {\mathcal {C}}}
2569:
48:, in 1895, and it extends the original
1558:Furthermore, it is inductive. For any
1042:{\displaystyle a<b\in \mathbb {R} }
367:, and in this context, it is known as
7:
2729:
2727:
2575:
2573:
2114:, we may form a partition of length
1011:. Cousin's lemma is now stated as:
2747:. You can help Knowledge (XXG) by
2366:
2363:
2360:
1748:
1745:
1742:
104:In modern terms, it is stated as:
14:
2461:and obtain a valid partition. So
1272:states that any inductive subset
821:{\displaystyle 1\leq j\leq \ell }
2731:
2719:, American Mathematical Society.
359:In Henstock–Kurzweil integration
172:. Then there exists a partition
86:{\displaystyle \mathbb {R} ^{n}}
2714:Graduate Studies in Mathematics
2710:A Modern Theory of Integration
2589:Gordon, Russell (1994-08-01).
2547:
2535:
2480:
2468:
2435:
2432:
2413:
2370:
2235:
2222:
2045:
2033:
1966:
1953:
1878:
1862:
1849:
1830:
1791:
1788:
1782:
1752:
1682:
1679:
1673:
1655:
1609:{\displaystyle [a,r)\subset S}
1597:
1585:
1555:for any associated partition.
1536:
1533:
1520:
1495:
1489:
1477:
1411:
1399:
1311:
1299:
1249:
1237:
1228:{\displaystyle [a,r)\subset S}
1216:
1204:
1184:
1172:
1077:
1074:
1062:
954:
942:
912:
899:
867:
835:
740:
728:
676:
673:
661:
487:
475:
441:
438:
426:
400:
388:
285:
253:
157:{\displaystyle {\mathcal {C}}}
125:{\displaystyle {\mathcal {C}}}
1:
2326:{\displaystyle t_{n+1}=x_{n}}
365:Henstock–Kurzweil integration
351:Cousin's lemma is studied in
2333:. In both cases, we can set
2248:. In the first case, we set
1270:The open induction principle
1140:the open induction principle
2794:Mathematical analysis stubs
2686:Hildebrandt, T. H. (1925).
20:, a branch of mathematics,
2815:
2726:
2186:{\displaystyle r>x_{n}}
2107:{\displaystyle x_{n}<b}
1917:{\displaystyle x_{n}<b}
1392:-fine tagged partition on
1328:Proof using open induction
1165:of a closed real interval
1142:, which reads as follows:
2626:"Theories of Integration"
2492:{\displaystyle \subset S}
2280:{\displaystyle t_{n+1}=r}
1979:, so we can just replace
1372:such that there exists a
1324:must be the entire set.
1261:{\displaystyle \subset S}
1134:Cousin's theorem has an
168:and length smaller than
2630:Series in Real Analysis
2579:Hildebrandt 1925, p. 29
2287:, in the second we set
2026:and get a partition of
1443:{\displaystyle s\geq r}
1385:{\displaystyle \delta }
1116:{\displaystyle \delta }
787:{\displaystyle \delta }
701:and a tagged partition
2743:–related article is a
2708:Bartle, R. G. (2001).
2554:
2513:
2493:
2455:
2327:
2281:
2242:
2187:
2154:
2134:
2108:
2072:
2052:
2020:
2000:
1973:
1918:
1885:
1798:
1715:
1695:
1636:
1635:{\displaystyle r>a}
1610:
1572:
1549:
1464:
1444:
1418:
1386:
1366:
1346:
1318:
1286:
1262:
1229:
1191:
1159:
1117:
1096:
1043:
1005:
985:
961:
960:{\displaystyle B(x,r)}
926:
822:
788:
767:
747:
715:
695:
639:
494:
460:
407:
302:
227:
158:
126:
99:Borel–Lebesgue theorem
87:
2741:mathematical analysis
2693:Raman, M. J. (1997).
2555:
2514:
2494:
2456:
2328:
2282:
2243:
2188:
2155:
2135:
2109:
2073:
2053:
2021:
2001:
1999:{\displaystyle x_{n}}
1974:
1919:
1886:
1799:
1716:
1696:
1637:
1611:
1573:
1550:
1465:
1445:
1419:
1387:
1367:
1352:be the set of points
1347:
1319:
1287:
1263:
1230:
1192:
1160:
1118:
1097:
1044:
1006:
986:
962:
927:
823:
789:
768:
748:
716:
696:
640:
501:is a finite sequence
495:
461:
408:
303:
228:
159:
127:
88:
2526:
2503:
2465:
2337:
2291:
2252:
2197:
2164:
2144:
2118:
2085:
2062:
2030:
2010:
1983:
1928:
1924:. In the first case
1895:
1808:
1725:
1705:
1646:
1620:
1582:
1562:
1474:
1454:
1428:
1396:
1376:
1356:
1336:
1296:
1276:
1234:
1201:
1169:
1149:
1130:Proof of the theorem
1107:
1053:
1019:
995:
975:
936:
832:
800:
778:
757:
725:
705:
652:
508:
472:
468:tagged partition of
417:
385:
237:
176:
144:
112:
68:
2522:By open induction,
2133:{\displaystyle n+1}
1049:, then every gauge
353:reverse mathematics
136:∈ , there exists a
50:Heine–Borel theorem
2672:Bartle 2001, p. 11
2553:{\displaystyle S=}
2550:
2509:
2499:in all cases, and
2489:
2451:
2408:
2323:
2277:
2238:
2183:
2150:
2130:
2104:
2068:
2048:
2016:
1996:
1969:
1914:
1881:
1794:
1777:
1711:
1691:
1632:
1606:
1568:
1545:
1460:
1440:
1414:
1382:
1362:
1342:
1314:
1282:
1258:
1225:
1187:
1155:
1113:
1092:
1039:
1001:
981:
957:
922:
818:
784:
763:
743:
711:
691:
635:
490:
456:
403:
298:
223:
154:
122:
83:
2756:
2755:
2651:978-981-4368-99-5
2610:978-0-8218-3805-1
2512:{\displaystyle S}
2407:
2153:{\displaystyle r}
2071:{\displaystyle r}
2019:{\displaystyle b}
1776:
1714:{\displaystyle n}
1571:{\displaystyle r}
1463:{\displaystyle S}
1365:{\displaystyle r}
1345:{\displaystyle S}
1285:{\displaystyle S}
1158:{\displaystyle S}
1004:{\displaystyle x}
984:{\displaystyle r}
766:{\displaystyle P}
714:{\displaystyle P}
2806:
2777:
2770:
2763:
2735:
2728:
2673:
2670:
2664:
2663:
2621:
2615:
2614:
2586:
2580:
2577:
2559:
2557:
2556:
2551:
2518:
2516:
2515:
2510:
2498:
2496:
2495:
2490:
2460:
2458:
2457:
2452:
2450:
2449:
2431:
2430:
2409:
2400:
2394:
2393:
2369:
2355:
2354:
2332:
2330:
2329:
2324:
2322:
2321:
2309:
2308:
2286:
2284:
2283:
2278:
2270:
2269:
2247:
2245:
2244:
2239:
2234:
2233:
2215:
2214:
2192:
2190:
2189:
2184:
2182:
2181:
2159:
2157:
2156:
2151:
2139:
2137:
2136:
2131:
2113:
2111:
2110:
2105:
2097:
2096:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2051:{\displaystyle }
2049:
2025:
2023:
2022:
2017:
2005:
2003:
2002:
1997:
1995:
1994:
1978:
1976:
1975:
1970:
1965:
1964:
1946:
1945:
1923:
1921:
1920:
1915:
1907:
1906:
1890:
1888:
1887:
1882:
1877:
1876:
1861:
1860:
1842:
1841:
1820:
1819:
1803:
1801:
1800:
1795:
1778:
1769:
1751:
1737:
1736:
1720:
1718:
1717:
1712:
1700:
1698:
1697:
1692:
1641:
1639:
1638:
1633:
1615:
1613:
1612:
1607:
1577:
1575:
1574:
1569:
1554:
1552:
1551:
1546:
1532:
1531:
1513:
1512:
1469:
1467:
1466:
1461:
1449:
1447:
1446:
1441:
1423:
1421:
1420:
1417:{\displaystyle }
1415:
1391:
1389:
1388:
1383:
1371:
1369:
1368:
1363:
1351:
1349:
1348:
1343:
1323:
1321:
1320:
1317:{\displaystyle }
1315:
1291:
1289:
1288:
1283:
1267:
1265:
1264:
1259:
1232:
1231:
1226:
1196:
1194:
1193:
1190:{\displaystyle }
1188:
1164:
1162:
1161:
1156:
1122:
1120:
1119:
1114:
1101:
1099:
1098:
1093:
1091:
1090:
1085:
1048:
1046:
1045:
1040:
1038:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
966:
964:
963:
958:
931:
929:
928:
923:
921:
920:
911:
910:
892:
891:
882:
881:
866:
865:
853:
852:
827:
825:
824:
819:
793:
791:
790:
785:
772:
770:
769:
764:
752:
750:
749:
746:{\displaystyle }
744:
720:
718:
717:
712:
700:
698:
697:
692:
690:
689:
684:
644:
642:
641:
636:
625:
624:
612:
611:
599:
598:
574:
573:
561:
560:
548:
547:
535:
534:
499:
497:
496:
493:{\displaystyle }
491:
465:
463:
462:
457:
455:
454:
449:
412:
410:
409:
406:{\displaystyle }
404:
373:fineness theorem
307:
305:
304:
299:
297:
296:
284:
283:
271:
270:
249:
248:
232:
230:
229:
224:
222:
221:
220:
202:
201:
189:
188:
163:
161:
160:
155:
153:
152:
131:
129:
128:
123:
121:
120:
92:
90:
89:
84:
82:
81:
76:
22:Cousin's theorem
2814:
2813:
2809:
2808:
2807:
2805:
2804:
2803:
2784:
2783:
2782:
2781:
2724:
2722:
2682:
2677:
2676:
2671:
2667:
2652:
2623:
2622:
2618:
2611:
2601:10.1090/gsm/004
2588:
2587:
2583:
2578:
2571:
2566:
2524:
2523:
2501:
2500:
2463:
2462:
2441:
2416:
2379:
2340:
2335:
2334:
2313:
2294:
2289:
2288:
2255:
2250:
2249:
2225:
2206:
2195:
2194:
2173:
2162:
2161:
2142:
2141:
2116:
2115:
2088:
2083:
2082:
2060:
2059:
2028:
2027:
2008:
2007:
1986:
1981:
1980:
1956:
1937:
1926:
1925:
1898:
1893:
1892:
1868:
1852:
1833:
1811:
1806:
1805:
1728:
1723:
1722:
1703:
1702:
1644:
1643:
1618:
1617:
1580:
1579:
1560:
1559:
1523:
1504:
1472:
1471:
1452:
1451:
1426:
1425:
1394:
1393:
1374:
1373:
1354:
1353:
1334:
1333:
1330:
1294:
1293:
1274:
1273:
1199:
1198:
1167:
1166:
1147:
1146:
1145:An open subset
1132:
1105:
1104:
1080:
1051:
1050:
1017:
1016:
993:
992:
973:
972:
934:
933:
902:
883:
857:
838:
830:
829:
798:
797:
776:
775:
755:
754:
723:
722:
703:
702:
679:
650:
649:
616:
603:
584:
565:
552:
539:
526:
506:
505:
470:
469:
444:
415:
414:
383:
382:
361:
334:
325:
318:
275:
256:
240:
235:
234:
212:
193:
180:
174:
173:
142:
141:
110:
109:
71:
66:
65:
12:
11:
5:
2812:
2810:
2802:
2801:
2796:
2786:
2785:
2780:
2779:
2772:
2765:
2757:
2754:
2753:
2736:
2721:
2720:
2706:
2691:
2683:
2681:
2678:
2675:
2674:
2665:
2650:
2616:
2609:
2581:
2568:
2567:
2565:
2562:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2519:is inductive.
2508:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2448:
2444:
2440:
2437:
2434:
2429:
2426:
2423:
2419:
2415:
2412:
2406:
2403:
2397:
2392:
2389:
2386:
2382:
2378:
2375:
2372:
2368:
2365:
2362:
2358:
2353:
2350:
2347:
2343:
2320:
2316:
2312:
2307:
2304:
2301:
2297:
2276:
2273:
2268:
2265:
2262:
2258:
2237:
2232:
2228:
2224:
2221:
2218:
2213:
2209:
2205:
2202:
2180:
2176:
2172:
2169:
2149:
2140:that includes
2129:
2126:
2123:
2103:
2100:
2095:
2091:
2067:
2058:that includes
2047:
2044:
2041:
2038:
2035:
2015:
1993:
1989:
1968:
1963:
1959:
1955:
1952:
1949:
1944:
1940:
1936:
1933:
1913:
1910:
1905:
1901:
1880:
1875:
1871:
1867:
1864:
1859:
1855:
1851:
1848:
1845:
1840:
1836:
1832:
1829:
1826:
1823:
1818:
1814:
1804:. Then either
1793:
1790:
1787:
1784:
1781:
1775:
1772:
1766:
1763:
1760:
1757:
1754:
1750:
1747:
1744:
1740:
1735:
1731:
1710:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1631:
1628:
1625:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1567:
1544:
1541:
1538:
1535:
1530:
1526:
1522:
1519:
1516:
1511:
1507:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1459:
1439:
1436:
1433:
1413:
1410:
1407:
1404:
1401:
1381:
1361:
1341:
1329:
1326:
1313:
1310:
1307:
1304:
1301:
1281:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1186:
1183:
1180:
1177:
1174:
1154:
1136:intuitionistic
1131:
1128:
1127:
1126:
1112:
1089:
1084:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1037:
1033:
1030:
1027:
1024:
1000:
980:
956:
953:
950:
947:
944:
941:
919:
914:
909:
905:
901:
898:
895:
890:
886:
880:
875:
872:
869:
864:
860:
856:
851:
848:
845:
841:
837:
817:
814:
811:
808:
805:
783:
762:
742:
739:
736:
733:
730:
710:
688:
683:
678:
675:
672:
669:
666:
663:
660:
657:
648:Given a gauge
646:
645:
634:
631:
628:
623:
619:
615:
610:
606:
602:
597:
594:
591:
587:
583:
580:
577:
572:
568:
564:
559:
555:
551:
546:
542:
538:
533:
529:
525:
522:
519:
516:
513:
489:
486:
483:
480:
477:
453:
448:
443:
440:
437:
434:
431:
428:
425:
422:
402:
399:
396:
393:
390:
369:Cousin's lemma
360:
357:
349:
348:
330:
323:
316:
295:
290:
287:
282:
278:
274:
269:
266:
263:
259:
255:
252:
247:
243:
219:
215:
211:
208:
205:
200:
196:
192:
187:
183:
151:
140:>0 so that
119:
95:Henri Lebesgue
80:
75:
56:for arbitrary
46:Henri Poincaré
42:
41:
13:
10:
9:
6:
4:
3:
2:
2811:
2800:
2799:Real analysis
2797:
2795:
2792:
2791:
2789:
2778:
2773:
2771:
2766:
2764:
2759:
2758:
2752:
2750:
2746:
2742:
2737:
2734:
2730:
2725:
2718:
2715:
2711:
2707:
2704:
2700:
2696:
2692:
2689:
2685:
2684:
2679:
2669:
2666:
2661:
2657:
2653:
2647:
2643:
2639:
2635:
2631:
2627:
2620:
2617:
2612:
2606:
2602:
2598:
2594:
2593:
2585:
2582:
2576:
2574:
2570:
2563:
2561:
2544:
2541:
2538:
2532:
2529:
2520:
2506:
2486:
2483:
2477:
2474:
2471:
2446:
2442:
2438:
2427:
2424:
2421:
2417:
2410:
2404:
2401:
2395:
2390:
2387:
2384:
2380:
2376:
2373:
2356:
2351:
2348:
2345:
2341:
2318:
2314:
2310:
2305:
2302:
2299:
2295:
2274:
2271:
2266:
2263:
2260:
2256:
2230:
2226:
2219:
2216:
2211:
2207:
2203:
2200:
2178:
2174:
2170:
2167:
2147:
2127:
2124:
2121:
2101:
2098:
2093:
2089:
2079:
2065:
2042:
2039:
2036:
2013:
1991:
1987:
1961:
1957:
1950:
1947:
1942:
1938:
1934:
1931:
1911:
1908:
1903:
1899:
1873:
1869:
1865:
1857:
1853:
1846:
1843:
1838:
1834:
1827:
1824:
1821:
1816:
1812:
1785:
1779:
1773:
1770:
1764:
1761:
1758:
1755:
1738:
1733:
1729:
1708:
1688:
1685:
1676:
1670:
1667:
1664:
1661:
1658:
1652:
1649:
1629:
1626:
1623:
1603:
1600:
1594:
1591:
1588:
1565:
1556:
1542:
1539:
1528:
1524:
1517:
1514:
1509:
1505:
1501:
1498:
1492:
1486:
1483:
1480:
1457:
1437:
1434:
1431:
1408:
1405:
1402:
1379:
1359:
1339:
1327:
1325:
1308:
1305:
1302:
1279:
1271:
1255:
1252:
1246:
1243:
1240:
1222:
1219:
1213:
1210:
1207:
1181:
1178:
1175:
1152:
1143:
1141:
1137:
1129:
1124:
1110:
1087:
1071:
1068:
1065:
1059:
1056:
1031:
1028:
1025:
1022:
1014:
1013:
1012:
998:
978:
970:
951:
948:
945:
939:
907:
903:
896:
893:
888:
884:
873:
870:
862:
858:
854:
849:
846:
843:
839:
815:
812:
809:
806:
803:
795:
781:
760:
737:
734:
731:
708:
686:
670:
667:
664:
658:
655:
629:
626:
621:
617:
613:
608:
604:
600:
595:
592:
589:
585:
581:
578:
575:
570:
566:
562:
557:
553:
549:
544:
540:
536:
531:
527:
523:
520:
514:
511:
504:
503:
502:
500:
484:
481:
478:
451:
435:
432:
429:
423:
420:
397:
394:
391:
381:
376:
374:
370:
366:
358:
356:
354:
346:
342:
338:
333:
329:
322:
315:
311:
288:
280:
276:
272:
267:
264:
261:
257:
250:
245:
241:
217:
213:
209:
206:
203:
198:
194:
190:
185:
181:
171:
167:
139:
135:
107:
106:
105:
102:
100:
96:
78:
63:
59:
55:
51:
47:
39:
35:
31:
27:
26:
25:
24:states that:
23:
19:
18:real analysis
2749:expanding it
2738:
2723:
2716:
2709:
2694:
2687:
2668:
2642:10.1142/8291
2633:
2629:
2619:
2591:
2584:
2521:
2080:
1557:
1331:
1144:
1138:proof using
1133:
1103:
967:denotes the
774:
647:
467:
379:
377:
372:
368:
362:
350:
344:
340:
336:
331:
327:
326:< ⋯ <
320:
313:
309:
169:
165:
137:
133:
103:
98:
43:
38:neighborhood
21:
15:
1450:. The set
991:centred at
796:if for all
64:subsets of
54:compactness
2788:Categories
2680:References
1578:, suppose
1125:partition.
971:of radius
828:, we have
466:, while a
339:for all 1≤
2703:1006.4131
2660:1793-1134
2484:⊂
2411:δ
2220:δ
1951:δ
1866:−
1847:δ
1828:−
1780:δ
1765:−
1686:⊂
1671:δ
1653:∈
1601:⊂
1540:⊂
1518:δ
1493:∩
1435:≥
1424:for some
1380:δ
1253:⊂
1220:⊂
1111:δ
1078:→
1057:δ
1032:∈
969:open ball
897:δ
871:⊆
847:−
816:ℓ
813:≤
807:≤
782:δ
753:, we say
677:→
656:δ
633:⟩
622:ℓ
609:ℓ
593:−
590:ℓ
579:⋯
518:⟨
442:→
421:δ
289:∈
265:−
207:⋯
932:, where
380:gauge on
371:or the
97:as the
62:compact
34:bounded
2658:
2648:
2607:
1102:has a
58:covers
30:closed
2739:This
2699:arXiv
2564:Notes
2006:with
1721:with
1123:-fine
794:-fine
319:<
2745:stub
2656:ISSN
2646:ISBN
2605:ISBN
2439:>
2204:<
2171:>
2099:<
1935:<
1909:<
1822:>
1739:>
1627:>
1332:Let
1026:<
614:<
601:<
582:<
576:<
563:<
550:<
537:<
308:and
108:Let
32:and
2638:doi
2597:doi
2193:or
2081:If
2078:.
1891:or
1642:or
1292:of
1015:If
773:is
721:of
60:of
52:on
16:In
2790::
2717:32
2712:,
2654:.
2644:.
2636:.
2634:13
2632:.
2628:.
2603:.
2572:^
2560:.
1268:.
378:A
375:.
2776:e
2769:t
2762:v
2751:.
2705:.
2701::
2662:.
2640::
2613:.
2599::
2548:]
2545:b
2542:,
2539:a
2536:[
2533:=
2530:S
2507:S
2487:S
2481:]
2478:r
2475:,
2472:a
2469:[
2447:n
2443:x
2436:)
2433:)
2428:1
2425:+
2422:n
2418:t
2414:(
2405:2
2402:1
2396:+
2391:1
2388:+
2385:n
2381:t
2377:,
2374:b
2371:(
2367:n
2364:i
2361:m
2357:=
2352:1
2349:+
2346:n
2342:x
2319:n
2315:x
2311:=
2306:1
2303:+
2300:n
2296:t
2275:r
2272:=
2267:1
2264:+
2261:n
2257:t
2236:)
2231:n
2227:x
2223:(
2217:+
2212:n
2208:x
2201:r
2179:n
2175:x
2168:r
2148:r
2128:1
2125:+
2122:n
2102:b
2094:n
2090:x
2066:r
2046:]
2043:b
2040:,
2037:a
2034:[
2014:b
1992:n
1988:x
1967:)
1962:n
1958:t
1954:(
1948:+
1943:n
1939:t
1932:b
1912:b
1904:n
1900:x
1879:)
1874:n
1870:x
1863:)
1858:n
1854:t
1850:(
1844:+
1839:n
1835:t
1831:(
1825:b
1817:n
1813:x
1792:)
1789:)
1786:r
1783:(
1774:2
1771:1
1762:r
1759:,
1756:a
1753:(
1749:x
1746:a
1743:m
1734:n
1730:x
1709:n
1689:S
1683:)
1680:)
1677:a
1674:(
1668:+
1665:a
1662:,
1659:a
1656:[
1650:r
1630:a
1624:r
1604:S
1598:)
1595:r
1592:,
1589:a
1586:[
1566:r
1543:S
1537:)
1534:)
1529:n
1525:t
1521:(
1515:+
1510:n
1506:t
1502:,
1499:a
1496:[
1490:]
1487:b
1484:,
1481:a
1478:[
1458:S
1438:r
1432:s
1412:]
1409:s
1406:,
1403:a
1400:[
1360:r
1340:S
1312:]
1309:b
1306:,
1303:a
1300:[
1280:S
1256:S
1250:]
1247:r
1244:,
1241:a
1238:[
1223:S
1217:)
1214:r
1211:,
1208:a
1205:[
1185:]
1182:b
1179:,
1176:a
1173:[
1153:S
1088:+
1083:R
1075:]
1072:b
1069:,
1066:a
1063:[
1060::
1036:R
1029:b
1023:a
999:x
979:r
955:)
952:r
949:,
946:x
943:(
940:B
918:)
913:)
908:j
904:t
900:(
894:,
889:j
885:t
879:(
874:B
868:)
863:j
859:x
855:,
850:1
844:j
840:x
836:(
810:j
804:1
761:P
741:]
738:b
735:,
732:a
729:[
709:P
687:+
682:R
674:]
671:b
668:,
665:a
662:[
659::
630:b
627:=
618:x
605:t
596:1
586:x
571:2
567:t
558:1
554:x
545:1
541:t
532:0
528:x
524:=
521:a
515:=
512:P
488:]
485:b
482:,
479:a
476:[
452:+
447:R
439:]
436:b
433:,
430:a
427:[
424::
401:]
398:b
395:,
392:a
389:[
347:.
345:n
343:≤
341:i
337:b
335:=
332:n
328:x
324:1
321:x
317:0
314:x
312:=
310:a
294:C
286:]
281:i
277:x
273:,
268:1
262:i
258:x
254:[
251:=
246:i
242:I
218:n
214:I
210:,
204:,
199:2
195:I
191:,
186:1
182:I
170:δ
166:x
150:C
138:δ
134:x
118:C
79:n
74:R
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