269:
792:
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
1694:
2467:
2106:
1325:
113:
1648:
1574:
527:, 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.
2142:
956:
1726:
1164:
1354:
1128:
843:
318:
2380:
2266:
1260:
867:
786:
2043:
980:
457:
2008:
1848:
1596:
897:
2297:
2229:
1401:
1287:
1093:
724:
2793:
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
2185:
582:
1980:
417:
2751:
1441:
1421:
1374:
1206:
638:
923:
685:
609:
484:
2811:
2791:
2771:
2699:
2165:
1542:
1049:
1942:
1889:
1770:
1492:
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
1463:, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into
2719:
1510:
1226:
1186:
1020:
1000:
811:
658:
549:
358:
338:
289:
191:
1809:
1467:
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
3020:
199:
2989:
2955:
2871:
165:
is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the
1653:
2623:
193:
1327:
of all functions from the real line to itself, endowed with the product topology, is a separable
Hausdorff space of cardinality
2533:
Every separable metric space is isometric to a subset of C(), the separable Banach space of continuous functions â
138:
on a separable space whose image is a subset of a
Hausdorff space is determined by its values on the countable dense subset.
1816:
2355:, p. 49); if the space was a Hausdorff space then the space constructed that it embeds into is also a Hausdorff space.
1444:
531:
We can construct an example of a separable topological space that is not second countable. Consider any uncountable set
2383:
762:
2310:
is not separable; note however that this space has very important applications in mathematics, physics and engineering.
2937:
2505:
1460:
2431:
1095:. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection
2055:
3015:
2202:
1299:
505:
of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
1892:
67:
1607:
1468:
134:, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every
1551:
2910:
2657:
2553:
2111:
2933:
928:
383:
123:
2145:
1699:
1133:
1330:
1098:
819:
294:
2672:
2361:
2244:
2188:
2050:
2011:
1231:
848:
767:
2613:
2024:
961:
422:
2321:
1812:
1773:
870:
751:
502:
142:
135:
1991:
1822:
1579:
876:
2943:
2662:
2275:
2207:
1456:
1379:
1265:
1054:
789:
690:
162:
2358:
The set of all real-valued continuous functions on a separable space has a cardinality equal to
2170:
554:
1958:
1955:. It follows that any separable, infinite-dimensional Hilbert space is isometric to the space
1168:
The same arguments establish a more general result: suppose that a
Hausdorff topological space
389:
2985:
2951:
2929:
2867:
2724:
2307:
1952:
513:
491:
46:
1426:
1406:
1359:
1191:
614:
2897:
2402:
2325:
1696:
whose Ï-algebra is countably generated and whose measure is Ï-finite, are separable for any
902:
739:
520:
487:
170:
58:
35:
2999:
2965:
2922:
2881:
2617:
663:
587:
490:
is separable if and only if it is second countable, which is the case if and only if it is
462:
2995:
2961:
2947:
2918:
2877:
2863:
2796:
2776:
2756:
2684:
2412:
2150:
1986:
1900:
1518:
1025:
758:
747:
361:
131:
31:
1906:
1853:
1850:
of polynomials in one variable with rational coefficients is a countable dense subset of
1734:
2676:
2981:
2855:
2817:
2704:
2232:
1495:
1211:
1171:
1005:
985:
796:
643:
534:
368:
343:
323:
274:
176:
2974:
1782:
761:, separable Hausdorff space (in particular, a separable metric space) has at most the
3009:
2645:
2542:
2409:
1948:
1777:
1545:
743:
54:
2832:
2386:. This follows since such functions are determined by their values on dense subsets.
726:
is open. Therefore, the space is separable but there cannot have a countable base.
17:
2568:
2538:
2523:
2516:
2501:
2497:
2474:
2398:
2341:
2303:
2269:
2239:
2046:
1896:
1486:
869:
is the cardinality of the continuum. For this closure is characterized in terms of
734:
The property of separability does not in and of itself give any limitations on the
2888:
Kleiber, Martin; Pervin, William J. (1969), "A generalized Banach-Mazur theorem",
2018:
2649:
750:. The "trouble" with the trivial topology is its poor separation properties: its
2348:
2347:
In fact, every topological space is a subspace of a separable space of the same
2329:
735:
127:
116:
42:
2917:, Mathematical Expositions, No. 7, Toronto, Ont.: University of Toronto Press,
2902:
158:
146:
1292:
The product of at most continuum many separable spaces is a separable space (
2572:
1464:
166:
264:{\displaystyle {\boldsymbol {r}}=(r_{1},\ldots ,r_{n})\in \mathbb {Q} ^{n}}
2393:
is a separable space having an uncountable closed discrete subspace, then
2512:
1602:
62:
584:, and define the topology to be the collection of all sets that contain
126:, separability is a "limitation on size", not necessarily in terms of
2667:
1985:
An example of a separable space that is not second-countable is the
2351:. A construction adding at most countably many points is given in (
1689:{\displaystyle \left\langle X,{\mathcal {M}},\mu \right\rangle }
173:
form a countable dense subset. Similarly the set of all length-
145:, which is in general stronger but equivalent on the class of
2650:"Properties of the class of measure separable compact spaces"
2552:
Every separable metric space is isometric to a subset of the
2437:
2129:
2030:
1670:
967:
119:
of the space contains at least one element of the sequence.
2587:, the space of real continuous functions on the product of
2389:
From the above property, one can deduce the following: If
523:
of at most continuum many separable spaces is separable (
516:
of a second-countable space need not be second countable.
508:
Any continuous image of a separable space is separable (
367:
A simple example of a space that is not separable is a
271:, is a countable dense subset of the set of all length-
2820:
this metric space is separable as a topological space.
2799:
2779:
2759:
2727:
2707:
2687:
2434:
2364:
2278:
2247:
2210:
2173:
2153:
2114:
2058:
2027:
1994:
1961:
1909:
1856:
1825:
1785:
1737:
1702:
1656:
1610:
1582:
1554:
1521:
1498:
1429:
1409:
1382:
1362:
1333:
1302:
1268:
1234:
1214:
1194:
1174:
1136:
1101:
1057:
1028:
1008:
988:
964:
931:
905:
879:
851:
822:
799:
770:
693:
666:
646:
617:
590:
557:
537:
465:
425:
392:
346:
326:
297:
277:
202:
179:
70:
2324:
of a separable space need not be separable (see the
1376:
is any infinite cardinal, then a product of at most
816:
A separable
Hausdorff space has cardinality at most
2973:
2805:
2785:
2765:
2745:
2713:
2693:
2461:
2374:
2291:
2260:
2223:
2179:
2159:
2136:
2100:
2037:
2002:
1974:
1936:
1883:
1842:
1803:
1764:
1720:
1688:
1642:
1590:
1568:
1536:
1504:
1435:
1415:
1395:
1368:
1348:
1319:
1296:, p. 109, Th 16.4c). In particular the space
1281:
1254:
1220:
1200:
1180:
1158:
1122:
1087:
1043:
1014:
994:
974:
950:
917:
891:
861:
837:
805:
780:
718:
679:
652:
632:
603:
576:
543:
478:
451:
411:
352:
332:
312:
283:
263:
185:
115:of elements of the space such that every nonempty
107:
738:of a topological space: any set endowed with the
141:Contrast separability with the related notion of
2753:is the Boolean algebra of all Borel sets modulo
2340:, Th 16.4b). Also every subspace of a separable
1815:is a separable space, since it follows from the
2045:that is a separable space when considered as a
1951:is separable if and only if it has a countable
27:Topological space with a dense countable subset
2462:{\displaystyle {\mathcal {C}}(X,\mathbb {R} )}
486:gives a countable dense subset. Conversely, a
2946:reprint of 1978 ed.), Berlin, New York:
2629:
2592:
2101:{\displaystyle \rho (A,B)=\mu (A\triangle B)}
8:
2336:subspace of a separable space is separable (
2010:, the set of real numbers equipped with the
713:
694:
406:
393:
85:
71:
2526:; this is known as the Fréchet embedding. (
1320:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
742:is separable, as well as second countable,
2504:. This is established in the proof of the
2352:
1512:-dimensional Euclidean space is separable.
1423:has itself a dense subset of size at most
1403:spaces with dense subsets of size at most
958:if and only if there exists a filter base
2901:
2798:
2778:
2758:
2726:
2706:
2686:
2666:
2452:
2451:
2436:
2435:
2433:
2366:
2365:
2363:
2283:
2277:
2252:
2246:
2215:
2209:
2172:
2152:
2128:
2127:
2113:
2057:
2029:
2028:
2026:
1996:
1995:
1993:
1966:
1960:
1908:
1855:
1827:
1826:
1824:
1784:
1736:
1701:
1669:
1668:
1655:
1615:
1609:
1584:
1583:
1581:
1562:
1561:
1553:
1520:
1497:
1428:
1408:
1387:
1381:
1361:
1339:
1338:
1332:
1311:
1310:
1309:
1305:
1304:
1301:
1273:
1267:
1244:
1239:
1233:
1213:
1193:
1173:
1137:
1135:
1100:
1076:
1068:
1067:
1062:
1056:
1027:
1007:
987:
966:
965:
963:
938:
930:
904:
878:
853:
852:
850:
828:
827:
821:
798:
772:
771:
769:
701:
692:
671:
665:
645:
623:
618:
616:
595:
589:
562:
556:
536:
497:To further compare these two properties:
470:
464:
443:
430:
424:
400:
391:
345:
325:
304:
300:
299:
296:
276:
255:
251:
250:
237:
218:
203:
201:
178:
108:{\displaystyle \{x_{n}\}_{n=1}^{\infty }}
99:
88:
78:
69:
2546:
2527:
2272:, is not separable. The same holds for
2268:of all bounded real sequences, with the
1899:is isometrically isomorphic to a closed
1643:{\displaystyle L^{p}\left(X,\mu \right)}
1455:Separability is especially important in
2605:
2522:of all bounded real sequences with the
2469:of continuous real-valued functions on
2337:
1569:{\displaystyle K\subseteq \mathbb {R} }
1293:
1188:contains a dense subset of cardinality
524:
509:
378:Separability versus second countability
204:
660:is the smallest closed set containing
611:(or are empty). Then, the closure of
2834:Geometric embeddings of metric spaces
2137:{\displaystyle A,B\in {\mathcal {F}}}
157:Any topological space that is itself
7:
1445:HewittâMarczewskiâPondiczery theorem
951:{\displaystyle z\in {\overline {Y}}}
2515:to a subset of the (non-separable)
2367:
1544:of all continuous functions from a
1489:(or metrizable space) is separable.
1340:
854:
829:
773:
2284:
2253:
2174:
2089:
1721:{\displaystyle 1\leq p<\infty }
1715:
1159:{\displaystyle {\overline {Y}}=X.}
419:is a countable base, choosing any
374:Further examples are given below.
100:
25:
2491:Embedding separable metric spaces
1817:Weierstrass approximation theorem
1349:{\displaystyle 2^{\mathfrak {c}}}
1123:{\displaystyle S(Y)\rightarrow X}
838:{\displaystyle 2^{\mathfrak {c}}}
3021:Properties of topological spaces
2511:Every separable metric space is
2496:Every separable metric space is
2418:, the following are equivalent:
1774:continuous real-valued functions
1051:of such filter bases is at most
313:{\displaystyle \mathbb {R} ^{n}}
130:(though, in the presence of the
61:subset; that is, there exists a
2831:Heinonen, Juha (January 2003),
2624:Springer Science+Business Media
2375:{\displaystyle {\mathfrak {c}}}
2261:{\displaystyle \ell ^{\infty }}
1255:{\displaystyle 2^{2^{\kappa }}}
862:{\displaystyle {\mathfrak {c}}}
781:{\displaystyle {\mathfrak {c}}}
2740:
2728:
2591:copies of the unit interval. (
2579:is isometric to a subspace of
2575:equal to an infinite cardinal
2456:
2442:
2308:functions of bounded variation
2095:
2083:
2074:
2062:
2038:{\displaystyle {\mathcal {F}}}
1931:
1928:
1916:
1913:
1878:
1875:
1863:
1860:
1837:
1831:
1798:
1786:
1759:
1756:
1744:
1741:
1531:
1525:
1114:
1111:
1105:
1077:
1069:
1038:
1032:
975:{\displaystyle {\mathcal {B}}}
452:{\displaystyle x_{n}\in U_{n}}
243:
211:
1:
1982:of square-summable sequences.
1022:. The cardinality of the set
687:), but every set of the form
2384:cardinality of the continuum
2231:, equipped with its natural
2003:{\displaystyle \mathbb {S} }
1843:{\displaystyle \mathbb {Q} }
1591:{\displaystyle \mathbb {R} }
1142:
943:
892:{\displaystyle Y\subseteq X}
371:of uncountable cardinality.
2939:Counterexamples in Topology
2506:Urysohn metrization theorem
2292:{\displaystyle L^{\infty }}
2224:{\displaystyle \omega _{1}}
1895:asserts that any separable
1396:{\displaystyle 2^{\kappa }}
1282:{\displaystyle 2^{\kappa }}
1088:{\displaystyle 2^{2^{|Y|}}}
719:{\displaystyle \{x_{0},x\}}
3037:
2180:{\displaystyle \triangle }
1289:if it is first countable.
577:{\displaystyle x_{0}\in X}
29:
2972:Willard, Stephen (1970),
2903:10.1017/S0004972700041411
2890:Bull. Austral. Math. Soc.
2721:, the measure algebra of
2630:
2593:Kleiber & Pervin 1969
2203:first uncountable ordinal
1975:{\displaystyle \ell ^{2}}
982:consisting of subsets of
412:{\displaystyle \{U_{n}\}}
291:vectors of real numbers,
2746:{\displaystyle (X,\mu )}
2019:separable σ-algebra
1461:constructive mathematics
1451:Constructive mathematics
1262:and cardinality at most
1228:has cardinality at most
754:is the one-point space.
30:Not to be confused with
2658:Fundamenta Mathematicae
2561:For nonseparable spaces
2554:Urysohn universal space
1650:, over a measure space
1436:{\displaystyle \kappa }
1416:{\displaystyle \kappa }
1369:{\displaystyle \kappa }
1201:{\displaystyle \kappa }
633:{\displaystyle {x_{0}}}
2934:Seebach, J. Arthur Jr.
2807:
2787:
2767:
2747:
2715:
2701:is a Borel measure on
2695:
2463:
2401:. This shows that the
2376:
2293:
2262:
2225:
2181:
2161:
2138:
2102:
2039:
2004:
1976:
1938:
1885:
1844:
1805:
1766:
1722:
1690:
1644:
1592:
1570:
1538:
1506:
1437:
1417:
1397:
1370:
1350:
1321:
1283:
1256:
1222:
1202:
1182:
1160:
1124:
1089:
1045:
1016:
996:
976:
952:
919:
918:{\displaystyle z\in X}
893:
871:limits of filter bases
863:
839:
807:
782:
720:
681:
654:
634:
605:
578:
545:
480:
453:
413:
384:second-countable space
354:
334:
314:
285:
265:
187:
124:axioms of countability
109:
2808:
2788:
2768:
2748:
2716:
2696:
2464:
2377:
2294:
2263:
2226:
2182:
2162:
2139:
2103:
2040:
2005:
1977:
1939:
1886:
1845:
1806:
1767:
1723:
1691:
1645:
1593:
1571:
1539:
1507:
1438:
1418:
1398:
1371:
1356:. More generally, if
1351:
1322:
1284:
1257:
1223:
1203:
1183:
1161:
1125:
1090:
1046:
1017:
997:
977:
953:
920:
894:
864:
840:
808:
783:
763:continuum cardinality
721:
682:
680:{\displaystyle x_{0}}
655:
635:
606:
604:{\displaystyle x_{0}}
579:
546:
512:, Th. 16.4a); even a
481:
479:{\displaystyle U_{n}}
454:
414:
355:
335:
315:
286:
266:
196:of rational numbers,
188:
110:
2862:, Berlin, New York:
2806:{\displaystyle \mu }
2797:
2786:{\displaystyle \mu }
2777:
2766:{\displaystyle \mu }
2757:
2725:
2705:
2694:{\displaystyle \mu }
2685:
2432:
2425:is second countable.
2362:
2276:
2245:
2208:
2196:Non-separable spaces
2189:symmetric difference
2171:
2160:{\displaystyle \mu }
2151:
2112:
2056:
2025:
2021:is a σ-algebra
2012:lower limit topology
1992:
1959:
1907:
1893:BanachâMazur theorem
1854:
1823:
1783:
1735:
1700:
1654:
1608:
1580:
1552:
1537:{\displaystyle C(K)}
1519:
1496:
1427:
1407:
1380:
1360:
1331:
1300:
1266:
1232:
1212:
1192:
1172:
1134:
1099:
1055:
1044:{\displaystyle S(Y)}
1026:
1006:
986:
962:
929:
903:
877:
849:
820:
797:
768:
691:
664:
644:
640:is the whole space (
615:
588:
555:
535:
463:
423:
390:
344:
324:
295:
275:
200:
177:
68:
18:Separable (topology)
2677:1994math......8201D
2500:to a subset of the
2235:, is not separable.
2144:and a given finite
1937:{\displaystyle C()}
1884:{\displaystyle C()}
1813:uniform convergence
1811:with the metric of
1765:{\displaystyle C()}
1469:HahnâBanach theorem
788:. In such a space,
752:Kolmogorov quotient
459:from the non-empty
143:second countability
136:continuous function
104:
2930:Steen, Lynn Arthur
2911:SierpiĆski, WacĆaw
2803:
2783:
2763:
2743:
2711:
2691:
2459:
2372:
2289:
2258:
2221:
2177:
2157:
2134:
2098:
2035:
2000:
1972:
1934:
1881:
1840:
1801:
1762:
1718:
1686:
1640:
1588:
1566:
1534:
1502:
1457:numerical analysis
1433:
1413:
1393:
1366:
1346:
1317:
1279:
1252:
1218:
1198:
1178:
1156:
1120:
1085:
1041:
1012:
1002:that converges to
992:
972:
948:
915:
889:
859:
835:
803:
778:
716:
677:
650:
630:
601:
574:
541:
476:
449:
409:
350:
330:
310:
281:
261:
183:
163:countably infinite
105:
84:
2991:978-0-201-08707-9
2957:978-0-486-68735-3
2873:978-0-387-90125-1
2714:{\displaystyle X}
2644:DĆŸamonja, Mirna;
2541:. This is due to
1953:orthonormal basis
1576:to the real line
1505:{\displaystyle n}
1221:{\displaystyle X}
1181:{\displaystyle X}
1145:
1015:{\displaystyle z}
995:{\displaystyle Y}
946:
806:{\displaystyle X}
653:{\displaystyle X}
544:{\displaystyle X}
386:is separable: if
353:{\displaystyle n}
333:{\displaystyle n}
284:{\displaystyle n}
186:{\displaystyle n}
53:if it contains a
47:topological space
16:(Redirected from
3028:
3016:General topology
3002:
2979:
2976:General Topology
2968:
2925:
2915:General topology
2906:
2905:
2884:
2860:General Topology
2847:
2846:
2844:
2839:
2823:
2822:
2812:
2810:
2809:
2804:
2792:
2790:
2789:
2784:
2772:
2770:
2769:
2764:
2752:
2750:
2749:
2744:
2720:
2718:
2717:
2712:
2700:
2698:
2697:
2692:
2670:
2654:
2641:
2635:
2633:
2632:
2627:
2610:
2590:
2586:
2578:
2468:
2466:
2465:
2460:
2455:
2441:
2440:
2403:Sorgenfrey plane
2381:
2379:
2378:
2373:
2371:
2370:
2326:Sorgenfrey plane
2298:
2296:
2295:
2290:
2288:
2287:
2267:
2265:
2264:
2259:
2257:
2256:
2230:
2228:
2227:
2222:
2220:
2219:
2186:
2184:
2183:
2178:
2166:
2164:
2163:
2158:
2143:
2141:
2140:
2135:
2133:
2132:
2107:
2105:
2104:
2099:
2044:
2042:
2041:
2036:
2034:
2033:
2009:
2007:
2006:
2001:
1999:
1981:
1979:
1978:
1973:
1971:
1970:
1943:
1941:
1940:
1935:
1890:
1888:
1887:
1882:
1849:
1847:
1846:
1841:
1830:
1810:
1808:
1807:
1804:{\displaystyle }
1802:
1771:
1769:
1768:
1763:
1727:
1725:
1724:
1719:
1695:
1693:
1692:
1687:
1685:
1681:
1674:
1673:
1649:
1647:
1646:
1641:
1639:
1635:
1620:
1619:
1597:
1595:
1594:
1589:
1587:
1575:
1573:
1572:
1567:
1565:
1543:
1541:
1540:
1535:
1511:
1509:
1508:
1503:
1480:Separable spaces
1475:Further examples
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1392:
1391:
1375:
1373:
1372:
1367:
1355:
1353:
1352:
1347:
1345:
1344:
1343:
1326:
1324:
1323:
1318:
1316:
1315:
1314:
1308:
1288:
1286:
1285:
1280:
1278:
1277:
1261:
1259:
1258:
1253:
1251:
1250:
1249:
1248:
1227:
1225:
1224:
1219:
1207:
1205:
1204:
1199:
1187:
1185:
1184:
1179:
1165:
1163:
1162:
1157:
1146:
1138:
1129:
1127:
1126:
1121:
1094:
1092:
1091:
1086:
1084:
1083:
1082:
1081:
1080:
1072:
1050:
1048:
1047:
1042:
1021:
1019:
1018:
1013:
1001:
999:
998:
993:
981:
979:
978:
973:
971:
970:
957:
955:
954:
949:
947:
939:
924:
922:
921:
916:
898:
896:
895:
890:
868:
866:
865:
860:
858:
857:
844:
842:
841:
836:
834:
833:
832:
812:
810:
809:
804:
787:
785:
784:
779:
777:
776:
740:trivial topology
725:
723:
722:
717:
706:
705:
686:
684:
683:
678:
676:
675:
659:
657:
656:
651:
639:
637:
636:
631:
629:
628:
627:
610:
608:
607:
602:
600:
599:
583:
581:
580:
575:
567:
566:
550:
548:
547:
542:
488:metrizable space
485:
483:
482:
477:
475:
474:
458:
456:
455:
450:
448:
447:
435:
434:
418:
416:
415:
410:
405:
404:
359:
357:
356:
351:
339:
337:
336:
331:
319:
317:
316:
311:
309:
308:
303:
290:
288:
287:
282:
270:
268:
267:
262:
260:
259:
254:
242:
241:
223:
222:
207:
192:
190:
189:
184:
171:rational numbers
114:
112:
111:
106:
103:
98:
83:
82:
36:Separation axiom
21:
3036:
3035:
3031:
3030:
3029:
3027:
3026:
3025:
3006:
3005:
2992:
2971:
2958:
2948:Springer-Verlag
2928:
2909:
2887:
2874:
2864:Springer-Verlag
2856:Kelley, John L.
2854:
2842:
2840:
2837:
2830:
2827:
2826:
2795:
2794:
2775:
2774:
2773:-null sets. If
2755:
2754:
2723:
2722:
2703:
2702:
2683:
2682:
2652:
2643:
2642:
2638:
2612:
2611:
2607:
2602:
2588:
2580:
2576:
2493:
2486:
2430:
2429:
2413:Hausdorff space
2360:
2359:
2353:SierpiĆski 1952
2317:
2279:
2274:
2273:
2248:
2243:
2242:
2211:
2206:
2205:
2198:
2169:
2168:
2149:
2148:
2110:
2109:
2054:
2053:
2023:
2022:
1990:
1989:
1987:Sorgenfrey line
1962:
1957:
1956:
1905:
1904:
1901:linear subspace
1852:
1851:
1821:
1820:
1781:
1780:
1733:
1732:
1698:
1697:
1661:
1657:
1652:
1651:
1625:
1621:
1611:
1606:
1605:
1603:Lebesgue spaces
1578:
1577:
1550:
1549:
1517:
1516:
1494:
1493:
1482:
1477:
1453:
1425:
1424:
1405:
1404:
1383:
1378:
1377:
1358:
1357:
1334:
1329:
1328:
1303:
1298:
1297:
1269:
1264:
1263:
1240:
1235:
1230:
1229:
1210:
1209:
1190:
1189:
1170:
1169:
1132:
1131:
1097:
1096:
1063:
1058:
1053:
1052:
1024:
1023:
1004:
1003:
984:
983:
960:
959:
927:
926:
901:
900:
875:
874:
847:
846:
823:
818:
817:
795:
794:
766:
765:
759:first-countable
732:
697:
689:
688:
667:
662:
661:
642:
641:
619:
613:
612:
591:
586:
585:
558:
553:
552:
533:
532:
466:
461:
460:
439:
426:
421:
420:
396:
388:
387:
380:
362:Euclidean space
342:
341:
322:
321:
320:; so for every
298:
293:
292:
273:
272:
249:
233:
214:
198:
197:
175:
174:
169:, in which the
155:
132:Hausdorff axiom
122:Like the other
74:
66:
65:
39:
32:Separated space
28:
23:
22:
15:
12:
11:
5:
3034:
3032:
3024:
3023:
3018:
3008:
3007:
3004:
3003:
2990:
2982:Addison-Wesley
2969:
2956:
2926:
2907:
2896:(2): 169â173,
2885:
2872:
2849:
2848:
2825:
2824:
2802:
2782:
2762:
2742:
2739:
2736:
2733:
2730:
2710:
2690:
2646:Kunen, Kenneth
2636:
2619:Measure Theory
2614:Donald L. Cohn
2604:
2603:
2601:
2598:
2597:
2596:
2558:
2557:
2550:
2531:
2509:
2492:
2489:
2488:
2487:
2485:
2484:
2483:is metrizable.
2478:
2458:
2454:
2450:
2447:
2444:
2439:
2426:
2419:
2406:
2405:is not normal.
2387:
2369:
2356:
2345:
2316:
2313:
2312:
2311:
2300:
2286:
2282:
2255:
2251:
2236:
2233:order topology
2218:
2214:
2197:
2194:
2193:
2192:
2176:
2156:
2131:
2126:
2123:
2120:
2117:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2032:
2015:
1998:
1983:
1969:
1965:
1945:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1839:
1836:
1833:
1829:
1800:
1797:
1794:
1791:
1788:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1729:
1717:
1714:
1711:
1708:
1705:
1684:
1680:
1677:
1672:
1667:
1664:
1660:
1638:
1634:
1631:
1628:
1624:
1618:
1614:
1599:
1586:
1564:
1560:
1557:
1533:
1530:
1527:
1524:
1513:
1501:
1490:
1485:Every compact
1481:
1478:
1476:
1473:
1452:
1449:
1432:
1412:
1390:
1386:
1365:
1342:
1337:
1313:
1307:
1276:
1272:
1247:
1243:
1238:
1217:
1197:
1177:
1155:
1152:
1149:
1144:
1141:
1119:
1116:
1113:
1110:
1107:
1104:
1079:
1075:
1071:
1066:
1061:
1040:
1037:
1034:
1031:
1011:
991:
969:
945:
942:
937:
934:
914:
911:
908:
888:
885:
882:
856:
831:
826:
802:
775:
731:
728:
715:
712:
709:
704:
700:
696:
674:
670:
649:
626:
622:
598:
594:
573:
570:
565:
561:
540:
529:
528:
517:
506:
473:
469:
446:
442:
438:
433:
429:
408:
403:
399:
395:
379:
376:
369:discrete space
364:is separable.
349:
329:
307:
302:
280:
258:
253:
248:
245:
240:
236:
232:
229:
226:
221:
217:
213:
210:
206:
182:
154:
153:First examples
151:
102:
97:
94:
91:
87:
81:
77:
73:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3033:
3022:
3019:
3017:
3014:
3013:
3011:
3001:
2997:
2993:
2987:
2983:
2978:
2977:
2970:
2967:
2963:
2959:
2953:
2949:
2945:
2941:
2940:
2935:
2931:
2927:
2924:
2920:
2916:
2912:
2908:
2904:
2899:
2895:
2891:
2886:
2883:
2879:
2875:
2869:
2865:
2861:
2857:
2853:
2852:
2851:
2836:
2835:
2829:
2828:
2821:
2819:
2816:
2800:
2780:
2760:
2737:
2734:
2731:
2708:
2688:
2678:
2674:
2669:
2664:
2660:
2659:
2651:
2647:
2640:
2637:
2625:
2621:
2620:
2615:
2609:
2606:
2599:
2594:
2584:
2574:
2570:
2566:
2565:
2564:
2562:
2555:
2551:
2548:
2547:Heinonen 2003
2544:
2543:Stefan Banach
2540:
2539:supremum norm
2536:
2532:
2529:
2528:Heinonen 2003
2525:
2524:supremum norm
2521:
2518:
2514:
2510:
2507:
2503:
2499:
2495:
2494:
2490:
2482:
2479:
2477:is separable.
2476:
2475:supremum norm
2472:
2448:
2445:
2427:
2424:
2421:
2420:
2417:
2414:
2411:
2407:
2404:
2400:
2396:
2392:
2388:
2385:
2357:
2354:
2350:
2346:
2344:is separable.
2343:
2339:
2335:
2332:), but every
2331:
2327:
2323:
2319:
2318:
2314:
2309:
2305:
2301:
2280:
2271:
2270:supremum norm
2249:
2241:
2237:
2234:
2216:
2212:
2204:
2200:
2199:
2195:
2190:
2154:
2147:
2124:
2121:
2118:
2115:
2092:
2086:
2080:
2077:
2071:
2068:
2065:
2059:
2052:
2048:
2020:
2016:
2013:
1988:
1984:
1967:
1963:
1954:
1950:
1949:Hilbert space
1946:
1925:
1922:
1919:
1910:
1902:
1898:
1894:
1872:
1869:
1866:
1857:
1834:
1819:that the set
1818:
1814:
1795:
1792:
1789:
1779:
1778:unit interval
1775:
1753:
1750:
1747:
1738:
1730:
1712:
1709:
1706:
1703:
1682:
1678:
1675:
1665:
1662:
1658:
1636:
1632:
1629:
1626:
1622:
1616:
1612:
1604:
1600:
1598:is separable.
1558:
1555:
1547:
1528:
1522:
1514:
1499:
1491:
1488:
1484:
1483:
1479:
1474:
1472:
1470:
1466:
1462:
1458:
1450:
1448:
1446:
1430:
1410:
1388:
1384:
1363:
1335:
1295:
1290:
1274:
1270:
1245:
1241:
1236:
1215:
1195:
1175:
1166:
1153:
1150:
1147:
1139:
1117:
1108:
1102:
1073:
1064:
1059:
1035:
1029:
1009:
989:
940:
935:
932:
912:
909:
906:
886:
883:
880:
872:
824:
814:
800:
791:
764:
760:
755:
753:
749:
745:
744:quasi-compact
741:
737:
729:
727:
710:
707:
702:
698:
672:
668:
647:
624:
620:
596:
592:
571:
568:
563:
559:
538:
526:
522:
518:
515:
511:
507:
504:
501:An arbitrary
500:
499:
498:
495:
493:
489:
471:
467:
444:
440:
436:
431:
427:
401:
397:
385:
377:
375:
372:
370:
365:
363:
360:-dimensional
347:
327:
305:
278:
256:
246:
238:
234:
230:
227:
224:
219:
215:
208:
195:
180:
172:
168:
164:
160:
152:
150:
148:
144:
139:
137:
133:
129:
125:
120:
118:
95:
92:
89:
79:
75:
64:
60:
56:
52:
48:
44:
37:
33:
19:
2975:
2938:
2914:
2893:
2889:
2859:
2850:
2841:, retrieved
2833:
2814:
2680:
2668:math/9408201
2656:
2639:
2618:
2608:
2582:
2569:metric space
2560:
2559:
2534:
2519:
2517:Banach space
2502:Hilbert cube
2498:homeomorphic
2480:
2470:
2422:
2415:
2394:
2390:
2342:metric space
2338:Willard 1970
2333:
2304:Banach space
2240:Banach space
2047:metric space
1897:Banach space
1487:metric space
1454:
1294:Willard 1970
1291:
1167:
815:
756:
733:
551:, pick some
530:
525:Willard 1970
510:Willard 1970
496:
381:
373:
366:
156:
140:
121:
50:
40:
2631:Proposition
2537:, with the
2349:cardinality
2330:Moore plane
736:cardinality
730:Cardinality
128:cardinality
117:open subset
43:mathematics
3010:Categories
2843:6 February
2600:References
2428:The space
2397:cannot be
2315:Properties
2191:operator).
2187:being the
2167:(and with
1731:The space
1515:The space
1465:algorithms
147:metrizable
49:is called
2936:(1995) ,
2815:separable
2801:μ
2781:μ
2761:μ
2738:μ
2689:μ
2513:isometric
2473:with the
2285:∞
2254:∞
2250:ℓ
2213:ω
2175:△
2155:μ
2125:∈
2090:△
2081:μ
2060:ρ
1964:ℓ
1716:∞
1707:≤
1679:μ
1633:μ
1559:⊆
1431:κ
1411:κ
1389:κ
1364:κ
1275:κ
1246:κ
1196:κ
1143:¯
1115:→
944:¯
936:∈
910:∈
884:⊆
748:connected
569:∈
437:∈
247:∈
228:…
167:real line
101:∞
55:countable
51:separable
2913:(1952),
2858:(1975),
2648:(1995).
2616:(2013).
2328:and the
2322:subspace
1683:⟩
1659:⟨
845:, where
514:quotient
503:subspace
492:Lindelöf
149:spaces.
63:sequence
3000:0264581
2966:0507446
2923:0050870
2882:0370454
2673:Bibcode
2661:: 262.
2573:density
2410:compact
2146:measure
1776:on the
1548:subset
1546:compact
1208:. Then
925:, then
790:closure
521:product
194:vectors
2998:
2988:
2964:
2954:
2921:
2880:
2870:
2634:3.4.5.
2408:For a
2399:normal
2382:, the
2051:metric
1891:. The
746:, and
159:finite
2944:Dover
2838:(PDF)
2663:arXiv
2653:(PDF)
2049:with
1130:when
873:: if
59:dense
2986:ISBN
2952:ISBN
2868:ISBN
2845:2009
2581:C(,
2334:open
2302:The
2238:The
2201:The
2108:for
1713:<
1601:The
1459:and
899:and
382:Any
45:, a
2898:doi
2818:iff
2813:is
2681:If
2571:of
2545:. (
2306:of
1903:of
1772:of
1447:).
161:or
41:In
34:or
3012::
2996:MR
2994:,
2984:,
2980:,
2962:MR
2960:,
2950:,
2932:;
2919:MR
2892:,
2878:MR
2876:,
2866:,
2679:.
2671:.
2655:.
2628:,
2622:.
2567:A
2563::
2320:A
2017:A
1947:A
1471:.
813:.
757:A
519:A
494:.
340:,
57:,
2942:(
2900::
2894:1
2741:)
2735:,
2732:X
2729:(
2709:X
2675::
2665::
2626:.
2595:)
2589:α
2585:)
2583:R
2577:α
2556:.
2549:)
2535:R
2530:)
2520:l
2508:.
2481:X
2471:X
2457:)
2453:R
2449:,
2446:X
2443:(
2438:C
2423:X
2416:X
2395:X
2391:X
2368:c
2299:.
2281:L
2217:1
2130:F
2122:B
2119:,
2116:A
2096:)
2093:B
2087:A
2084:(
2078:=
2075:)
2072:B
2069:,
2066:A
2063:(
2031:F
2014:.
1997:S
1968:2
1944:.
1932:)
1929:]
1926:1
1923:,
1920:0
1917:[
1914:(
1911:C
1879:)
1876:]
1873:1
1870:,
1867:0
1864:[
1861:(
1858:C
1838:]
1835:x
1832:[
1828:Q
1799:]
1796:1
1793:,
1790:0
1787:[
1760:)
1757:]
1754:1
1751:,
1748:0
1745:[
1742:(
1739:C
1728:.
1710:p
1704:1
1676:,
1671:M
1666:,
1663:X
1637:)
1630:,
1627:X
1623:(
1617:p
1613:L
1585:R
1563:R
1556:K
1532:)
1529:K
1526:(
1523:C
1500:n
1443:(
1385:2
1341:c
1336:2
1312:R
1306:R
1271:2
1242:2
1237:2
1216:X
1176:X
1154:.
1151:X
1148:=
1140:Y
1118:X
1112:)
1109:Y
1106:(
1103:S
1078:|
1074:Y
1070:|
1065:2
1060:2
1039:)
1036:Y
1033:(
1030:S
1010:z
990:Y
968:B
941:Y
933:z
913:X
907:z
887:X
881:Y
855:c
830:c
825:2
801:X
774:c
714:}
711:x
708:,
703:0
699:x
695:{
673:0
669:x
648:X
625:0
621:x
597:0
593:x
572:X
564:0
560:x
539:X
472:n
468:U
445:n
441:U
432:n
428:x
407:}
402:n
398:U
394:{
348:n
328:n
306:n
301:R
279:n
257:n
252:Q
244:)
239:n
235:r
231:,
225:,
220:1
216:r
212:(
209:=
205:r
181:n
96:1
93:=
90:n
86:}
80:n
76:x
72:{
38:.
20:)
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