258:
781:
is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of
1683:
1433:
2456:
2095:
1314:
102:
1637:
1563:
516:, p. 109, Th 16.4c). A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable.
2131:
945:
1715:
1153:
1343:
1117:
832:
307:
2369:
2255:
1249:
856:
775:
2032:
969:
446:
1997:
1837:
1585:
886:
2286:
2218:
1390:
1276:
1082:
713:
2782:
is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that
2174:
571:
1969:
406:
2740:
1430:
1410:
1363:
1195:
627:
912:
674:
598:
473:
2800:
2780:
2760:
2688:
2154:
1531:
1038:
1931:
1878:
1759:
1481:
Any topological space that is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that
1452:, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into
2708:
1499:
1215:
1175:
1009:
989:
800:
647:
538:
347:
327:
278:
180:
1798:
1456:
for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the
2602:
188:
154:
is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the
1642:
3009:
1316:
of all functions from the real line to itself, endowed with the product topology, is a separable
Hausdorff space of cardinality
2522:
Every separable metric space is isometric to a subset of C(), the separable Banach space of continuous functions â
2978:
2944:
2860:
127:
on a separable space whose image is a subset of a
Hausdorff space is determined by its values on the countable dense subset.
2344:, p. 49); if the space was a Hausdorff space then the space constructed that it embeds into is also a Hausdorff space.
520:
We can construct an example of a separable topological space that is not second countable. Consider any uncountable set
2299:
is not separable; note however that this space has very important applications in mathematics, physics and engineering.
2420:
1084:. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection
2612:
2044:
182:
1288:
494:
of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
56:
1596:
123:, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every
1805:
1540:
2372:
2100:
751:
917:
2926:
2494:
1449:
1688:
1122:
1319:
1087:
808:
283:
3004:
2350:
2233:
2191:
1220:
837:
756:
2013:
950:
411:
1881:
1980:
1811:
1568:
865:
2646:
2542:
2264:
2196:
1457:
1368:
1254:
1043:
679:
2347:
The set of all real-valued continuous functions on a separable space has a cardinality equal to
2159:
543:
2922:
2899:
2007:
1947:
1944:. It follows that any separable, infinite-dimensional Hilbert space is isometric to the space
1157:
The same arguments establish a more general result: suppose that a
Hausdorff topological space
378:
372:
112:
2713:
2134:
1415:
1395:
1348:
1180:
603:
1685:
whose Ï-algebra is countably generated and whose measure is Ï-finite, are separable for any
891:
2988:
2954:
2911:
2870:
2661:
2606:
2177:
2039:
2000:
652:
576:
479:
is separable if and only if it is second countable, which is the case if and only if it is
451:
2785:
2765:
2745:
2673:
2139:
1507:
1014:
8:
2310:
1895:
1842:
1839:
of polynomials in one variable with rational coefficients is a countable dense subset of
1801:
1762:
1723:
859:
740:
491:
131:
124:
2665:
2932:
2693:
2651:
1484:
1445:
1200:
1160:
994:
974:
785:
778:
632:
523:
332:
312:
263:
165:
151:
2963:
1771:
750:, separable Hausdorff space (in particular, a separable metric space) has at most the
2974:
2940:
2918:
2856:
2296:
1941:
502:
35:
2821:
2375:. This follows since such functions are determined by their values on dense subsets.
715:
is open. Therefore, the space is separable but there cannot have a countable base.
480:
2886:
2391:
2314:
858:
is the cardinality of the continuum. For this closure is characterized in terms of
728:
723:
The property of separability does not in and of itself give any limitations on the
509:
476:
159:
47:
24:
2877:
Kleiber, Martin; Pervin, William J. (1969), "A generalized Banach-Mazur theorem",
2984:
2950:
2936:
2907:
2866:
2852:
2638:
2401:
1975:
1889:
747:
739:. The "trouble" with the trivial topology is its poor separation properties: its
736:
350:
120:
20:
2336:
In fact, every topological space is a subspace of a separable space of the same
2970:
2906:, Mathematical Expositions, No. 7, Toronto, Ont.: University of Toronto Press,
2844:
2806:
2221:
357:
2891:
2998:
2634:
2531:
2398:
1937:
1766:
1534:
1281:
The product of at most continuum many separable spaces is a separable space (
732:
43:
2557:
2527:
2512:
2505:
2490:
2486:
2463:
2387:
2330:
2292:
2258:
2228:
2035:
1885:
1475:
253:{\displaystyle {\boldsymbol {r}}=(r_{1},\ldots ,r_{n})\in \mathbb {Q} ^{n}}
2382:
is a separable space having an uncountable closed discrete subspace, then
2337:
2318:
724:
116:
105:
31:
573:, and define the topology to be the collection of all sets that contain
147:
135:
115:, separability is a "limitation on size", not necessarily in terms of
2656:
2561:
1453:
155:
1974:
An example of a separable space that is not second-countable is the
2501:
2340:. A construction adding at most countably many points is given in (
1591:
51:
1678:{\displaystyle \left\langle X,{\mathcal {M}},\mu \right\rangle }
162:
form a countable dense subset. Similarly the set of all length-
134:, which is in general stronger but equivalent on the class of
2639:"Properties of the class of measure separable compact spaces"
2541:
Every separable metric space is isometric to a subset of the
108:
of the space contains at least one element of the sequence.
2576:, the space of real continuous functions on the product of
2378:
From the above property, one can deduce the following: If
366:
512:
of at most continuum many separable spaces is separable (
505:
of a second-countable space need not be second countable.
497:
Any continuous image of a separable space is separable (
356:
A simple example of a space that is not separable is a
260:, is a countable dense subset of the set of all length-
2809:
this metric space is separable as a topological space.
2788:
2768:
2748:
2716:
2696:
2676:
2423:
2353:
2267:
2236:
2199:
2162:
2142:
2103:
2047:
2016:
1983:
1950:
1898:
1845:
1814:
1774:
1726:
1691:
1645:
1599:
1571:
1543:
1510:
1487:
1418:
1398:
1371:
1351:
1322:
1291:
1257:
1223:
1203:
1183:
1163:
1125:
1090:
1046:
1017:
997:
977:
953:
920:
894:
868:
840:
811:
788:
759:
682:
655:
635:
606:
579:
546:
526:
454:
414:
381:
335:
315:
286:
266:
191:
168:
59:
2479:
2313:
of a separable space need not be separable (see the
1365:
is any infinite cardinal, then a product of at most
805:
A separable
Hausdorff space has cardinality at most
2962:
2794:
2774:
2754:
2734:
2702:
2682:
2450:
2363:
2280:
2249:
2212:
2168:
2148:
2125:
2089:
2026:
1991:
1963:
1925:
1872:
1831:
1792:
1753:
1709:
1677:
1631:
1579:
1557:
1525:
1493:
1424:
1404:
1384:
1357:
1337:
1308:
1285:, p. 109, Th 16.4c). In particular the space
1270:
1243:
1209:
1189:
1169:
1147:
1111:
1076:
1032:
1003:
983:
963:
939:
906:
880:
850:
826:
794:
769:
707:
668:
641:
621:
592:
565:
532:
467:
440:
400:
341:
321:
301:
272:
252:
174:
104:of elements of the space such that every nonempty
96:
727:of a topological space: any set endowed with the
130:Contrast separability with the related notion of
2996:
2742:is the Boolean algebra of all Borel sets modulo
2329:, Th 16.4b). Also every subspace of a separable
1804:is a separable space, since it follows from the
2034:that is a separable space when considered as a
1940:is separable if and only if it has a countable
16:Topological space with a dense countable subset
2601:
2451:{\displaystyle {\mathcal {C}}(X,\mathbb {R} )}
475:gives a countable dense subset. Conversely, a
2935:reprint of 1978 ed.), Berlin, New York:
2876:
2632:
2618:
2581:
2090:{\displaystyle \rho (A,B)=\mu (A\triangle B)}
2917:
2325:subspace of a separable space is separable (
1999:, the set of real numbers equipped with the
702:
683:
395:
382:
74:
60:
2515:; this is known as the Fréchet embedding. (
1439:
1309:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
731:is separable, as well as second countable,
2898:
2493:. This is established in the proof of the
2341:
1501:-dimensional Euclidean space is separable.
1412:has itself a dense subset of size at most
1392:spaces with dense subsets of size at most
947:if and only if there exists a filter base
2890:
2655:
2441:
1985:
1816:
1573:
1551:
1300:
1294:
486:To further compare these two properties:
289:
240:
97:{\displaystyle \{x_{n}\}_{n=1}^{\infty }}
2819:
2535:
2516:
2261:, is not separable. The same holds for
2257:of all bounded real sequences, with the
1888:is isometrically isomorphic to a closed
1632:{\displaystyle L^{p}\left(X,\mu \right)}
1444:Separability is especially important in
2960:
2511:of all bounded real sequences with the
2458:of continuous real-valued functions on
2326:
2184:
1558:{\displaystyle K\subseteq \mathbb {R} }
1282:
1177:contains a dense subset of cardinality
513:
498:
367:Separability versus second countability
193:
2997:
2843:
649:is the smallest closed set containing
600:(or are empty). Then, the closure of
2823:Geometric embeddings of metric spaces
2126:{\displaystyle A,B\in {\mathcal {F}}}
146:Any topological space that is itself
1434:HewittâMarczewskiâPondiczery theorem
940:{\displaystyle z\in {\overline {Y}}}
2504:to a subset of the (non-separable)
2356:
1533:of all continuous functions from a
1478:(or metrizable space) is separable.
1468:
1463:
1329:
843:
818:
762:
13:
2426:
2273:
2242:
2163:
2118:
2078:
2019:
1710:{\displaystyle 1\leq p<\infty }
1704:
1659:
1148:{\displaystyle {\overline {Y}}=X.}
956:
408:is a countable base, choosing any
363:Further examples are given below.
89:
14:
3021:
2480:Embedding separable metric spaces
1806:Weierstrass approximation theorem
1338:{\displaystyle 2^{\mathfrak {c}}}
1112:{\displaystyle S(Y)\rightarrow X}
827:{\displaystyle 2^{\mathfrak {c}}}
141:
3010:Properties of topological spaces
2500:Every separable metric space is
2485:Every separable metric space is
2407:, the following are equivalent:
1763:continuous real-valued functions
1040:of such filter bases is at most
302:{\displaystyle \mathbb {R} ^{n}}
119:(though, in the presence of the
50:subset; that is, there exists a
2820:Heinonen, Juha (January 2003),
2613:Springer Science+Business Media
2364:{\displaystyle {\mathfrak {c}}}
2250:{\displaystyle \ell ^{\infty }}
1244:{\displaystyle 2^{2^{\kappa }}}
851:{\displaystyle {\mathfrak {c}}}
770:{\displaystyle {\mathfrak {c}}}
2729:
2717:
2626:
2595:
2580:copies of the unit interval. (
2568:is isometric to a subspace of
2564:equal to an infinite cardinal
2445:
2431:
2297:functions of bounded variation
2084:
2072:
2063:
2051:
2027:{\displaystyle {\mathcal {F}}}
1920:
1917:
1905:
1902:
1867:
1864:
1852:
1849:
1826:
1820:
1787:
1775:
1748:
1745:
1733:
1730:
1520:
1514:
1103:
1100:
1094:
1066:
1058:
1027:
1021:
964:{\displaystyle {\mathcal {B}}}
718:
441:{\displaystyle x_{n}\in U_{n}}
232:
200:
1:
2588:
2303:
1971:of square-summable sequences.
1011:. The cardinality of the set
676:), but every set of the form
2373:cardinality of the continuum
2220:, equipped with its natural
1992:{\displaystyle \mathbb {S} }
1832:{\displaystyle \mathbb {Q} }
1580:{\displaystyle \mathbb {R} }
1131:
932:
881:{\displaystyle Y\subseteq X}
360:of uncountable cardinality.
7:
2928:Counterexamples in Topology
2495:Urysohn metrization theorem
2281:{\displaystyle L^{\infty }}
2213:{\displaystyle \omega _{1}}
1884:asserts that any separable
1385:{\displaystyle 2^{\kappa }}
1271:{\displaystyle 2^{\kappa }}
1077:{\displaystyle 2^{2^{|Y|}}}
708:{\displaystyle \{x_{0},x\}}
10:
3026:
2169:{\displaystyle \triangle }
1278:if it is first countable.
566:{\displaystyle x_{0}\in X}
18:
2961:Willard, Stephen (1970),
2892:10.1017/S0004972700041411
2879:Bull. Austral. Math. Soc.
2710:, the measure algebra of
2619:
2582:Kleiber & Pervin 1969
2192:first uncountable ordinal
1964:{\displaystyle \ell ^{2}}
971:consisting of subsets of
401:{\displaystyle \{U_{n}\}}
280:vectors of real numbers,
2735:{\displaystyle (X,\mu )}
2008:separable σ-algebra
1450:constructive mathematics
1440:Constructive mathematics
1251:and cardinality at most
1217:has cardinality at most
743:is the one-point space.
19:Not to be confused with
2647:Fundamenta Mathematicae
2550:For nonseparable spaces
2543:Urysohn universal space
1639:, over a measure space
1425:{\displaystyle \kappa }
1405:{\displaystyle \kappa }
1358:{\displaystyle \kappa }
1190:{\displaystyle \kappa }
622:{\displaystyle {x_{0}}}
2923:Seebach, J. Arthur Jr.
2796:
2776:
2756:
2736:
2704:
2690:is a Borel measure on
2684:
2452:
2390:. This shows that the
2365:
2282:
2251:
2214:
2170:
2150:
2127:
2091:
2028:
1993:
1965:
1927:
1874:
1833:
1794:
1755:
1711:
1679:
1633:
1581:
1559:
1527:
1495:
1426:
1406:
1386:
1359:
1339:
1310:
1272:
1245:
1211:
1191:
1171:
1149:
1113:
1078:
1034:
1005:
985:
965:
941:
908:
907:{\displaystyle z\in X}
882:
860:limits of filter bases
852:
828:
796:
771:
709:
670:
643:
623:
594:
567:
534:
469:
442:
402:
373:second-countable space
343:
323:
303:
274:
254:
176:
113:axioms of countability
98:
2797:
2777:
2757:
2737:
2705:
2685:
2453:
2366:
2283:
2252:
2215:
2171:
2151:
2128:
2092:
2029:
1994:
1966:
1928:
1875:
1834:
1795:
1756:
1712:
1680:
1634:
1582:
1560:
1528:
1496:
1427:
1407:
1387:
1360:
1345:. More generally, if
1340:
1311:
1273:
1246:
1212:
1192:
1172:
1150:
1114:
1079:
1035:
1006:
986:
966:
942:
909:
883:
853:
829:
797:
772:
752:continuum cardinality
710:
671:
669:{\displaystyle x_{0}}
644:
624:
595:
593:{\displaystyle x_{0}}
568:
535:
501:, Th. 16.4a); even a
470:
468:{\displaystyle U_{n}}
443:
403:
344:
324:
304:
275:
255:
185:of rational numbers,
177:
99:
2851:, Berlin, New York:
2795:{\displaystyle \mu }
2786:
2775:{\displaystyle \mu }
2766:
2755:{\displaystyle \mu }
2746:
2714:
2694:
2683:{\displaystyle \mu }
2674:
2421:
2414:is second countable.
2351:
2265:
2234:
2197:
2185:Non-separable spaces
2178:symmetric difference
2160:
2149:{\displaystyle \mu }
2140:
2101:
2045:
2014:
2010:is a σ-algebra
2001:lower limit topology
1981:
1948:
1896:
1882:BanachâMazur theorem
1843:
1812:
1772:
1724:
1689:
1643:
1597:
1569:
1541:
1526:{\displaystyle C(K)}
1508:
1485:
1416:
1396:
1369:
1349:
1320:
1289:
1255:
1221:
1201:
1181:
1161:
1123:
1088:
1044:
1033:{\displaystyle S(Y)}
1015:
995:
975:
951:
918:
892:
866:
838:
809:
786:
757:
680:
653:
633:
629:is the whole space (
604:
577:
544:
524:
452:
412:
379:
333:
313:
284:
264:
189:
166:
57:
2666:1994math......8201D
2489:to a subset of the
2224:, is not separable.
2133:and a given finite
1926:{\displaystyle C()}
1873:{\displaystyle C()}
1802:uniform convergence
1800:with the metric of
1754:{\displaystyle C()}
1458:HahnâBanach theorem
777:. In such a space,
741:Kolmogorov quotient
448:from the non-empty
132:second countability
125:continuous function
93:
2919:Steen, Lynn Arthur
2900:SierpiĆski, WacĆaw
2792:
2772:
2752:
2732:
2700:
2680:
2448:
2361:
2278:
2247:
2210:
2166:
2146:
2123:
2087:
2024:
1989:
1961:
1923:
1870:
1829:
1790:
1751:
1707:
1675:
1629:
1577:
1555:
1523:
1491:
1446:numerical analysis
1422:
1402:
1382:
1355:
1335:
1306:
1268:
1241:
1207:
1187:
1167:
1145:
1109:
1074:
1030:
1001:
991:that converges to
981:
961:
937:
904:
878:
848:
824:
792:
767:
705:
666:
639:
619:
590:
563:
530:
465:
438:
398:
339:
319:
299:
270:
250:
172:
152:countably infinite
94:
73:
2980:978-0-201-08707-9
2946:978-0-486-68735-3
2862:978-0-387-90125-1
2703:{\displaystyle X}
2633:DĆŸamonja, Mirna;
2530:. This is due to
1942:orthonormal basis
1565:to the real line
1494:{\displaystyle n}
1210:{\displaystyle X}
1170:{\displaystyle X}
1134:
1004:{\displaystyle z}
984:{\displaystyle Y}
935:
795:{\displaystyle X}
642:{\displaystyle X}
533:{\displaystyle X}
375:is separable: if
342:{\displaystyle n}
322:{\displaystyle n}
273:{\displaystyle n}
175:{\displaystyle n}
42:if it contains a
36:topological space
3017:
3005:General topology
2991:
2968:
2965:General Topology
2957:
2914:
2904:General topology
2895:
2894:
2873:
2849:General Topology
2836:
2835:
2833:
2828:
2812:
2811:
2801:
2799:
2798:
2793:
2781:
2779:
2778:
2773:
2761:
2759:
2758:
2753:
2741:
2739:
2738:
2733:
2709:
2707:
2706:
2701:
2689:
2687:
2686:
2681:
2659:
2643:
2630:
2624:
2622:
2621:
2616:
2599:
2579:
2575:
2567:
2457:
2455:
2454:
2449:
2444:
2430:
2429:
2392:Sorgenfrey plane
2370:
2368:
2367:
2362:
2360:
2359:
2315:Sorgenfrey plane
2287:
2285:
2284:
2279:
2277:
2276:
2256:
2254:
2253:
2248:
2246:
2245:
2219:
2217:
2216:
2211:
2209:
2208:
2175:
2173:
2172:
2167:
2155:
2153:
2152:
2147:
2132:
2130:
2129:
2124:
2122:
2121:
2096:
2094:
2093:
2088:
2033:
2031:
2030:
2025:
2023:
2022:
1998:
1996:
1995:
1990:
1988:
1970:
1968:
1967:
1962:
1960:
1959:
1932:
1930:
1929:
1924:
1879:
1877:
1876:
1871:
1838:
1836:
1835:
1830:
1819:
1799:
1797:
1796:
1793:{\displaystyle }
1791:
1760:
1758:
1757:
1752:
1716:
1714:
1713:
1708:
1684:
1682:
1681:
1676:
1674:
1670:
1663:
1662:
1638:
1636:
1635:
1630:
1628:
1624:
1609:
1608:
1586:
1584:
1583:
1578:
1576:
1564:
1562:
1561:
1556:
1554:
1532:
1530:
1529:
1524:
1500:
1498:
1497:
1492:
1469:Separable spaces
1464:Further examples
1431:
1429:
1428:
1423:
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1381:
1380:
1364:
1362:
1361:
1356:
1344:
1342:
1341:
1336:
1334:
1333:
1332:
1315:
1313:
1312:
1307:
1305:
1304:
1303:
1297:
1277:
1275:
1274:
1269:
1267:
1266:
1250:
1248:
1247:
1242:
1240:
1239:
1238:
1237:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1176:
1174:
1173:
1168:
1154:
1152:
1151:
1146:
1135:
1127:
1118:
1116:
1115:
1110:
1083:
1081:
1080:
1075:
1073:
1072:
1071:
1070:
1069:
1061:
1039:
1037:
1036:
1031:
1010:
1008:
1007:
1002:
990:
988:
987:
982:
970:
968:
967:
962:
960:
959:
946:
944:
943:
938:
936:
928:
913:
911:
910:
905:
887:
885:
884:
879:
857:
855:
854:
849:
847:
846:
833:
831:
830:
825:
823:
822:
821:
801:
799:
798:
793:
776:
774:
773:
768:
766:
765:
729:trivial topology
714:
712:
711:
706:
695:
694:
675:
673:
672:
667:
665:
664:
648:
646:
645:
640:
628:
626:
625:
620:
618:
617:
616:
599:
597:
596:
591:
589:
588:
572:
570:
569:
564:
556:
555:
539:
537:
536:
531:
477:metrizable space
474:
472:
471:
466:
464:
463:
447:
445:
444:
439:
437:
436:
424:
423:
407:
405:
404:
399:
394:
393:
348:
346:
345:
340:
328:
326:
325:
320:
308:
306:
305:
300:
298:
297:
292:
279:
277:
276:
271:
259:
257:
256:
251:
249:
248:
243:
231:
230:
212:
211:
196:
181:
179:
178:
173:
160:rational numbers
103:
101:
100:
95:
92:
87:
72:
71:
25:Separation axiom
3025:
3024:
3020:
3019:
3018:
3016:
3015:
3014:
2995:
2994:
2981:
2947:
2937:Springer-Verlag
2863:
2853:Springer-Verlag
2845:Kelley, John L.
2831:
2829:
2826:
2816:
2815:
2787:
2784:
2783:
2767:
2764:
2763:
2762:-null sets. If
2747:
2744:
2743:
2715:
2712:
2711:
2695:
2692:
2691:
2675:
2672:
2671:
2641:
2631:
2627:
2600:
2596:
2591:
2577:
2569:
2565:
2482:
2475:
2440:
2425:
2424:
2422:
2419:
2418:
2402:Hausdorff space
2355:
2354:
2352:
2349:
2348:
2342:SierpiĆski 1952
2306:
2272:
2268:
2266:
2263:
2262:
2241:
2237:
2235:
2232:
2231:
2204:
2200:
2198:
2195:
2194:
2187:
2161:
2158:
2157:
2141:
2138:
2137:
2117:
2116:
2102:
2099:
2098:
2046:
2043:
2042:
2018:
2017:
2015:
2012:
2011:
1984:
1982:
1979:
1978:
1976:Sorgenfrey line
1955:
1951:
1949:
1946:
1945:
1897:
1894:
1893:
1890:linear subspace
1844:
1841:
1840:
1815:
1813:
1810:
1809:
1773:
1770:
1769:
1725:
1722:
1721:
1690:
1687:
1686:
1658:
1657:
1650:
1646:
1644:
1641:
1640:
1614:
1610:
1604:
1600:
1598:
1595:
1594:
1592:Lebesgue spaces
1572:
1570:
1567:
1566:
1550:
1542:
1539:
1538:
1509:
1506:
1505:
1486:
1483:
1482:
1471:
1466:
1442:
1417:
1414:
1413:
1397:
1394:
1393:
1376:
1372:
1370:
1367:
1366:
1350:
1347:
1346:
1328:
1327:
1323:
1321:
1318:
1317:
1299:
1298:
1293:
1292:
1290:
1287:
1286:
1262:
1258:
1256:
1253:
1252:
1233:
1229:
1228:
1224:
1222:
1219:
1218:
1202:
1199:
1198:
1182:
1179:
1178:
1162:
1159:
1158:
1126:
1124:
1121:
1120:
1089:
1086:
1085:
1065:
1057:
1056:
1052:
1051:
1047:
1045:
1042:
1041:
1016:
1013:
1012:
996:
993:
992:
976:
973:
972:
955:
954:
952:
949:
948:
927:
919:
916:
915:
893:
890:
889:
867:
864:
863:
842:
841:
839:
836:
835:
817:
816:
812:
810:
807:
806:
787:
784:
783:
761:
760:
758:
755:
754:
748:first-countable
721:
690:
686:
681:
678:
677:
660:
656:
654:
651:
650:
634:
631:
630:
612:
608:
607:
605:
602:
601:
584:
580:
578:
575:
574:
551:
547:
545:
542:
541:
525:
522:
521:
459:
455:
453:
450:
449:
432:
428:
419:
415:
413:
410:
409:
389:
385:
380:
377:
376:
369:
351:Euclidean space
334:
331:
330:
314:
311:
310:
309:; so for every
293:
288:
287:
285:
282:
281:
265:
262:
261:
244:
239:
238:
226:
222:
207:
203:
192:
190:
187:
186:
167:
164:
163:
158:, in which the
144:
121:Hausdorff axiom
111:Like the other
88:
77:
67:
63:
58:
55:
54:
28:
21:Separated space
17:
12:
11:
5:
3023:
3013:
3012:
3007:
2993:
2992:
2979:
2971:Addison-Wesley
2958:
2945:
2915:
2896:
2885:(2): 169â173,
2874:
2861:
2838:
2837:
2814:
2813:
2791:
2771:
2751:
2731:
2728:
2725:
2722:
2719:
2699:
2679:
2635:Kunen, Kenneth
2625:
2608:Measure Theory
2603:Donald L. Cohn
2593:
2592:
2590:
2587:
2586:
2585:
2547:
2546:
2539:
2520:
2498:
2481:
2478:
2477:
2476:
2474:
2473:
2472:is metrizable.
2467:
2447:
2443:
2439:
2436:
2433:
2428:
2415:
2408:
2395:
2394:is not normal.
2376:
2358:
2345:
2334:
2305:
2302:
2301:
2300:
2289:
2275:
2271:
2244:
2240:
2225:
2222:order topology
2207:
2203:
2186:
2183:
2182:
2181:
2165:
2145:
2120:
2115:
2112:
2109:
2106:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2021:
2004:
1987:
1972:
1958:
1954:
1934:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1828:
1825:
1822:
1818:
1789:
1786:
1783:
1780:
1777:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1718:
1706:
1703:
1700:
1697:
1694:
1673:
1669:
1666:
1661:
1656:
1653:
1649:
1627:
1623:
1620:
1617:
1613:
1607:
1603:
1588:
1575:
1553:
1549:
1546:
1522:
1519:
1516:
1513:
1502:
1490:
1479:
1474:Every compact
1470:
1467:
1465:
1462:
1441:
1438:
1421:
1401:
1379:
1375:
1354:
1331:
1326:
1302:
1296:
1265:
1261:
1236:
1232:
1227:
1206:
1186:
1166:
1144:
1141:
1138:
1133:
1130:
1108:
1105:
1102:
1099:
1096:
1093:
1068:
1064:
1060:
1055:
1050:
1029:
1026:
1023:
1020:
1000:
980:
958:
934:
931:
926:
923:
903:
900:
897:
877:
874:
871:
845:
820:
815:
791:
764:
720:
717:
704:
701:
698:
693:
689:
685:
663:
659:
638:
615:
611:
587:
583:
562:
559:
554:
550:
529:
518:
517:
506:
495:
462:
458:
435:
431:
427:
422:
418:
397:
392:
388:
384:
368:
365:
358:discrete space
353:is separable.
338:
318:
296:
291:
269:
247:
242:
237:
234:
229:
225:
221:
218:
215:
210:
206:
202:
199:
195:
171:
143:
142:First examples
140:
91:
86:
83:
80:
76:
70:
66:
62:
15:
9:
6:
4:
3:
2:
3022:
3011:
3008:
3006:
3003:
3002:
3000:
2990:
2986:
2982:
2976:
2972:
2967:
2966:
2959:
2956:
2952:
2948:
2942:
2938:
2934:
2930:
2929:
2924:
2920:
2916:
2913:
2909:
2905:
2901:
2897:
2893:
2888:
2884:
2880:
2875:
2872:
2868:
2864:
2858:
2854:
2850:
2846:
2842:
2841:
2840:
2825:
2824:
2818:
2817:
2810:
2808:
2805:
2789:
2769:
2749:
2726:
2723:
2720:
2697:
2677:
2667:
2663:
2658:
2653:
2649:
2648:
2640:
2636:
2629:
2614:
2610:
2609:
2604:
2598:
2594:
2583:
2573:
2563:
2559:
2555:
2554:
2553:
2551:
2544:
2540:
2537:
2536:Heinonen 2003
2533:
2532:Stefan Banach
2529:
2528:supremum norm
2525:
2521:
2518:
2517:Heinonen 2003
2514:
2513:supremum norm
2510:
2507:
2503:
2499:
2496:
2492:
2488:
2484:
2483:
2471:
2468:
2466:is separable.
2465:
2464:supremum norm
2461:
2437:
2434:
2416:
2413:
2410:
2409:
2406:
2403:
2400:
2396:
2393:
2389:
2385:
2381:
2377:
2374:
2346:
2343:
2339:
2335:
2333:is separable.
2332:
2328:
2324:
2321:), but every
2320:
2316:
2312:
2308:
2307:
2298:
2294:
2290:
2269:
2260:
2259:supremum norm
2238:
2230:
2226:
2223:
2205:
2201:
2193:
2189:
2188:
2179:
2143:
2136:
2113:
2110:
2107:
2104:
2081:
2075:
2069:
2066:
2060:
2057:
2054:
2048:
2041:
2037:
2009:
2005:
2002:
1977:
1973:
1956:
1952:
1943:
1939:
1938:Hilbert space
1935:
1914:
1911:
1908:
1899:
1891:
1887:
1883:
1861:
1858:
1855:
1846:
1823:
1808:that the set
1807:
1803:
1784:
1781:
1778:
1768:
1767:unit interval
1764:
1742:
1739:
1736:
1727:
1719:
1701:
1698:
1695:
1692:
1671:
1667:
1664:
1654:
1651:
1647:
1625:
1621:
1618:
1615:
1611:
1605:
1601:
1593:
1589:
1587:is separable.
1547:
1544:
1536:
1517:
1511:
1503:
1488:
1480:
1477:
1473:
1472:
1461:
1459:
1455:
1451:
1447:
1437:
1435:
1419:
1399:
1377:
1373:
1352:
1324:
1284:
1279:
1263:
1259:
1234:
1230:
1225:
1204:
1184:
1164:
1155:
1142:
1139:
1136:
1128:
1106:
1097:
1091:
1062:
1053:
1048:
1024:
1018:
998:
978:
929:
924:
921:
901:
898:
895:
875:
872:
869:
861:
813:
803:
789:
780:
753:
749:
744:
742:
738:
734:
733:quasi-compact
730:
726:
716:
699:
696:
691:
687:
661:
657:
636:
613:
609:
585:
581:
560:
557:
552:
548:
527:
515:
511:
507:
504:
500:
496:
493:
490:An arbitrary
489:
488:
487:
484:
482:
478:
460:
456:
433:
429:
425:
420:
416:
390:
386:
374:
364:
361:
359:
354:
352:
349:-dimensional
336:
316:
294:
267:
245:
235:
227:
223:
219:
216:
213:
208:
204:
197:
184:
169:
161:
157:
153:
149:
139:
137:
133:
128:
126:
122:
118:
114:
109:
107:
84:
81:
78:
68:
64:
53:
49:
45:
41:
37:
33:
26:
22:
2964:
2927:
2903:
2882:
2878:
2848:
2839:
2830:, retrieved
2822:
2803:
2669:
2657:math/9408201
2645:
2628:
2607:
2597:
2571:
2558:metric space
2549:
2548:
2523:
2508:
2506:Banach space
2491:Hilbert cube
2487:homeomorphic
2469:
2459:
2411:
2404:
2383:
2379:
2331:metric space
2327:Willard 1970
2322:
2293:Banach space
2229:Banach space
2036:metric space
1886:Banach space
1476:metric space
1443:
1283:Willard 1970
1280:
1156:
804:
745:
722:
540:, pick some
519:
514:Willard 1970
499:Willard 1970
485:
370:
362:
355:
145:
129:
110:
39:
29:
2620:Proposition
2526:, with the
2338:cardinality
2319:Moore plane
725:cardinality
719:Cardinality
117:cardinality
106:open subset
32:mathematics
2999:Categories
2832:6 February
2589:References
2417:The space
2386:cannot be
2304:Properties
2180:operator).
2176:being the
2156:(and with
1720:The space
1504:The space
1454:algorithms
136:metrizable
38:is called
2925:(1995) ,
2804:separable
2790:μ
2770:μ
2750:μ
2727:μ
2678:μ
2502:isometric
2462:with the
2274:∞
2243:∞
2239:ℓ
2202:ω
2164:△
2144:μ
2114:∈
2079:△
2070:μ
2049:ρ
1953:ℓ
1705:∞
1696:≤
1668:μ
1622:μ
1548:⊆
1420:κ
1400:κ
1378:κ
1353:κ
1264:κ
1235:κ
1185:κ
1132:¯
1104:→
933:¯
925:∈
899:∈
873:⊆
737:connected
558:∈
426:∈
236:∈
217:…
156:real line
90:∞
44:countable
40:separable
2902:(1952),
2847:(1975),
2637:(1995).
2605:(2013).
2317:and the
2311:subspace
1672:⟩
1648:⟨
834:, where
503:quotient
492:subspace
481:Lindelöf
138:spaces.
52:sequence
2989:0264581
2955:0507446
2912:0050870
2871:0370454
2662:Bibcode
2650:: 262.
2562:density
2399:compact
2135:measure
1765:on the
1537:subset
1535:compact
1197:. Then
914:, then
779:closure
510:product
183:vectors
2987:
2977:
2953:
2943:
2910:
2869:
2859:
2623:3.4.5.
2397:For a
2388:normal
2371:, the
2040:metric
1880:. The
735:, and
148:finite
2933:Dover
2827:(PDF)
2652:arXiv
2642:(PDF)
2038:with
1119:when
862:: if
48:dense
2975:ISBN
2941:ISBN
2857:ISBN
2834:2009
2570:C(,
2323:open
2291:The
2227:The
2190:The
2097:for
1702:<
1590:The
1448:and
888:and
371:Any
34:, a
2887:doi
2807:iff
2802:is
2670:If
2560:of
2534:. (
2295:of
1892:of
1761:of
1436:).
150:or
30:In
23:or
3001::
2985:MR
2983:,
2973:,
2969:,
2951:MR
2949:,
2939:,
2921:;
2908:MR
2881:,
2867:MR
2865:,
2855:,
2668:.
2660:.
2644:.
2617:,
2611:.
2556:A
2552::
2309:A
2006:A
1936:A
1460:.
802:.
746:A
508:A
483:.
329:,
46:,
2931:(
2889::
2883:1
2730:)
2724:,
2721:X
2718:(
2698:X
2664::
2654::
2615:.
2584:)
2578:α
2574:)
2572:R
2566:α
2545:.
2538:)
2524:R
2519:)
2509:l
2497:.
2470:X
2460:X
2446:)
2442:R
2438:,
2435:X
2432:(
2427:C
2412:X
2405:X
2384:X
2380:X
2357:c
2288:.
2270:L
2206:1
2119:F
2111:B
2108:,
2105:A
2085:)
2082:B
2076:A
2073:(
2067:=
2064:)
2061:B
2058:,
2055:A
2052:(
2020:F
2003:.
1986:S
1957:2
1933:.
1921:)
1918:]
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27:.
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