3073:'s construction of the real numbers is similar to the above construction; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances. The additional subtlety to contend with is that it is not logically permissible to use the completeness of the real numbers in their own construction. Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a
2699:
2523:
3097:
for more details). One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the
3185:, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the
2581:
2405:
3067:). This defines an isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.
3031:
898:
707:
105:
is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the
3517:
is called complete. One can furthermore construct a completion for an arbitrary uniform space similar to the completion of metric spaces. The most general situation in which Cauchy nets apply is
3241:, an alternative to relying on a metric structure for defining completeness and constructing the completion of a space is to use a group structure. This is most often seen in the context of
2925:
2877:
967:
2042:
2694:{\textstyle \operatorname {int} \left(\bigcap _{i\in \mathbb {N} }\operatorname {cl} S_{i}\right)=\operatorname {int} \operatorname {cl} \left(\bigcap _{i\in \mathbb {N} }S_{i}\right).}
2518:{\textstyle \operatorname {cl} \left(\bigcup _{i\in \mathbb {N} }\operatorname {int} S_{i}\right)=\operatorname {cl} \operatorname {int} \left(\bigcup _{i\in \mathbb {N} }S_{i}\right).}
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If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an
1086:
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3196:. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section
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This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit
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Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of
3539: – in algebra, any of several related functors on rings and modules that result in complete topological rings and modules
3245:, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points
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to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval
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on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of
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is complete; for example the given sequence does have a limit in this interval, namely zero.
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3720:. Ramanujan, M.S. (trans.). Oxford: Clarendon Press; New York: Oxford University Press.
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becomes a complete metric space if we define the distance between the sequences
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yet no rational number has this property. However, considered as a sequence of
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A common generalisation of these definitions can be found in the context of a
3637:
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arise by completing the rational numbers with respect to a different metric.
3474:. If every Cauchy net (or equivalently every Cauchy filter) has a limit in
2804:
by this property (among all complete metric spaces isometrically containing
1530:
526:
3102:
give just one choice of Cauchy sequence in the relevant equivalence class.
1112:(with the metric given by the absolute difference) are complete, and so is
3035:(This limit exists because the real numbers are complete.) This is only a
3176:
3063:(i.e., the equivalence class containing the sequence with constant value
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30:"Cauchy completion" redirects here. For the use in category theory, see
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is a Banach space, and so a complete metric space, with respect to the
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is Cauchy, but does not have a limit in the given space. However the
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791:, is not complete. Consider for instance the sequence defined by
1154:. However, the supremum norm does not give a norm on the space C
1148:
continuous real-valued functions on a closed and bounded interval
893:{\displaystyle x_{n+1}={\frac {x_{n}}{2}}+{\frac {1}{x_{n}}}.}
3548: – Topological space with a notion of uniform properties
702:{\displaystyle \operatorname {diam} \left(F_{n}\right)\to 0,}
411:
if any of the following equivalent conditions are satisfied:
3189:
is purely topological, it applies to these spaces as well.
2777:, then there exists a unique uniformly continuous function
1891:
is a complete metric space. Here we define the distance in
3085:, and is the unique totally ordered complete field (up to
3622:. River Edge, N.J. London: World Scientific. p. 33.
1130:
may or may not be complete; those that are complete are
3550:
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3533: – Concept in general topology and analysis
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1977:
2920:{\displaystyle y_{\bullet }=\left(y_{n}\right)}
2872:{\displaystyle x_{\bullet }=\left(x_{n}\right)}
962:{\displaystyle x={\frac {x}{2}}+{\frac {1}{x}}}
3192:Completely metrizable spaces are often called
3164:, meaning that a complete metric space can be
3595:Introduction to Metric and Topological Spaces
3204:for a wider class of topological spaces, the
2037:{\displaystyle d(f,g)\equiv \sup\{d:x\in X\}}
8:
3464:in the definition of completeness by Cauchy
2031:
1980:
1210:whose topology can be induced by a complete
3572: – Theorem in order and lattice theory
3081:. This field is complete, admits a natural
3055:with the equivalence class of sequences in
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2241:. The fixed-point theorem is often used to
2210:says that every complete metric space is a
1521:if there is no such index. This space is
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3620:Convex analysis in general vector spaces
3716:Meise, Reinhold; Vogt, Dietmar (1997).
3582:
3362:{\displaystyle d(x,y)<\varepsilon ,}
3093:as the field of real numbers (see also
1208:locally convex topological vector space
779:of rational numbers, with the standard
3460:It is also possible to replace Cauchy
3211:A topological space homeomorphic to a
629:{\displaystyle F_{n+1}\subseteq F_{n}}
110:of a given space, as explained below.
3200:). Indeed, some authors use the term
1081:{\displaystyle x_{n}={\tfrac {1}{n}}}
7:
2237:on a complete metric space admits a
361:{\displaystyle d(x_{m},x_{n})<r.}
276:such that for all positive integers
3718:Introduction to functional analysis
3077:that has the rational numbers as a
2322:{\displaystyle S_{1},S_{2},\ldots }
2283:be a complete metric space and let
1622:is compact and therefore complete.
559:tending to 0, has a non-empty
417:Every Cauchy sequence of points in
3412:via subtraction in the comparison
3215:complete metric space is called a
3156:Completeness is a property of the
2931:, we may define their distance as
1592:. This is a generalization of the
1387:{\displaystyle \left(y_{n}\right)}
1352:{\displaystyle \left(x_{n}\right)}
1298:is an arbitrary set, then the set
1267:-adic metric in the same way that
25:
3705:, "Real and Functional Analysis"
3555:Complete topological vector space
3229:Uniform space § Completeness
2761:is any complete metric space and
3223:Alternatives and generalizations
3198:Alternatives and generalizations
3095:Construction of the real numbers
1568:; completeness follows from the
1443:is the smallest index for which
1202:can be given the structure of a
3561:Ekeland's variational principle
3285:is gauged not by a real number
3237:can also be defined in general
3206:completely uniformizable spaces
2831:. For any two Cauchy sequences
2823:can be constructed as a set of
2738:{\displaystyle {\overline {M}}}
2222:subsets of the space has empty
1657:be a complete metric space. If
1416:{\displaystyle {\tfrac {1}{N}}}
3347:
3335:
2190:
2178:
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79:is not complete, because e.g.
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3152:Topologically complete spaces
2767:uniformly continuous function
1761:is a complete subspace, then
709:then there is a unique point
525:Every decreasing sequence of
3298:{\displaystyle \varepsilon }
3182:completely metrizable spaces
2730:
2329:be a sequence of subsets of
1926:in terms of the distance in
1754:{\displaystyle A\subseteq X}
1676:{\displaystyle A\subseteq X}
437:has a limit that is also in
3650:Kelley, Problem 6.L, p. 208
1166:of continuous functions on
1029:{\displaystyle {\sqrt {2}}}
502:(that is, to some point of
97:{\displaystyle {\sqrt {2}}}
3771:
3226:
2718:(which is also denoted as
2231:Banach fixed-point theorem
2125:{\displaystyle C_{b}(X,M)}
1009:, it does converge to the
29:
3692:(Wiley, New York, 1978).
3440:{\displaystyle x-y\in N.}
3243:topological vector spaces
3172:, which is not complete.
2203:and hence also complete.
1863:of all bounded functions
590:is closed and non-empty,
462:Every Cauchy sequence in
304:{\displaystyle m,n>N,}
2753:. It has the following
2247:inverse function theorem
2168:is a closed subspace of
2161:{\displaystyle f:X\to M}
1533:number of copies of the
998:{\displaystyle x^{2}=2,}
2827:of Cauchy sequences in
1275:with the usual metric.
817:{\displaystyle x_{1}=1}
3618:Zalinescu, C. (2002).
3570:Knaster–Tarski theorem
3546:Complete uniform space
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2261: —
2235:contraction mapping
1594:Heine–Borel theorem
1588:it is complete and
1188:compact convergence
1180:unbounded functions
735:common to all sets
18:Complete (topology)
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3490:{\displaystyle X,}
3487:
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3371:open neighbourhood
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3325:in the comparison
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3295:
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3239:topological groups
3144:, the result is a
3121:{\displaystyle p,}
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3023:
2986:
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2819:The completion of
2755:universal property
2745:), which contains
2735:
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2437:
2392:
2372:
2345:{\displaystyle X.}
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2218:of countably many
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2136:bounded functions
2132:consisting of all
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1566:geodesic manifolds
1552:{\displaystyle S.}
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1349:
1314:
1288:
1252:{\displaystyle p.}
1249:
1078:
1076:
1026:
995:
959:
916:{\displaystyle x,}
913:
890:
814:
755:
725:
699:
652:{\displaystyle n,}
649:
626:
580:
548:{\displaystyle X,}
545:
512:
492:
472:
453:{\displaystyle X.}
450:
427:
397:
358:
301:
266:
243:
210:
175:
94:
3604:978-0-19-853161-6
3510:{\displaystyle X}
3405:{\displaystyle 0}
3385:{\displaystyle N}
3318:{\displaystyle d}
3278:{\displaystyle y}
3258:{\displaystyle x}
3100:decimal expansion
2977:
2733:
2655:
2596:
2571:{\displaystyle X}
2479:
2420:
2395:{\displaystyle X}
2276:{\displaystyle X}
2083:{\displaystyle M}
2068:topological space
2059:{\displaystyle X}
1939:{\displaystyle M}
1884:{\displaystyle M}
1821:{\displaystyle M}
1797:{\displaystyle X}
1774:{\displaystyle A}
1696:{\displaystyle A}
1609:{\displaystyle S}
1514:{\displaystyle 0}
1436:{\displaystyle N}
1410:
1317:{\displaystyle S}
1291:{\displaystyle S}
1075:
1024:
1011:irrational number
957:
944:
885:
865:
515:{\displaystyle X}
495:{\displaystyle X}
475:{\displaystyle X}
430:{\displaystyle X}
269:{\displaystyle N}
92:
16:(Redirected from
3762:
3731:
3682:
3669:General Topology
3651:
3648:
3642:
3641:
3615:
3609:
3608:
3587:
3566:
3551:
3542:
3516:
3514:
3513:
3508:
3496:
3494:
3493:
3488:
3446:
3444:
3443:
3438:
3411:
3409:
3408:
3403:
3391:
3389:
3388:
3383:
3368:
3366:
3365:
3360:
3324:
3322:
3321:
3316:
3304:
3302:
3301:
3296:
3284:
3282:
3281:
3276:
3264:
3262:
3261:
3256:
3235:Cauchy sequences
3171:
3134:
3127:
3125:
3124:
3119:
3032:
3030:
3029:
3024:
3022:
3018:
3017:
3016:
3004:
3003:
2985:
2973:
2969:
2968:
2967:
2955:
2954:
2926:
2924:
2923:
2918:
2916:
2912:
2911:
2895:
2894:
2878:
2876:
2875:
2870:
2868:
2864:
2863:
2847:
2846:
2744:
2742:
2741:
2736:
2734:
2726:
2700:
2698:
2697:
2692:
2687:
2683:
2682:
2681:
2671:
2670:
2634:
2630:
2629:
2628:
2612:
2611:
2577:
2575:
2574:
2569:
2553:
2551:
2550:
2545:
2543:
2542:
2524:
2522:
2521:
2516:
2511:
2507:
2506:
2505:
2495:
2494:
2458:
2454:
2453:
2452:
2436:
2435:
2401:
2399:
2398:
2393:
2381:
2379:
2378:
2373:
2371:
2370:
2351:
2349:
2348:
2343:
2328:
2326:
2325:
2320:
2312:
2311:
2299:
2298:
2282:
2280:
2279:
2274:
2262:
2259:
2202:
2200:
2199:
2194:
2167:
2165:
2164:
2159:
2131:
2129:
2128:
2123:
2106:
2105:
2089:
2087:
2086:
2081:
2065:
2063:
2062:
2057:
2043:
2041:
2040:
2035:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1890:
1888:
1887:
1882:
1870:
1866:
1862:
1860:
1859:
1854:
1827:
1825:
1824:
1819:
1803:
1801:
1800:
1795:
1781:is also closed.
1780:
1778:
1777:
1772:
1760:
1758:
1757:
1752:
1734:
1732:
1731:
1726:
1702:
1700:
1699:
1694:
1682:
1680:
1679:
1674:
1656:
1654:
1653:
1648:
1621:
1615:
1613:
1612:
1607:
1558:
1556:
1555:
1550:
1520:
1518:
1517:
1512:
1500:
1498:
1497:
1492:
1490:
1489:
1469:
1467:
1466:
1461:
1459:
1458:
1442:
1440:
1439:
1434:
1422:
1420:
1419:
1414:
1412:
1403:
1393:
1391:
1390:
1385:
1383:
1379:
1378:
1358:
1356:
1355:
1350:
1348:
1344:
1343:
1323:
1321:
1320:
1315:
1303:
1297:
1295:
1294:
1289:
1258:
1256:
1255:
1250:
1201:
1177:
1165:
1145:
1095:
1087:
1085:
1084:
1079:
1077:
1068:
1062:
1061:
1045:
1035:
1033:
1032:
1027:
1025:
1020:
1004:
1002:
1001:
996:
985:
984:
968:
966:
965:
960:
958:
950:
945:
937:
923:then by solving
922:
920:
919:
914:
899:
897:
896:
891:
886:
884:
883:
871:
866:
861:
860:
851:
846:
845:
823:
821:
820:
815:
807:
806:
764:
762:
761:
756:
751:
750:
734:
732:
731:
726:
708:
706:
705:
700:
689:
685:
684:
658:
656:
655:
650:
635:
633:
632:
627:
625:
624:
612:
611:
589:
587:
586:
581:
579:
578:
554:
552:
551:
546:
521:
519:
518:
513:
501:
499:
498:
493:
481:
479:
478:
473:
459:
457:
456:
451:
436:
434:
433:
428:
406:
404:
403:
398:
367:
365:
364:
359:
345:
344:
332:
331:
310:
308:
307:
302:
275:
273:
272:
267:
252:
250:
249:
244:
219:
217:
216:
211:
184:
182:
181:
176:
168:
167:
155:
154:
142:
141:
103:
101:
100:
95:
93:
88:
77:rational numbers
71:
68:that is also in
63:
47:
32:Karoubi envelope
21:
3770:
3769:
3765:
3764:
3763:
3761:
3760:
3759:
3745:Metric geometry
3735:
3734:
3728:
3715:
3686:Kreyszig, Erwin
3679:
3665:Kelley, John L.
3663:
3660:
3655:
3654:
3649:
3645:
3630:
3617:
3616:
3612:
3605:
3589:
3588:
3584:
3579:
3564:
3549:
3540:
3527:
3499:
3498:
3476:
3475:
3414:
3413:
3394:
3393:
3374:
3373:
3327:
3326:
3307:
3306:
3305:via the metric
3287:
3286:
3267:
3266:
3247:
3246:
3231:
3225:
3169:
3160:and not of the
3154:
3130:
3107:
3106:
3008:
2995:
2994:
2990:
2959:
2946:
2945:
2941:
2933:
2932:
2903:
2899:
2886:
2881:
2880:
2855:
2851:
2838:
2833:
2832:
2720:
2719:
2708:
2703:
2673:
2654:
2650:
2620:
2595:
2591:
2580:
2579:
2560:
2559:
2534:
2529:
2528:
2497:
2478:
2474:
2444:
2419:
2415:
2404:
2403:
2384:
2383:
2362:
2357:
2356:
2331:
2330:
2303:
2290:
2285:
2284:
2265:
2264:
2260:
2257:
2214:. That is, the
2170:
2169:
2138:
2137:
2097:
2092:
2091:
2072:
2071:
2048:
2047:
1951:
1950:
1928:
1927:
1893:
1892:
1873:
1872:
1868:
1864:
1830:
1829:
1810:
1809:
1786:
1785:
1763:
1762:
1737:
1736:
1705:
1704:
1685:
1684:
1659:
1658:
1627:
1626:
1617:
1598:
1597:
1590:totally bounded
1578:
1538:
1537:
1503:
1502:
1481:
1476:
1475:
1450:
1445:
1444:
1425:
1424:
1396:
1395:
1370:
1366:
1361:
1360:
1335:
1331:
1326:
1325:
1306:
1305:
1299:
1280:
1279:
1238:
1237:
1225:
1191:
1167:
1155:
1135:
1114:Euclidean space
1110:complex numbers
1093:
1090:closed interval
1053:
1048:
1047:
1043:
1014:
1013:
976:
971:
970:
925:
924:
902:
901:
875:
852:
831:
826:
825:
798:
793:
792:
773:
742:
737:
736:
711:
710:
676:
672:
661:
660:
638:
637:
616:
597:
592:
591:
570:
565:
564:
534:
533:
504:
503:
484:
483:
464:
463:
439:
438:
419:
418:
377:
376:
375:A metric space
336:
323:
312:
311:
278:
277:
258:
257:
229:
228:
190:
189:
159:
146:
133:
128:
127:
119:Cauchy sequence
116:
82:
81:
69:
61:
58:Cauchy sequence
45:
35:
28:
27:Metric geometry
23:
22:
15:
12:
11:
5:
3768:
3766:
3758:
3757:
3755:Uniform spaces
3752:
3747:
3737:
3736:
3733:
3732:
3726:
3713:
3700:
3683:
3677:
3659:
3656:
3653:
3652:
3643:
3628:
3610:
3603:
3581:
3580:
3578:
3575:
3574:
3573:
3567:
3558:
3552:
3543:
3534:
3526:
3523:
3506:
3486:
3483:
3436:
3433:
3430:
3427:
3424:
3421:
3401:
3381:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3314:
3294:
3274:
3254:
3227:Main article:
3224:
3221:
3179:one considers
3153:
3150:
3117:
3114:
3083:total ordering
3059:converging to
3021:
3015:
3011:
3007:
3002:
2998:
2993:
2989:
2984:
2980:
2976:
2972:
2966:
2962:
2958:
2953:
2949:
2944:
2940:
2915:
2910:
2906:
2902:
2898:
2893:
2889:
2867:
2862:
2858:
2854:
2850:
2845:
2841:
2797:is determined
2751:dense subspace
2732:
2729:
2707:
2704:
2702:
2701:
2690:
2686:
2680:
2676:
2669:
2665:
2662:
2658:
2653:
2649:
2646:
2643:
2640:
2637:
2633:
2627:
2623:
2619:
2616:
2610:
2606:
2603:
2599:
2594:
2590:
2587:
2567:
2541:
2537:
2525:
2514:
2510:
2504:
2500:
2493:
2489:
2486:
2482:
2477:
2473:
2470:
2467:
2464:
2461:
2457:
2451:
2447:
2443:
2440:
2434:
2430:
2427:
2423:
2418:
2414:
2411:
2391:
2369:
2365:
2341:
2338:
2318:
2315:
2310:
2306:
2302:
2297:
2293:
2272:
2251:
2233:states that a
2192:
2189:
2186:
2183:
2180:
2177:
2157:
2154:
2151:
2148:
2145:
2121:
2118:
2115:
2112:
2109:
2104:
2100:
2079:
2055:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1935:
1915:
1912:
1909:
1906:
1903:
1900:
1880:
1852:
1849:
1846:
1843:
1840:
1837:
1817:
1793:
1770:
1750:
1747:
1744:
1724:
1721:
1718:
1715:
1712:
1692:
1672:
1669:
1666:
1646:
1643:
1640:
1637:
1634:
1605:
1586:if and only if
1577:
1574:
1548:
1545:
1535:discrete space
1510:
1488:
1484:
1457:
1453:
1432:
1409:
1406:
1382:
1377:
1373:
1369:
1347:
1342:
1338:
1334:
1313:
1287:
1248:
1245:
1221:
1134:. The space C
1121:usual distance
1074:
1071:
1065:
1060:
1056:
1023:
994:
991:
988:
983:
979:
956:
953:
948:
943:
940:
935:
932:
912:
909:
889:
882:
878:
874:
869:
864:
859:
855:
849:
844:
841:
838:
834:
813:
810:
805:
801:
785:absolute value
772:
769:
768:
767:
766:
765:
754:
749:
745:
724:
721:
718:
698:
695:
692:
688:
683:
679:
675:
671:
668:
648:
645:
623:
619:
615:
610:
607:
604:
600:
577:
573:
544:
541:
530:closed subsets
523:
511:
491:
471:
460:
449:
446:
426:
396:
393:
390:
387:
384:
371:Complete space
357:
354:
351:
348:
343:
339:
335:
330:
326:
322:
319:
300:
297:
294:
291:
288:
285:
265:
242:
239:
236:
209:
206:
203:
200:
197:
174:
171:
166:
162:
158:
153:
149:
145:
140:
136:
115:
112:
91:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3767:
3756:
3753:
3751:
3748:
3746:
3743:
3742:
3740:
3729:
3727:0-19-851485-9
3723:
3719:
3714:
3712:
3711:0-387-94001-4
3708:
3704:
3701:
3699:
3698:0-471-03729-X
3695:
3691:
3687:
3684:
3680:
3678:0-387-90125-6
3674:
3670:
3666:
3662:
3661:
3657:
3647:
3644:
3639:
3635:
3631:
3629:981-238-067-1
3625:
3621:
3614:
3611:
3606:
3600:
3596:
3592:
3586:
3583:
3576:
3571:
3568:
3562:
3559:
3556:
3553:
3547:
3544:
3538:
3535:
3532:
3529:
3528:
3524:
3522:
3520:
3519:Cauchy spaces
3504:
3484:
3481:
3473:
3469:
3468:
3463:
3458:
3456:
3452:
3451:uniform space
3447:
3434:
3431:
3428:
3425:
3422:
3419:
3399:
3379:
3372:
3356:
3353:
3350:
3344:
3341:
3338:
3332:
3312:
3292:
3272:
3252:
3244:
3240:
3236:
3230:
3222:
3220:
3218:
3214:
3209:
3207:
3203:
3199:
3195:
3190:
3188:
3184:
3183:
3178:
3173:
3167:
3163:
3159:
3151:
3149:
3147:
3146:Hilbert space
3143:
3138:
3136:
3135:-adic numbers
3133:
3115:
3112:
3103:
3101:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3068:
3066:
3062:
3058:
3054:
3050:
3046:
3042:
3038:
3033:
3019:
3013:
3009:
3005:
3000:
2996:
2991:
2987:
2982:
2974:
2970:
2964:
2960:
2956:
2951:
2947:
2942:
2938:
2930:
2913:
2908:
2904:
2900:
2896:
2891:
2887:
2865:
2860:
2856:
2852:
2848:
2843:
2839:
2830:
2826:
2822:
2817:
2815:
2811:
2807:
2803:
2800:
2796:
2793:. The space
2792:
2789:that extends
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2727:
2717:
2713:
2705:
2688:
2684:
2678:
2674:
2663:
2660:
2656:
2651:
2647:
2644:
2641:
2638:
2635:
2631:
2625:
2621:
2617:
2614:
2604:
2601:
2597:
2592:
2588:
2585:
2565:
2557:
2539:
2535:
2526:
2512:
2508:
2502:
2498:
2487:
2484:
2480:
2475:
2471:
2468:
2465:
2462:
2459:
2455:
2449:
2445:
2441:
2438:
2428:
2425:
2421:
2416:
2412:
2409:
2389:
2382:is closed in
2367:
2363:
2354:
2353:
2352:
2339:
2336:
2316:
2313:
2308:
2304:
2300:
2295:
2291:
2270:
2250:
2248:
2244:
2240:
2236:
2232:
2227:
2225:
2221:
2220:nowhere dense
2217:
2213:
2209:
2204:
2187:
2184:
2181:
2175:
2155:
2149:
2146:
2143:
2135:
2116:
2113:
2110:
2102:
2098:
2077:
2069:
2053:
2044:
2028:
2025:
2022:
2019:
2010:
2004:
2001:
1995:
1989:
1983:
1974:
1968:
1965:
1962:
1956:
1949:
1948:supremum norm
1933:
1910:
1907:
1904:
1898:
1878:
1847:
1844:
1841:
1835:
1815:
1807:
1791:
1782:
1768:
1748:
1745:
1742:
1719:
1716:
1713:
1690:
1670:
1667:
1664:
1641:
1638:
1635:
1623:
1620:
1603:
1595:
1591:
1587:
1583:
1576:Some theorems
1575:
1573:
1571:
1567:
1563:
1559:
1546:
1543:
1536:
1532:
1528:
1524:
1508:
1486:
1482:
1473:
1455:
1451:
1430:
1407:
1404:
1380:
1375:
1371:
1367:
1345:
1340:
1336:
1332:
1311:
1302:
1285:
1276:
1274:
1270:
1266:
1262:
1246:
1243:
1236:
1232:
1231:-adic numbers
1230:
1224:
1220:
1215:
1213:
1209:
1205:
1204:Fréchet space
1199:
1195:
1189:
1185:
1181:
1175:
1171:
1163:
1159:
1153:
1152:supremum norm
1149:
1143:
1139:
1133:
1132:Banach spaces
1129:
1126:
1122:
1118:
1115:
1111:
1107:
1103:
1098:
1096:
1094:[0,1]
1091:
1072:
1069:
1063:
1058:
1054:
1042:
1041:open interval
1037:
1021:
1012:
1008:
992:
989:
986:
981:
977:
954:
951:
946:
941:
938:
933:
930:
910:
907:
887:
880:
876:
872:
867:
862:
857:
853:
847:
842:
839:
836:
832:
811:
808:
803:
799:
790:
786:
783:given by the
782:
778:
770:
752:
747:
743:
722:
719:
716:
696:
693:
686:
681:
677:
673:
669:
666:
646:
643:
621:
617:
613:
608:
605:
602:
598:
575:
571:
562:
558:
542:
539:
531:
528:
524:
509:
489:
482:converges in
469:
461:
447:
444:
424:
416:
415:
414:
413:
412:
410:
391:
388:
385:
373:
372:
368:
355:
352:
349:
341:
337:
333:
328:
324:
317:
298:
295:
292:
289:
286:
283:
263:
256:
240:
237:
234:
227:
223:
204:
201:
198:
188:
172:
169:
164:
160:
156:
151:
147:
143:
138:
134:
126:
121:
120:
113:
111:
109:
104:
89:
78:
73:
67:
60:of points in
59:
55:
51:
44:
40:
33:
19:
3717:
3689:
3671:. Springer.
3668:
3646:
3619:
3613:
3594:
3585:
3531:Cauchy space
3465:
3461:
3459:
3448:
3232:
3217:Polish space
3210:
3201:
3193:
3191:
3180:
3174:
3166:homeomorphic
3161:
3157:
3155:
3139:
3131:
3105:For a prime
3104:
3090:
3069:
3064:
3060:
3056:
3052:
3048:
3044:
3037:pseudometric
3034:
2928:
2828:
2820:
2818:
2813:
2809:
2805:
2794:
2790:
2786:
2782:
2778:
2774:
2770:
2762:
2758:
2746:
2715:
2711:
2709:
2258:(C. Ursescu)
2252:
2228:
2205:
2045:
1783:
1624:
1618:
1579:
1560:
1523:homeomorphic
1300:
1277:
1272:
1268:
1264:
1260:
1235:prime number
1228:
1222:
1218:
1216:
1197:
1193:
1173:
1169:
1161:
1157:
1141:
1137:
1116:
1105:
1101:
1099:
1038:
1007:real numbers
969:necessarily
776:
774:
561:intersection
408:
374:
370:
369:
221:
187:metric space
122:
118:
117:
107:
74:
54:Cauchy space
53:
49:
43:metric space
36:
3703:Lang, Serge
3453:, where an
3087:isomorphism
2239:fixed point
2212:Baire space
1119:, with the
226:real number
56:) if every
3739:Categories
3658:References
3470:or Cauchy
3369:but by an
3089:). It is
2810:completion
2706:Completion
2134:continuous
1271:completes
1217:The space
1100:The space
789:difference
775:The space
636:for every
220:is called
114:Definition
108:completion
48:is called
3638:285163112
3462:sequences
3455:entourage
3429:∈
3423:−
3354:ε
3293:ε
3213:separable
2965:∙
2952:∙
2892:∙
2844:∙
2731:¯
2664:∈
2657:⋂
2648:
2642:
2618:
2605:∈
2598:⋂
2589:
2488:∈
2481:⋃
2472:
2466:
2442:
2429:∈
2422:⋃
2413:
2317:…
2153:→
2026:∈
1975:≡
1946:with the
1746:⊆
1668:⊆
1531:countable
1263:with the
720:∈
691:→
670:
614:⊆
557:diameters
527:non-empty
173:…
3750:Topology
3667:(1975).
3593:(1975).
3525:See also
3177:topology
3162:topology
3079:subfield
2802:isometry
2527:If each
2355:If each
2224:interior
1472:distinct
1214:metric.
1184:topology
771:Examples
409:complete
125:sequence
50:complete
3472:filters
3091:defined
2765:is any
2254:Theorem
1527:product
1525:to the
787:of the
255:integer
3724:
3709:
3696:
3675:
3636:
3626:
3601:
3233:Since
3158:metric
3071:Cantor
2256:
1580:Every
1423:where
1394:to be
781:metric
222:Cauchy
64:has a
52:(or a
3577:Notes
3497:then
3170:(0,1)
3075:field
2799:up to
2781:from
2769:from
2757:: if
2749:as a
2578:then
2402:then
2243:prove
2216:union
2066:is a
1867:from
1804:is a
1529:of a
1474:from
1144:]
1136:[
1044:(0,1)
563:: if
555:with
185:in a
66:limit
3722:ISBN
3707:ISBN
3694:ISBN
3673:ISBN
3634:OCLC
3624:ISBN
3599:ISBN
3467:nets
3351:<
3265:and
3128:the
2879:and
2556:open
2263:Let
2245:the
2229:The
2206:The
2070:and
1808:and
1625:Let
1359:and
1206:: a
1039:The
824:and
667:diam
659:and
350:<
293:>
238:>
41:, a
3392:of
3175:In
3051:of
2979:lim
2927:in
2812:of
2785:to
2773:to
2639:int
2586:int
2558:in
2554:is
2469:int
2439:int
2046:If
1978:sup
1871:to
1806:set
1784:If
1616:of
1501:or
1470:is
1278:If
1226:of
1190:, C
1186:of
1146:of
1108:of
532:of
407:is
37:In
3741::
3688:,
3632:.
3597:.
3219:.
3208:.
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2816:.
2795:M'
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2779:f′
2716:M′
2645:cl
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123:A
72:.
3730:.
3681:.
3640:.
3607:.
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2689:.
2685:)
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2005:g
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1984:d
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