32:
956:
308:
662:
1401:
798:
594:
1159:
1048:
1089:
1234:
716:
499:
461:
749:
419:
238:
686:
1282:
1686:
1527:
369:
993:
1863:
1813:
1720:
1557:
1308:
1188:
1886:
1763:
1612:
1426:
1022:
526:
1837:
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1740:
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1459:
1208:
1133:
1113:
546:
193:
157:
247:
602:
2108:
1313:
2065:
96:
68:
2152:
2120:
2098:
2050:
49:
75:
1911:(FIP) is non-empty; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is
2005:
569:
115:
82:
1967:
53:
64:
1138:
1027:
1908:
1588:
1065:
1213:
691:
466:
428:
951:{\displaystyle \left\{\,\{w\in X:f(x)=f(w)\}~:~x\in X\,\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.}
725:
1959:
378:
2147:
1889:
1639:
216:
667:
42:
20:
1258:
137:
89:
1659:
1500:
342:
1991:
1953:
963:
1842:
1792:
1699:
1536:
1287:
1564:
172:
168:
1164:
8:
2157:
1924:
1689:
1651:
1560:
1486:
1482:
1059:
1051:
788:
311:
1868:
1745:
1594:
1408:
1004:
508:
1822:
1768:
1725:
1617:
1570:
1464:
1444:
1193:
1118:
1098:
531:
199:
178:
142:
2066:
A space is compact iff any family of closed sets having fip has non-empty intersection
16:
Equivalence relation expressing that two elements have the same image under a function
2162:
2126:
2116:
2094:
2046:
2001:
1963:
1945:
1896:
1693:
1253:
1092:
372:
1949:
1436:
319:
1786:
1245:
1490:
211:
2141:
2130:
2093:. Oxford Logic Guides. Vol. 49 (2nd ed.). Oxford University Press.
2038:
1900:
2086:
1530:
1494:
792:
502:
241:
1635:
315:
303:{\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.}
2069:
2021:
2019:
2017:
1904:
1816:
657:{\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.}
129:
175:
that roughly expresses the idea of "equivalent as far as the function
1997:
1567:
that is compatible with the algebraic structure), and the coimage of
1055:
719:
2014:
31:
997:
1249:
1638:
in the algebraic sense; this is the most general form of the
1405:
The study of the properties of this subset can shed light on
779:
if it is not fixed; that is, if its kernel is the empty set.
1239:
1871:
1845:
1825:
1795:
1771:
1748:
1728:
1702:
1662:
1620:
1597:
1573:
1539:
1503:
1467:
1447:
1411:
1316:
1290:
1261:
1216:
1196:
1167:
1141:
1121:
1101:
1068:
1030:
1007:
966:
801:
728:
694:
670:
605:
572:
534:
511:
469:
431:
381:
345:
250:
219:
181:
145:
1310:(or a variation) and may be defined symbolically as
2115:. New Jersey: World Scientific Publishing Company.
1993:
Universal
Algebra: Fundamentals and Selected Topics
1614:The bijection between the coimage and the image of
56:. Unsourced material may be challenged and removed.
2045:. New Delhi: Prentice-Hall of India. p. 169.
1880:
1857:
1831:
1807:
1777:
1757:
1734:
1714:
1680:
1626:
1606:
1579:
1551:
1521:
1473:
1453:
1420:
1395:
1302:
1276:
1248:, the kernel of a function may be thought of as a
1228:
1202:
1182:
1153:
1127:
1107:
1083:
1042:
1016:
987:
950:
743:
710:
680:
656:
588:
540:
520:
493:
455:
413:
363:
302:
232:
187:
151:
787:Like any equivalence relation, the kernel can be
751:is typically left undefined. A family is called
2139:
1944:
1938:
1927: – Family of sets representing "large" sets
1996:, Pure and Applied Mathematics, vol. 301,
589:{\displaystyle {\mathcal {B}}\neq \varnothing }
1396:{\displaystyle \ker f:=\{(x,x'):f(x)=f(x')\}.}
2107:
2025:
1903:if and only if the kernel of every family of
1387:
1329:
850:
808:
1892:topology, must also be a Hausdorff space.
548:is the equivalence relation thus defined.
1985:
1983:
1981:
1979:
1284:In this guise, the kernel may be denoted
871:
807:
795:, and the quotient set is the partition:
310:This definition is used in the theory of
293:
116:Learn how and when to remove this message
2037:
1989:
1430:
771:is not empty. A family is said to be
2140:
2085:
2031:
1976:
1154:{\displaystyle \operatorname {coim} f}
1043:{\displaystyle \operatorname {coim} f}
1865:is a closed set, then the coimage of
1084:{\displaystyle \operatorname {im} f;}
1240:As a subset of the Cartesian product
1229:{\displaystyle \operatorname {im} f}
711:{\displaystyle \cap {\mathcal {B}}.}
54:adding citations to reliable sources
25:
2113:Convergence Foundations Of Topology
1696:then the topological properties of
505:, that is, are the same element of
494:{\displaystyle f\left(x_{2}\right)}
456:{\displaystyle f\left(x_{1}\right)}
206:An unrelated notion is that of the
13:
744:{\displaystyle \ker \varnothing ,}
700:
673:
641:
614:
575:
286:
259:
222:
14:
2174:
1054:(in the set-theoretic sense of a
1050:(or a variation). The coimage is
735:
583:
414:{\displaystyle x_{1},x_{2}\in X}
30:
2079:
339:For the formal definition, let
233:{\displaystyle {\mathcal {B}},}
41:needs additional citations for
2059:
1672:
1645:
1513:
1384:
1373:
1364:
1358:
1349:
1332:
1177:
1171:
937:
931:
910:
904:
847:
841:
832:
826:
681:{\displaystyle {\mathcal {B}}}
355:
1:
1931:
1722:can shed light on the spaces
688:is also sometimes denoted by
325:
2153:Basic concepts in set theory
1909:finite intersection property
1485:of some fixed type (such as
782:
163:) may be taken to be either
7:
2111:; Mynard, Frédéric (2016).
1918:
240:which by definition is the
10:
2179:
1990:Bergman, Clifford (2011),
1960:Chelsea Publishing Company
1912:
1649:
1434:
1277:{\displaystyle X\times X.}
554:Kernel of a family of sets
371:be a function between two
314:to classify them as being
18:
2026:Dolecki & Mynard 2016
1839:is a Hausdorff space and
1640:first isomorphism theorem
2028:, pp. 27–29, 33–35.
1681:{\displaystyle f:X\to Y}
1522:{\displaystyle f:X\to Y}
1210:(which is an element of
1135:(which is an element of
364:{\displaystyle f:X\to Y}
65:"Kernel" set theory
1497:), and if the function
988:{\displaystyle X/=_{f}}
21:Kernel (disambiguation)
1882:
1859:
1858:{\displaystyle \ker f}
1833:
1809:
1808:{\displaystyle \ker f}
1779:
1759:
1736:
1716:
1715:{\displaystyle \ker f}
1682:
1628:
1608:
1581:
1553:
1552:{\displaystyle \ker f}
1523:
1475:
1455:
1422:
1397:
1304:
1303:{\displaystyle \ker f}
1278:
1230:
1204:
1184:
1155:
1129:
1109:
1085:
1044:
1018:
989:
952:
763:non-empty intersection
745:
712:
682:
658:
590:
542:
522:
495:
457:
415:
365:
304:
234:
189:
153:
1883:
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1810:
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1717:
1683:
1629:
1609:
1582:
1554:
1524:
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1305:
1279:
1231:
1205:
1185:
1156:
1130:
1110:
1086:
1045:
1019:
990:
953:
746:
713:
683:
659:
591:
543:
523:
496:
458:
416:
366:
305:
244:of all its elements:
235:
190:
154:
1869:
1843:
1823:
1793:
1769:
1746:
1726:
1700:
1660:
1618:
1595:
1571:
1565:equivalence relation
1537:
1501:
1483:algebraic structures
1465:
1445:
1431:Algebraic structures
1409:
1314:
1288:
1259:
1214:
1194:
1183:{\displaystyle f(x)}
1165:
1139:
1119:
1099:
1066:
1052:naturally isomorphic
1028:
1005:
964:
799:
759:and is said to have
726:
692:
668:
603:
570:
532:
509:
467:
429:
379:
343:
333:Kernel of a function
248:
217:
179:
169:equivalence relation
143:
50:improve this article
19:For other uses, see
1925:Filter (set theory)
1690:continuous function
1652:Filters in topology
1561:congruence relation
2000:, pp. 14–16,
1946:Mac Lane, Saunders
1881:{\displaystyle f,}
1878:
1855:
1829:
1805:
1775:
1758:{\displaystyle Y.}
1755:
1732:
1712:
1694:topological spaces
1678:
1624:
1607:{\displaystyle X.}
1604:
1577:
1549:
1519:
1471:
1451:
1421:{\displaystyle f.}
1418:
1393:
1300:
1274:
1226:
1200:
1180:
1151:
1125:
1105:
1091:specifically, the
1081:
1040:
1017:{\displaystyle f,}
1014:
985:
960:This quotient set
948:
741:
718:The kernel of the
708:
678:
654:
647:
586:
538:
521:{\displaystyle Y.}
518:
491:
453:
411:
361:
300:
292:
230:
198:the corresponding
185:
171:on the function's
161:equivalence kernel
149:
2122:978-981-4571-52-4
2100:978-0-19-923718-0
2052:978-81-203-2046-8
1950:Birkhoff, Garrett
1832:{\displaystyle X}
1819:. Conversely, if
1778:{\displaystyle Y}
1735:{\displaystyle X}
1627:{\displaystyle f}
1580:{\displaystyle f}
1474:{\displaystyle Y}
1454:{\displaystyle X}
1254:Cartesian product
1203:{\displaystyle Y}
1161:) corresponds to
1128:{\displaystyle X}
1108:{\displaystyle x}
1093:equivalence class
921:
915:
885:
879:
861:
855:
628:
627:
621:
541:{\displaystyle f}
273:
272:
266:
188:{\displaystyle f}
152:{\displaystyle f}
126:
125:
118:
100:
2170:
2148:Abstract algebra
2134:
2104:
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2012:
2010:
1987:
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1776:
1765:For example, if
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1713:
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1528:
1526:
1525:
1520:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1437:Kernel (algebra)
1427:
1425:
1424:
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1402:
1400:
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1383:
1348:
1309:
1307:
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1301:
1283:
1281:
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1227:
1209:
1207:
1206:
1201:
1189:
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1181:
1160:
1158:
1157:
1152:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1090:
1088:
1087:
1082:
1049:
1047:
1046:
1041:
1023:
1021:
1020:
1015:
1001:of the function
994:
992:
991:
986:
984:
983:
974:
957:
955:
954:
949:
944:
940:
919:
913:
903:
902:
883:
877:
876:
872:
859:
853:
777:
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765:
764:
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756:
750:
748:
747:
742:
717:
715:
714:
709:
704:
703:
687:
685:
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676:
663:
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619:
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587:
579:
578:
556:
555:
547:
545:
544:
539:
527:
525:
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519:
500:
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492:
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485:
462:
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459:
454:
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448:
447:
420:
418:
417:
412:
404:
403:
391:
390:
370:
368:
367:
362:
335:
334:
309:
307:
306:
301:
291:
290:
289:
270:
264:
263:
262:
239:
237:
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231:
226:
225:
194:
192:
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158:
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150:
121:
114:
110:
107:
101:
99:
58:
34:
26:
2178:
2177:
2173:
2172:
2171:
2169:
2168:
2167:
2138:
2137:
2123:
2109:Dolecki, Szymon
2101:
2091:Category Theory
2082:
2077:
2076:
2064:
2060:
2053:
2036:
2032:
2024:
2015:
2008:
1988:
1977:
1970:
1943:
1939:
1934:
1921:
1870:
1867:
1866:
1844:
1841:
1840:
1824:
1821:
1820:
1794:
1791:
1790:
1787:Hausdorff space
1770:
1767:
1766:
1747:
1744:
1743:
1727:
1724:
1723:
1701:
1698:
1697:
1661:
1658:
1657:
1654:
1648:
1619:
1616:
1615:
1596:
1593:
1592:
1572:
1569:
1568:
1538:
1535:
1534:
1502:
1499:
1498:
1466:
1463:
1462:
1446:
1443:
1442:
1439:
1433:
1410:
1407:
1406:
1376:
1341:
1315:
1312:
1311:
1289:
1286:
1285:
1260:
1257:
1256:
1246:binary relation
1242:
1215:
1212:
1211:
1195:
1192:
1191:
1166:
1163:
1162:
1140:
1137:
1136:
1120:
1117:
1116:
1100:
1097:
1096:
1067:
1064:
1063:
1029:
1026:
1025:
1006:
1003:
1002:
979:
975:
970:
965:
962:
961:
895:
891:
890:
886:
806:
802:
800:
797:
796:
785:
774:
773:
762:
761:
754:
753:
727:
724:
723:
699:
698:
693:
690:
689:
672:
671:
669:
666:
665:
640:
639:
632:
613:
612:
604:
601:
600:
574:
573:
571:
568:
567:
562:
561:
553:
552:
533:
530:
529:
510:
507:
506:
481:
477:
473:
468:
465:
464:
443:
439:
435:
430:
427:
426:
399:
395:
386:
382:
380:
377:
376:
344:
341:
340:
332:
331:
328:
285:
284:
277:
258:
257:
249:
246:
245:
221:
220:
218:
215:
214:
210:of a non-empty
180:
177:
176:
144:
141:
140:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
2176:
2166:
2165:
2160:
2155:
2150:
2136:
2135:
2121:
2105:
2099:
2081:
2078:
2075:
2074:
2058:
2051:
2039:Munkres, James
2030:
2013:
2006:
1975:
1968:
1962:, p. 33,
1936:
1935:
1933:
1930:
1929:
1928:
1920:
1917:
1905:closed subsets
1890:quotient space
1877:
1874:
1854:
1851:
1848:
1828:
1804:
1801:
1798:
1774:
1754:
1751:
1731:
1711:
1708:
1705:
1677:
1674:
1671:
1668:
1665:
1647:
1644:
1623:
1603:
1600:
1576:
1548:
1545:
1542:
1518:
1515:
1512:
1509:
1506:
1470:
1450:
1432:
1429:
1417:
1414:
1392:
1389:
1386:
1382:
1379:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1347:
1344:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1299:
1296:
1293:
1273:
1270:
1267:
1264:
1241:
1238:
1225:
1222:
1219:
1199:
1179:
1176:
1173:
1170:
1150:
1147:
1144:
1124:
1104:
1080:
1077:
1074:
1071:
1039:
1036:
1033:
1013:
1010:
995:is called the
982:
978:
973:
969:
947:
943:
939:
936:
933:
930:
927:
924:
918:
912:
909:
906:
901:
898:
894:
889:
882:
875:
870:
867:
864:
858:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
805:
784:
781:
778:
770:
766:
758:
740:
737:
734:
731:
707:
702:
697:
675:
664:The kernel of
653:
650:
643:
638:
635:
631:
624:
616:
611:
608:
585:
582:
577:
537:
528:The kernel of
517:
514:
489:
484:
480:
476:
472:
451:
446:
442:
438:
434:
410:
407:
402:
398:
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389:
385:
360:
357:
354:
351:
348:
327:
324:
299:
296:
288:
283:
280:
276:
269:
261:
256:
253:
229:
224:
212:family of sets
204:
203:
202:of the domain.
196:
184:
148:
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
2175:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2145:
2143:
2132:
2128:
2124:
2118:
2114:
2110:
2106:
2102:
2096:
2092:
2088:
2087:Awodey, Steve
2084:
2083:
2071:
2067:
2062:
2054:
2048:
2044:
2040:
2034:
2027:
2022:
2020:
2018:
2009:
2007:9781439851296
2003:
1999:
1995:
1994:
1986:
1984:
1982:
1980:
1971:
1965:
1961:
1957:
1956:
1951:
1947:
1941:
1937:
1926:
1923:
1922:
1916:
1914:
1910:
1906:
1902:
1898:
1893:
1891:
1888:if given the
1875:
1872:
1852:
1849:
1846:
1826:
1818:
1802:
1799:
1796:
1788:
1772:
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66:
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61:Find sources:
55:
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37:
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28:
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2080:Bibliography
2061:
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1992:
1954:
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1655:
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566:of a family
563:
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242:intersection
207:
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127:
112:
103:
93:
86:
79:
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48:Please help
43:verification
40:
1907:having the
1646:In topology
1636:isomorphism
375:. Elements
2158:Set theory
2142:Categories
2070:PlanetMath
1969:0821816462
1932:References
1817:closed set
1815:must be a
1650:See also:
1435:See also:
791:to form a
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326:Definition
130:set theory
76:newspapers
2131:945169917
2089:(2010) .
1998:CRC Press
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2043:Topology
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