2552:
2335:
2573:
2541:
2610:
2583:
2563:
1510:
1857:
1265:
1088:
577:
359:
1394:
428:
968:
326:
1614:
1771:
213:
1538:
645:
605:
537:
484:
456:
387:
300:
271:
243:
1347:
1200:
1049:
918:
109:
1677:
2015:
872:
818:
152:
1707:
1644:
1292:
1227:
1168:
1142:
1114:
1023:
996:
1423:
1794:
1561:
843:
510:
2613:
1989:
1969:
1949:
1919:
1899:
1312:
892:
793:
769:
749:
725:
705:
685:
665:
172:
129:
31:, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize
923:
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
2247:
1438:
2601:
2596:
2112:
1806:
2591:
1922:
1646:
Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if
2493:
2218:
47:
1881:
Given a compact
Hausdorff space, there is a unique proximity space whose corresponding topology is the given topology:
1878:, using the last property of proximal neighborhoods to create the infinite increasing chain used in proving the lemma.
2213:
1860:
2634:
2501:
2104:
1232:
1055:
2639:
542:
2572:
2300:
332:
2586:
1359:
393:
2521:
2516:
2442:
2319:
2307:
2280:
2240:
2187:
132:
2363:
2290:
929:
305:
2551:
1566:
1712:
180:
2511:
2463:
2437:
2285:
2208:
2147:
2042:
2018:
2562:
2358:
2192:
1515:
622:
582:
514:
461:
433:
364:
277:
248:
220:
1317:
1173:
1028:
897:
2556:
2506:
2427:
2417:
2295:
2275:
2167:
1871:
82:
2526:
1875:
1649:
1994:
2544:
2410:
2368:
2233:
2108:
2036:
32:
848:
803:
137:
2324:
2270:
2159:
2135:
2118:
1682:
1619:
1271:
1206:
1147:
1121:
1093:
1002:
975:
2179:
2092:
2383:
2378:
2175:
2139:
2122:
2088:
1864:
1399:
2186:
Vita, Luminita; Bridges, Douglas (2001). "A Constructive Theory of Point-Set
Nearness".
1776:
1543:
825:
492:
2473:
2405:
1974:
1954:
1934:
1904:
1884:
1297:
877:
778:
754:
734:
710:
690:
670:
650:
157:
114:
2628:
2483:
2393:
2373:
2103:. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 59. Cambridge:
2039: – Generalization of the notion of convergence that is found in general topology
1929:
55:
39:
2576:
2468:
2388:
2334:
2030:
1921:
if and only if their closures intersect. More generally, proximities classify the
2566:
2478:
2422:
2353:
2312:
2447:
2432:
2400:
2349:
2256:
20:
2171:
2163:
2064:
Frederic Riesz' contributions to the foundations of general topology
1867:. Proximity maps will be continuous between the induced topologies.
46:) but ignored at the time. It was rediscovered and axiomatized by
1863:. If the proximity space is separated, the resulting topology is
616:(but then Axioms 2 and 4 must be stated in a two-sided fashion).
1505:{\displaystyle f:(X,\delta )\to \left(X^{*},\delta ^{*}\right),}
2229:
1803:
Given a proximity space, one can define a topology by letting
62:) discovered a version of the same concept under the name of
2225:
1852:{\displaystyle A\mapsto \left\{x:\{x\}\;\delta \;A\right\}}
16:
Structure describing a notion of "nearness" between subsets
2130:
Riesz, F. (1909), "Stetigkeit und abstrakte
Mengenlehre",
1874:. This can be proven by imitating the usual proofs of
2099:
Naimpally, Somashekhar A.; Warrack, Brian D. (1970).
1997:
1977:
1957:
1937:
1907:
1887:
1809:
1779:
1715:
1685:
1652:
1622:
1569:
1546:
1518:
1441:
1402:
1362:
1320:
1300:
1274:
1235:
1209:
1176:
1150:
1124:
1096:
1058:
1031:
1005:
978:
932:
900:
880:
851:
828:
806:
781:
757:
737:
713:
693:
673:
653:
625:
585:
545:
517:
495:
464:
436:
396:
367:
335:
308:
280:
251:
223:
183:
160:
140:
117:
85:
2492:
2456:
2342:
2263:
2009:
1983:
1963:
1943:
1913:
1893:
1851:
1788:
1765:
1701:
1671:
1638:
1608:
1555:
1532:
1504:
1417:
1388:
1341:
1306:
1286:
1259:
1221:
1194:
1162:
1136:
1108:
1082:
1043:
1017:
990:
962:
912:
886:
866:
837:
812:
787:
763:
743:
719:
699:
679:
659:
639:
599:
571:
531:
504:
478:
450:
422:
381:
353:
320:
294:
265:
237:
207:
166:
146:
123:
103:
2079:EfremoviÄŤ, V. A. (1951), "Infinitesimal spaces",
2033: – Concept in general topology and analysis
2017:has nonempty intersection with every entourage.
54:, but not published until 1951. In the interim,
1435:is one that preserves nearness, that is, given
2241:
8:
1833:
1827:
1383:
1377:
1369:
1363:
1260:{\displaystyle X\setminus B\ll X\setminus A}
1083:{\displaystyle A\subseteq B\ll C\subseteq D}
610:Proximity without the first axiom is called
2068:Handbook of the History of General Topology
2609:
2582:
2248:
2234:
2226:
1951:induces a proximity relation by declaring
1840:
1836:
1593:
1582:
1526:
1522:
1376:
1372:
633:
629:
593:
589:
572:{\displaystyle B\;\delta \;(X\setminus E)}
553:
549:
525:
521:
472:
468:
444:
440:
404:
400:
375:
371:
288:
284:
259:
255:
231:
227:
2191:
2021:maps will then be proximally continuous.
1996:
1976:
1956:
1936:
1925:of a completely regular Hausdorff space.
1906:
1886:
1808:
1778:
1745:
1720:
1714:
1690:
1684:
1660:
1651:
1627:
1621:
1587:
1568:
1545:
1517:
1488:
1475:
1440:
1401:
1361:
1319:
1299:
1273:
1234:
1208:
1175:
1149:
1123:
1095:
1057:
1030:
1004:
977:
931:
899:
879:
850:
827:
805:
780:
756:
736:
712:
692:
672:
652:
624:
584:
544:
516:
494:
463:
435:
395:
366:
334:
307:
279:
250:
222:
182:
159:
139:
116:
84:
354:{\displaystyle A\cap B\neq \varnothing }
2055:
1251:
1239:
904:
560:
348:
315:
59:
2045: – Generalized topological space
1389:{\displaystyle \{x\}\;\delta \;\{y\}}
423:{\displaystyle A\;\delta \;(B\cup C)}
174:satisfying the following properties:
43:
7:
2066:, in C.E. Aull and R. Lowen (eds.),
963:{\displaystyle A,B,C,D\subseteq X}
321:{\displaystyle A\neq \varnothing }
14:
1870:The resulting topology is always
1609:{\displaystyle f\;\delta ^{*}\;f}
2608:
2581:
2571:
2561:
2550:
2540:
2539:
2333:
1766:{\displaystyle f^{-1}\ll f^{-1}}
208:{\displaystyle A,B,C\subseteq X}
2070:, Volume 1, 21-29, Kluwer 1997.
1294:implies that there exists some
1813:
1760:
1754:
1735:
1729:
1603:
1597:
1579:
1573:
1463:
1460:
1448:
566:
554:
417:
405:
98:
86:
1:
1533:{\displaystyle A\;\delta \;B}
640:{\displaystyle A\;\delta \;B}
600:{\displaystyle A\;\delta \;B}
532:{\displaystyle A\;\delta \;E}
479:{\displaystyle A\;\delta \;C}
451:{\displaystyle A\;\delta \;B}
382:{\displaystyle A\;\delta \;B}
295:{\displaystyle A\;\delta \;B}
266:{\displaystyle B\;\delta \;A}
238:{\displaystyle A\;\delta \;B}
38:The concept was described by
1352:A proximity space is called
1342:{\displaystyle A\ll E\ll B.}
1195:{\displaystyle A\ll B\cap C}
1044:{\displaystyle A\subseteq B}
913:{\displaystyle X\setminus B}
2214:Encyclopedia of Mathematics
2083:, New Series (in Russian),
1861:Kuratowski closure operator
104:{\displaystyle (X,\delta )}
2656:
2502:Banach fixed-point theorem
2105:Cambridge University Press
2081:Doklady Akademii Nauk SSSR
1672:{\displaystyle C\ll ^{*}D}
50:in 1934 under the name of
2535:
2331:
2010:{\displaystyle A\times B}
2146:Wallace, A. D. (1941),
867:{\displaystyle A\ll B,}
813:{\displaystyle \delta }
147:{\displaystyle \delta }
2557:Mathematics portal
2457:Metrics and properties
2443:Second-countable space
2132:Rom. 4. Math. Kongr. 2
2011:
1985:
1965:
1945:
1915:
1895:
1853:
1790:
1767:
1703:
1702:{\displaystyle X^{*},}
1673:
1640:
1639:{\displaystyle X^{*}.}
1610:
1557:
1534:
1506:
1419:
1390:
1343:
1308:
1288:
1287:{\displaystyle A\ll B}
1261:
1223:
1222:{\displaystyle A\ll B}
1196:
1164:
1163:{\displaystyle A\ll C}
1138:
1137:{\displaystyle A\ll B}
1110:
1109:{\displaystyle A\ll D}
1084:
1045:
1019:
1018:{\displaystyle A\ll B}
992:
991:{\displaystyle X\ll X}
964:
914:
888:
868:
839:
814:
789:
765:
745:
721:
701:
681:
661:
641:
601:
573:
533:
506:
480:
452:
424:
383:
355:
322:
296:
267:
239:
209:
168:
148:
125:
105:
2012:
1986:
1966:
1946:
1916:
1896:
1854:
1791:
1768:
1704:
1674:
1641:
1611:
1558:
1535:
1507:
1420:
1391:
1344:
1309:
1289:
1262:
1224:
1197:
1165:
1139:
1111:
1085:
1046:
1020:
993:
965:
915:
889:
869:
840:
815:
790:
766:
746:
722:
702:
682:
662:
642:
602:
574:
534:
507:
481:
453:
425:
384:
356:
323:
297:
268:
240:
210:
169:
149:
126:
106:
2512:Invariance of domain
2464:Euler characteristic
2438:Bundle (mathematics)
2043:Pretopological space
2019:Uniformly continuous
1995:
1975:
1955:
1935:
1905:
1885:
1807:
1777:
1713:
1683:
1650:
1620:
1567:
1544:
1516:
1439:
1418:{\displaystyle x=y.}
1400:
1360:
1318:
1298:
1272:
1233:
1207:
1174:
1148:
1122:
1094:
1056:
1029:
1003:
976:
930:
898:
878:
849:
826:
804:
779:
755:
735:
711:
691:
671:
651:
623:
583:
543:
515:
493:
462:
434:
394:
365:
333:
306:
278:
249:
221:
181:
158:
138:
115:
83:
2522:Tychonoff's theorem
2517:Poincaré conjecture
2271:General (point-set)
2148:"Separation spaces"
731:; otherwise we say
154:between subsets of
52:infinitesimal space
2507:De Rham cohomology
2428:Polyhedral complex
2418:Simplicial complex
2007:
1981:
1961:
1941:
1911:
1891:
1872:completely regular
1849:
1789:{\displaystyle X.}
1786:
1763:
1699:
1669:
1636:
1606:
1556:{\displaystyle X,}
1553:
1530:
1502:
1415:
1386:
1339:
1304:
1284:
1257:
1219:
1192:
1160:
1134:
1106:
1080:
1041:
1015:
988:
960:
910:
884:
864:
838:{\displaystyle A,}
835:
810:
785:
761:
741:
717:
697:
677:
657:
637:
597:
569:
529:
505:{\displaystyle E,}
502:
476:
448:
420:
379:
351:
318:
292:
263:
235:
205:
164:
144:
121:
101:
33:topological spaces
2635:Closure operators
2622:
2621:
2411:fundamental group
2209:"Proximity space"
2037:Convergence space
1984:{\displaystyle B}
1964:{\displaystyle A}
1944:{\displaystyle X}
1923:compactifications
1914:{\displaystyle B}
1894:{\displaystyle A}
1307:{\displaystyle E}
887:{\displaystyle A}
788:{\displaystyle B}
764:{\displaystyle B}
744:{\displaystyle A}
720:{\displaystyle B}
700:{\displaystyle A}
680:{\displaystyle B}
660:{\displaystyle A}
167:{\displaystyle X}
124:{\displaystyle X}
56:A. D. Wallace
40:Frigyes Riesz
2647:
2640:General topology
2612:
2611:
2585:
2584:
2575:
2565:
2555:
2554:
2543:
2542:
2337:
2250:
2243:
2236:
2227:
2222:
2197:
2195:
2182:
2142:
2126:
2101:Proximity Spaces
2095:
2071:
2060:
2016:
2014:
2013:
2008:
1990:
1988:
1987:
1982:
1970:
1968:
1967:
1962:
1950:
1948:
1947:
1942:
1920:
1918:
1917:
1912:
1900:
1898:
1897:
1892:
1858:
1856:
1855:
1850:
1848:
1844:
1795:
1793:
1792:
1787:
1772:
1770:
1769:
1764:
1753:
1752:
1728:
1727:
1708:
1706:
1705:
1700:
1695:
1694:
1678:
1676:
1675:
1670:
1665:
1664:
1645:
1643:
1642:
1637:
1632:
1631:
1615:
1613:
1612:
1607:
1592:
1591:
1562:
1560:
1559:
1554:
1539:
1537:
1536:
1531:
1511:
1509:
1508:
1503:
1498:
1494:
1493:
1492:
1480:
1479:
1424:
1422:
1421:
1416:
1395:
1393:
1392:
1387:
1348:
1346:
1345:
1340:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1266:
1264:
1263:
1258:
1228:
1226:
1225:
1220:
1201:
1199:
1198:
1193:
1169:
1167:
1166:
1161:
1143:
1141:
1140:
1135:
1115:
1113:
1112:
1107:
1089:
1087:
1086:
1081:
1050:
1048:
1047:
1042:
1024:
1022:
1021:
1016:
997:
995:
994:
989:
969:
967:
966:
961:
926:For all subsets
919:
917:
916:
911:
893:
891:
890:
885:
873:
871:
870:
865:
844:
842:
841:
836:
819:
817:
816:
811:
794:
792:
791:
786:
770:
768:
767:
762:
750:
748:
747:
742:
726:
724:
723:
718:
706:
704:
703:
698:
686:
684:
683:
678:
666:
664:
663:
658:
646:
644:
643:
638:
606:
604:
603:
598:
578:
576:
575:
570:
538:
536:
535:
530:
511:
509:
508:
503:
485:
483:
482:
477:
457:
455:
454:
449:
429:
427:
426:
421:
388:
386:
385:
380:
360:
358:
357:
352:
327:
325:
324:
319:
301:
299:
298:
293:
272:
270:
269:
264:
244:
242:
241:
236:
214:
212:
211:
206:
177:For all subsets
173:
171:
170:
165:
153:
151:
150:
145:
130:
128:
127:
122:
110:
108:
107:
102:
64:separation space
27:, also called a
2655:
2654:
2650:
2649:
2648:
2646:
2645:
2644:
2625:
2624:
2623:
2618:
2549:
2531:
2527:Urysohn's lemma
2488:
2452:
2338:
2329:
2301:low-dimensional
2259:
2254:
2207:
2204:
2185:
2164:10.2307/1969257
2145:
2129:
2115:
2098:
2078:
2075:
2074:
2061:
2057:
2052:
2027:
1993:
1992:
1991:if and only if
1973:
1972:
1953:
1952:
1933:
1932:
1903:
1902:
1883:
1882:
1876:Urysohn's lemma
1820:
1816:
1805:
1804:
1801:
1775:
1774:
1741:
1716:
1711:
1710:
1686:
1681:
1680:
1656:
1648:
1647:
1623:
1618:
1617:
1583:
1565:
1564:
1542:
1541:
1514:
1513:
1484:
1471:
1470:
1466:
1437:
1436:
1398:
1397:
1358:
1357:
1316:
1315:
1296:
1295:
1270:
1269:
1231:
1230:
1205:
1204:
1172:
1171:
1146:
1145:
1120:
1119:
1092:
1091:
1054:
1053:
1027:
1026:
1001:
1000:
974:
973:
928:
927:
896:
895:
876:
875:
874:if and only if
847:
846:
824:
823:
802:
801:
777:
776:
753:
752:
733:
732:
709:
708:
689:
688:
669:
668:
649:
648:
621:
620:
613:quasi-proximity
581:
580:
541:
540:
513:
512:
491:
490:
460:
459:
432:
431:
392:
391:
363:
362:
331:
330:
304:
303:
276:
275:
247:
246:
219:
218:
179:
178:
156:
155:
136:
135:
113:
112:
81:
80:
77:proximity space
72:
48:V. A. EfremoviÄŤ
25:proximity space
17:
12:
11:
5:
2653:
2651:
2643:
2642:
2637:
2627:
2626:
2620:
2619:
2617:
2616:
2606:
2605:
2604:
2599:
2594:
2579:
2569:
2559:
2547:
2536:
2533:
2532:
2530:
2529:
2524:
2519:
2514:
2509:
2504:
2498:
2496:
2490:
2489:
2487:
2486:
2481:
2476:
2474:Winding number
2471:
2466:
2460:
2458:
2454:
2453:
2451:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2414:
2413:
2408:
2406:homotopy group
2398:
2397:
2396:
2391:
2386:
2381:
2376:
2366:
2361:
2356:
2346:
2344:
2340:
2339:
2332:
2330:
2328:
2327:
2322:
2317:
2316:
2315:
2305:
2304:
2303:
2293:
2288:
2283:
2278:
2273:
2267:
2265:
2261:
2260:
2255:
2253:
2252:
2245:
2238:
2230:
2224:
2223:
2203:
2202:External links
2200:
2199:
2198:
2193:10.1.1.15.1415
2183:
2158:(3): 687–697,
2143:
2127:
2113:
2096:
2073:
2072:
2054:
2053:
2051:
2048:
2047:
2046:
2040:
2034:
2026:
2023:
2006:
2003:
2000:
1980:
1960:
1940:
1910:
1890:
1847:
1843:
1839:
1835:
1832:
1829:
1826:
1823:
1819:
1815:
1812:
1800:
1797:
1785:
1782:
1762:
1759:
1756:
1751:
1748:
1744:
1740:
1737:
1734:
1731:
1726:
1723:
1719:
1698:
1693:
1689:
1668:
1663:
1659:
1655:
1635:
1630:
1626:
1605:
1602:
1599:
1596:
1590:
1586:
1581:
1578:
1575:
1572:
1552:
1549:
1529:
1525:
1521:
1501:
1497:
1491:
1487:
1483:
1478:
1474:
1469:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1434:
1430:
1414:
1411:
1408:
1405:
1385:
1382:
1379:
1375:
1371:
1368:
1365:
1355:
1350:
1349:
1338:
1335:
1332:
1329:
1326:
1323:
1303:
1283:
1280:
1277:
1267:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1218:
1215:
1212:
1202:
1191:
1188:
1185:
1182:
1179:
1159:
1156:
1153:
1133:
1130:
1127:
1116:
1105:
1102:
1099:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1051:
1040:
1037:
1034:
1014:
1011:
1008:
998:
987:
984:
981:
959:
956:
953:
950:
947:
944:
941:
938:
935:
909:
906:
903:
883:
863:
860:
857:
854:
834:
831:
821:
809:
798:
784:
774:
760:
740:
730:
716:
696:
676:
656:
636:
632:
628:
615:
608:
607:
596:
592:
588:
568:
565:
562:
559:
556:
552:
548:
528:
524:
520:
501:
498:
487:
475:
471:
467:
447:
443:
439:
419:
416:
413:
410:
407:
403:
399:
389:
378:
374:
370:
350:
347:
344:
341:
338:
328:
317:
314:
311:
291:
287:
283:
273:
262:
258:
254:
234:
230:
226:
204:
201:
198:
195:
192:
189:
186:
163:
143:
120:
100:
97:
94:
91:
88:
79:
71:
68:
29:nearness space
15:
13:
10:
9:
6:
4:
3:
2:
2652:
2641:
2638:
2636:
2633:
2632:
2630:
2615:
2607:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2588:
2580:
2578:
2574:
2570:
2568:
2564:
2560:
2558:
2553:
2548:
2546:
2538:
2537:
2534:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2499:
2497:
2495:
2491:
2485:
2484:Orientability
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2461:
2459:
2455:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2412:
2409:
2407:
2404:
2403:
2402:
2399:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2371:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2351:
2348:
2347:
2345:
2341:
2336:
2326:
2323:
2321:
2320:Set-theoretic
2318:
2314:
2311:
2310:
2309:
2306:
2302:
2299:
2298:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2281:Combinatorial
2279:
2277:
2274:
2272:
2269:
2268:
2266:
2262:
2258:
2251:
2246:
2244:
2239:
2237:
2232:
2231:
2228:
2220:
2216:
2215:
2210:
2206:
2205:
2201:
2194:
2189:
2184:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2152:Ann. of Math.
2149:
2144:
2141:
2137:
2133:
2128:
2124:
2120:
2116:
2114:0-521-07935-7
2110:
2106:
2102:
2097:
2094:
2090:
2086:
2082:
2077:
2076:
2069:
2065:
2062:W. J. Thron,
2059:
2056:
2049:
2044:
2041:
2038:
2035:
2032:
2029:
2028:
2024:
2022:
2020:
2004:
2001:
1998:
1978:
1958:
1938:
1931:
1930:uniform space
1926:
1924:
1908:
1888:
1879:
1877:
1873:
1868:
1866:
1862:
1845:
1841:
1837:
1830:
1824:
1821:
1817:
1810:
1798:
1796:
1783:
1780:
1757:
1749:
1746:
1742:
1738:
1732:
1724:
1721:
1717:
1696:
1691:
1687:
1666:
1661:
1657:
1653:
1633:
1628:
1624:
1600:
1594:
1588:
1584:
1576:
1570:
1550:
1547:
1527:
1523:
1519:
1499:
1495:
1489:
1485:
1481:
1476:
1472:
1467:
1457:
1454:
1451:
1445:
1442:
1432:
1428:
1425:
1412:
1409:
1406:
1403:
1380:
1373:
1366:
1353:
1336:
1333:
1330:
1327:
1324:
1321:
1301:
1281:
1278:
1275:
1268:
1254:
1248:
1245:
1242:
1236:
1216:
1213:
1210:
1203:
1189:
1186:
1183:
1180:
1177:
1157:
1154:
1151:
1131:
1128:
1125:
1117:
1103:
1100:
1097:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1052:
1038:
1035:
1032:
1012:
1009:
1006:
999:
985:
982:
979:
972:
971:
970:
957:
954:
951:
948:
945:
942:
939:
936:
933:
924:
921:
907:
901:
881:
861:
858:
855:
852:
832:
829:
820:-neighborhood
807:
800:
796:
782:
772:
758:
738:
728:
714:
694:
674:
654:
634:
630:
626:
617:
614:
611:
594:
590:
586:
563:
557:
550:
546:
526:
522:
518:
499:
496:
488:
473:
469:
465:
445:
441:
437:
414:
411:
408:
401:
397:
390:
376:
372:
368:
345:
342:
339:
336:
329:
312:
309:
289:
285:
281:
274:
260:
256:
252:
232:
228:
224:
217:
216:
215:
202:
199:
196:
193:
190:
187:
184:
175:
161:
141:
134:
118:
95:
92:
89:
78:
75:
69:
67:
65:
61:
57:
53:
49:
45:
41:
36:
34:
30:
26:
22:
2614:Publications
2479:Chern number
2469:Betti number
2352: /
2343:Key concepts
2291:Differential
2212:
2155:
2151:
2131:
2100:
2084:
2080:
2067:
2063:
2058:
2031:Cauchy space
1927:
1880:
1869:
1802:
1433:proximal map
1426:
1351:
925:
922:
618:
612:
609:
176:
76:
73:
63:
51:
37:
28:
24:
18:
2577:Wikiversity
2494:Key results
2087:: 341–343,
920:are apart.
2629:Categories
2423:CW complex
2364:Continuity
2354:Closed set
2313:cohomology
2140:40.0098.07
2123:0206.24601
2050:References
1799:Properties
1314:such that
1170:) implies
579:) implies
70:Definition
2602:geometric
2597:algebraic
2448:Cobordism
2384:Hausdorff
2379:connected
2296:Geometric
2286:Continuum
2276:Algebraic
2219:EMS Press
2188:CiteSeerX
2134:: 18–24,
2002:×
1865:Hausdorff
1838:δ
1814:↦
1773:holds in
1747:−
1739:≪
1722:−
1692:∗
1679:holds in
1662:∗
1658:≪
1629:∗
1589:∗
1585:δ
1524:δ
1490:∗
1486:δ
1477:∗
1464:→
1458:δ
1429:proximity
1374:δ
1354:separated
1331:≪
1325:≪
1279:≪
1252:∖
1246:≪
1240:∖
1214:≪
1187:∩
1181:≪
1155:≪
1129:≪
1101:≪
1075:⊆
1069:≪
1063:⊆
1036:⊆
1010:≪
983:≪
955:⊆
905:∖
856:≪
808:δ
797:proximal-
775:. We say
631:δ
591:δ
561:∖
551:δ
523:δ
489:(For all
470:δ
442:δ
430:implies (
412:∪
402:δ
373:δ
349:∅
346:≠
340:∩
316:∅
313:≠
286:δ
257:δ
229:δ
200:⊆
142:δ
111:is a set
96:δ
2567:Wikibook
2545:Category
2433:Manifold
2401:Homotopy
2359:Interior
2350:Open set
2308:Homology
2257:Topology
2025:See also
1971:is near
1901:is near
1396:implies
1229:implies
1090:implies
1025:implies
845:written
729:proximal
667:is near
361:implies
302:implies
245:implies
133:relation
21:topology
2592:general
2394:uniform
2374:compact
2325:Digital
2221:, 2001
2180:0004756
2172:1969257
2093:0040748
647:we say
131:with a
58: (
42: (
2587:Topics
2389:metric
2264:Fields
2190:
2178:
2170:
2138:
2121:
2111:
2091:
2369:Space
2168:JSTOR
2154:, 2,
1859:be a
1709:then
1563:then
795:is a
773:apart
2109:ISBN
1144:and
894:and
771:are
751:and
727:are
707:and
60:1941
44:1909
23:, a
2160:doi
2136:JFM
2119:Zbl
1616:in
1540:in
1512:if
1431:or
1356:if
822:of
799:or
687:or
619:If
539:or
458:or
19:In
2631::
2217:,
2211:,
2176:MR
2174:,
2166:,
2156:42
2150:,
2117:.
2107:.
2089:MR
2085:76
1928:A
1427:A
74:A
66:.
35:.
2249:e
2242:t
2235:v
2196:.
2162::
2125:.
2005:B
1999:A
1979:B
1959:A
1939:X
1909:B
1889:A
1846:}
1842:A
1834:}
1831:x
1828:{
1825::
1822:x
1818:{
1811:A
1784:.
1781:X
1761:]
1758:D
1755:[
1750:1
1743:f
1736:]
1733:C
1730:[
1725:1
1718:f
1697:,
1688:X
1667:D
1654:C
1634:.
1625:X
1604:]
1601:B
1598:[
1595:f
1580:]
1577:A
1574:[
1571:f
1551:,
1548:X
1528:B
1520:A
1500:,
1496:)
1482:,
1473:X
1468:(
1461:)
1455:,
1452:X
1449:(
1446::
1443:f
1413:.
1410:y
1407:=
1404:x
1384:}
1381:y
1378:{
1370:}
1367:x
1364:{
1337:.
1334:B
1328:E
1322:A
1302:E
1282:B
1276:A
1255:A
1249:X
1243:B
1237:X
1217:B
1211:A
1190:C
1184:B
1178:A
1158:C
1152:A
1132:B
1126:A
1118:(
1104:D
1098:A
1078:D
1072:C
1066:B
1060:A
1039:B
1033:A
1013:B
1007:A
986:X
980:X
958:X
952:D
949:,
946:C
943:,
940:B
937:,
934:A
908:B
902:X
882:A
862:,
859:B
853:A
833:,
830:A
783:B
759:B
739:A
715:B
695:A
675:B
655:A
635:B
627:A
595:B
587:A
567:)
564:E
558:X
555:(
547:B
527:E
519:A
500:,
497:E
486:)
474:C
466:A
446:B
438:A
418:)
415:C
409:B
406:(
398:A
377:B
369:A
343:B
337:A
310:A
290:B
282:A
261:A
253:B
233:B
225:A
203:X
197:C
194:,
191:B
188:,
185:A
162:X
119:X
99:)
93:,
90:X
87:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.