3062:'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
2688:
2512:
3086:
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
3174:, 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
2570:
2394:
3056:). 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.
3020:
887:
696:
94:
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
3506:
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
3230:, 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
2914:
2866:
956:
2031:
2683:{\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).}
2507:{\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).}
172:
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3129:
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
1075:
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3185:. 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
3528: – in algebra, any of several related functors on rings and modules that result in complete topological rings and modules
3591:
3234:, but requires only the existence of a continuous "subtraction" operation. In this setting, the distance between two points
3157:
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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3546: – A TVS where points that get progressively closer to each other will always converge to a point
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3552: – theorem that asserts that there exist nearly optimal solutions to some optimization problems
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is a set of all pairs of points that are at no more than a particular "distance" from each other.
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is complete; for example the given sequence does have a limit in this interval, namely zero.
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is complete, though complete spaces need not be compact. In fact, a metric space is compact
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3015:{\displaystyle d\left(x_{\bullet },y_{\bullet }\right)=\lim _{n}d\left(x_{n},y_{n}\right)}
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3709:. 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
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arise by completing the rational numbers with respect to a different metric.
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3463:. If every Cauchy net (or equivalently every Cauchy filter) has a limit in
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by this property (among all complete metric spaces isometrically containing
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3205:
3154:
3091:
give just one choice of Cauchy sequence in the relevant equivalence class.
3059:
1511:
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1120:
1101:(with the metric given by the absolute difference) are complete, and so is
175:
31:
3024:(This limit exists because the real numbers are complete.) This is only a
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3052:(i.e., the equivalence class containing the sequence with constant value
2544:
2200:
995:
777:
214:
19:"Cauchy completion" redirects here. For the use in category theory, see
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1139:
is a Banach space, and so a complete metric space, with respect to the
1077:
is Cauchy, but does not have a limit in the given space. However the
1519:
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3165:
2790:
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780:, is not complete. Consider for instance the sequence defined by
1143:. However, the supremum norm does not give a norm on the space C
1137:
continuous real-valued functions on a closed and bounded interval
243:
882:{\displaystyle x_{n+1}={\frac {x_{n}}{2}}+{\frac {1}{x_{n}}}.}
3537: – Topological space with a notion of uniform properties
2787:
691:{\displaystyle \operatorname {diam} \left(F_{n}\right)\to 0,}
400:
if any of the following equivalent conditions are satisfied:
3178:
is purely topological, it applies to these spaces as well.
2766:, then there exists a unique uniformly continuous function
1880:
is a complete metric space. Here we define the distance in
3074:, and is the unique totally ordered complete field (up to
3611:. River Edge, N.J. London: World Scientific. p. 33.
1119:
may or may not be complete; those that are complete are
3539:
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1585:, which states that any closed and bounded subspace
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3137:containing the original space as a dense subspace.
3679:Introductory functional analysis with applications
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3522: – Concept in general topology and analysis
3140:
2967:
1966:
2909:{\displaystyle y_{\bullet }=\left(y_{n}\right)}
2861:{\displaystyle x_{\bullet }=\left(x_{n}\right)}
951:{\displaystyle x={\frac {x}{2}}+{\frac {1}{x}}}
3181:Completely metrizable spaces are often called
3153:, meaning that a complete metric space can be
3584:Introduction to Metric and Topological Spaces
3193:for a wider class of topological spaces, the
2026:{\displaystyle d(f,g)\equiv \sup\{d:x\in X\}}
3453:in the definition of completeness by Cauchy
2020:
1969:
1199:whose topology can be induced by a complete
3561: – Theorem in order and lattice theory
3070:. This field is complete, admits a natural
3044:with the equivalence class of sequences in
2230:. The fixed-point theorem is often used to
2199:says that every complete metric space is a
3578:
1510:if there is no such index. This space is
3704:
3606:
2656:
2597:
2480:
2421:
2079:is a complete metric space, then the set
1817:is a complete metric space, then the set
167:{\displaystyle x_{1},x_{2},x_{3},\ldots }
3609:Convex analysis in general vector spaces
3705:Meise, Reinhold; Vogt, Dietmar (1997).
3351:{\displaystyle d(x,y)<\varepsilon ,}
3082:as the field of real numbers (see also
1197:locally convex topological vector space
768:of rational numbers, with the standard
3726:
3652:
3449:It is also possible to replace Cauchy
3200:A topological space homeomorphic to a
618:{\displaystyle F_{n+1}\subseteq F_{n}}
99:of a given space, as explained below.
3189:). Indeed, some authors use the term
1070:{\displaystyle x_{n}={\tfrac {1}{n}}}
2226:on a complete metric space admits a
350:{\displaystyle d(x_{m},x_{n})<r.}
265:such that for all positive integers
3707:Introduction to functional analysis
3066:that has the rational numbers as a
2311:{\displaystyle S_{1},S_{2},\ldots }
2272:be a complete metric space and let
1611:is compact and therefore complete.
548:tending to 0, has a non-empty
406:Every Cauchy sequence of points in
13:
3401:via subtraction in the comparison
3204:complete metric space is called a
3145:Completeness is a property of the
2920:, we may define their distance as
1581:. This is a generalization of the
1376:{\displaystyle \left(y_{n}\right)}
1341:{\displaystyle \left(x_{n}\right)}
1287:is an arbitrary set, then the set
1256:-adic metric in the same way that
14:
3755:
3694:, "Real and Functional Analysis"
3544:Complete topological vector space
3218:Uniform space § Completeness
2750:is any complete metric space and
3212:Alternatives and generalizations
3187:Alternatives and generalizations
3084:Construction of the real numbers
1564:
1557:; completeness follows from the
1432:is the smallest index for which
1191:can be given the structure of a
3550:Ekeland's variational principle
3274:is gauged not by a real number
3226:can also be defined in general
3195:completely uniformizable spaces
2820:. For any two Cauchy sequences
2812:can be constructed as a set of
2727:{\displaystyle {\overline {M}}}
2211:subsets of the space has empty
1646:be a complete metric space. If
1405:{\displaystyle {\tfrac {1}{N}}}
3633:
3572:
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1553:which are complete are called
1093:of real numbers and the space
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309:
196:
184:
68:is not complete, because e.g.
1:
3646:
3141:Topologically complete spaces
2756:uniformly continuous function
2694:
1750:is a complete subspace, then
698:then there is a unique point
514:Every decreasing sequence of
102:
3287:{\displaystyle \varepsilon }
3171:completely metrizable spaces
2719:
2318:be a sequence of subsets of
1915:in terms of the distance in
1743:{\displaystyle A\subseteq X}
1665:{\displaystyle A\subseteq X}
426:has a limit that is also in
7:
3639:Kelley, Problem 6.L, p. 208
3513:
1155:of continuous functions on
1018:{\displaystyle {\sqrt {2}}}
759:
491:(that is, to some point of
86:{\displaystyle {\sqrt {2}}}
10:
3760:
3215:
2707:(which is also denoted as
2220:Banach fixed-point theorem
2114:{\displaystyle C_{b}(X,M)}
998:, it does converge to the
18:
3681:(Wiley, New York, 1978).
3429:{\displaystyle x-y\in N.}
3232:topological vector spaces
3161:, which is not complete.
2192:and hence also complete.
1852:of all bounded functions
579:is closed and non-empty,
451:Every Cauchy sequence in
293:{\displaystyle m,n>N,}
3565:
2742:. It has the following
2236:inverse function theorem
2157:is a closed subspace of
2150:{\displaystyle f:X\to M}
1522:number of copies of the
987:{\displaystyle x^{2}=2,}
2816:of Cauchy sequences in
1264:with the usual metric.
806:{\displaystyle x_{1}=1}
3607:Zalinescu, C. (2002).
3559:Knaster–Tarski theorem
3535:Complete uniform space
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2197:Baire category theorem
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2185:{\displaystyle B(X,M)}
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2115:
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1909:
1908:{\displaystyle B(X,M)}
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1845:{\displaystyle B(X,M)}
1811:
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1724:be a metric space. If
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1692:is also complete. Let
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747:{\displaystyle F_{n}.}
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235:{\displaystyle r>0}
213:if for every positive
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2699:For any metric space
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2536:{\displaystyle S_{i}}
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1171:. Instead, with the
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28:mathematical analysis
3526:Completion (algebra)
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1571:compact metric space
1551:Riemannian manifolds
1529:
1494:
1467:
1436:
1416:
1387:
1352:
1317:
1297:
1293:of all sequences in
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1222:is complete for any
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1114:infinite-dimensional
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242:there is a positive
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3131:inner product space
2814:equivalence classes
2250: —
2224:contraction mapping
1583:Heine–Borel theorem
1577:it is complete and
1177:compact convergence
1169:unbounded functions
724:common to all sets
3496:
3479:{\displaystyle X,}
3476:
3426:
3391:
3371:
3360:open neighbourhood
3348:
3314:in the comparison
3304:
3284:
3264:
3244:
3228:topological groups
3133:, the result is a
3110:{\displaystyle p,}
3107:
3012:
2975:
2906:
2858:
2808:The completion of
2744:universal property
2734:), which contains
2724:
2680:
2661:
2602:
2557:
2533:
2504:
2485:
2426:
2381:
2361:
2334:{\displaystyle X.}
2331:
2308:
2262:
2244:
2207:of countably many
2182:
2147:
2125:bounded functions
2121:consisting of all
2111:
2069:
2045:
2023:
1925:
1905:
1870:
1842:
1807:
1783:
1760:
1740:
1714:
1682:
1662:
1636:
1595:
1559:Hopf–Rinow theorem
1555:geodesic manifolds
1541:{\displaystyle S.}
1538:
1500:
1480:
1449:
1422:
1402:
1400:
1373:
1338:
1303:
1277:
1241:{\displaystyle p.}
1238:
1067:
1065:
1015:
984:
948:
905:{\displaystyle x,}
902:
879:
803:
744:
714:
688:
641:{\displaystyle n,}
638:
615:
569:
537:{\displaystyle X,}
534:
501:
481:
461:
442:{\displaystyle X.}
439:
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347:
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164:
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3593:978-0-19-853161-6
3499:{\displaystyle X}
3394:{\displaystyle 0}
3374:{\displaystyle N}
3307:{\displaystyle d}
3267:{\displaystyle y}
3247:{\displaystyle x}
3089:decimal expansion
2966:
2722:
2644:
2585:
2560:{\displaystyle X}
2468:
2409:
2384:{\displaystyle X}
2265:{\displaystyle X}
2242:
2072:{\displaystyle M}
2057:topological space
2048:{\displaystyle X}
1928:{\displaystyle M}
1873:{\displaystyle M}
1810:{\displaystyle M}
1786:{\displaystyle X}
1763:{\displaystyle A}
1685:{\displaystyle A}
1598:{\displaystyle S}
1503:{\displaystyle 0}
1425:{\displaystyle N}
1399:
1306:{\displaystyle S}
1280:{\displaystyle S}
1064:
1013:
1000:irrational number
946:
933:
874:
854:
504:{\displaystyle X}
484:{\displaystyle X}
464:{\displaystyle X}
419:{\displaystyle X}
258:{\displaystyle N}
81:
3751:
3720:
3671:
3658:General Topology
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3637:
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3630:
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3505:
3503:
3502:
3497:
3485:
3483:
3482:
3477:
3435:
3433:
3432:
3427:
3400:
3398:
3397:
3392:
3380:
3378:
3377:
3372:
3357:
3355:
3354:
3349:
3313:
3311:
3310:
3305:
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3253:
3251:
3250:
3245:
3224:Cauchy sequences
3160:
3123:
3116:
3114:
3113:
3108:
3021:
3019:
3018:
3013:
3011:
3007:
3006:
3005:
2993:
2992:
2974:
2962:
2958:
2957:
2956:
2944:
2943:
2915:
2913:
2912:
2907:
2905:
2901:
2900:
2884:
2883:
2867:
2865:
2864:
2859:
2857:
2853:
2852:
2836:
2835:
2733:
2731:
2730:
2725:
2723:
2715:
2689:
2687:
2686:
2681:
2676:
2672:
2671:
2670:
2660:
2659:
2623:
2619:
2618:
2617:
2601:
2600:
2566:
2564:
2563:
2558:
2542:
2540:
2539:
2534:
2532:
2531:
2513:
2511:
2510:
2505:
2500:
2496:
2495:
2494:
2484:
2483:
2447:
2443:
2442:
2441:
2425:
2424:
2390:
2388:
2387:
2382:
2370:
2368:
2367:
2362:
2360:
2359:
2340:
2338:
2337:
2332:
2317:
2315:
2314:
2309:
2301:
2300:
2288:
2287:
2271:
2269:
2268:
2263:
2251:
2248:
2191:
2189:
2188:
2183:
2156:
2154:
2153:
2148:
2120:
2118:
2117:
2112:
2095:
2094:
2078:
2076:
2075:
2070:
2054:
2052:
2051:
2046:
2032:
2030:
2029:
2024:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1879:
1877:
1876:
1871:
1859:
1855:
1851:
1849:
1848:
1843:
1816:
1814:
1813:
1808:
1792:
1790:
1789:
1784:
1770:is also closed.
1769:
1767:
1766:
1761:
1749:
1747:
1746:
1741:
1723:
1721:
1720:
1715:
1691:
1689:
1688:
1683:
1671:
1669:
1668:
1663:
1645:
1643:
1642:
1637:
1610:
1604:
1602:
1601:
1596:
1547:
1545:
1544:
1539:
1509:
1507:
1506:
1501:
1489:
1487:
1486:
1481:
1479:
1478:
1458:
1456:
1455:
1450:
1448:
1447:
1431:
1429:
1428:
1423:
1411:
1409:
1408:
1403:
1401:
1392:
1382:
1380:
1379:
1374:
1372:
1368:
1367:
1347:
1345:
1344:
1339:
1337:
1333:
1332:
1312:
1310:
1309:
1304:
1292:
1286:
1284:
1283:
1278:
1247:
1245:
1244:
1239:
1190:
1166:
1154:
1134:
1084:
1076:
1074:
1073:
1068:
1066:
1057:
1051:
1050:
1034:
1024:
1022:
1021:
1016:
1014:
1009:
993:
991:
990:
985:
974:
973:
957:
955:
954:
949:
947:
939:
934:
926:
912:then by solving
911:
909:
908:
903:
888:
886:
885:
880:
875:
873:
872:
860:
855:
850:
849:
840:
835:
834:
812:
810:
809:
804:
796:
795:
753:
751:
750:
745:
740:
739:
723:
721:
720:
715:
697:
695:
694:
689:
678:
674:
673:
647:
645:
644:
639:
624:
622:
621:
616:
614:
613:
601:
600:
578:
576:
575:
570:
568:
567:
543:
541:
540:
535:
510:
508:
507:
502:
490:
488:
487:
482:
470:
468:
467:
462:
448:
446:
445:
440:
425:
423:
422:
417:
395:
393:
392:
387:
356:
354:
353:
348:
334:
333:
321:
320:
299:
297:
296:
291:
264:
262:
261:
256:
241:
239:
238:
233:
208:
206:
205:
200:
173:
171:
170:
165:
157:
156:
144:
143:
131:
130:
92:
90:
89:
84:
82:
77:
66:rational numbers
60:
57:that is also in
52:
36:
21:Karoubi envelope
3759:
3758:
3754:
3753:
3752:
3750:
3749:
3748:
3734:Metric geometry
3724:
3723:
3717:
3675:Kreyszig, Erwin
3668:
3654:Kelley, John L.
3649:
3644:
3643:
3638:
3634:
3619:
3605:
3601:
3594:
3577:
3573:
3568:
3553:
3538:
3529:
3516:
3491:
3488:
3487:
3468:
3465:
3464:
3406:
3403:
3402:
3386:
3383:
3382:
3366:
3363:
3362:
3319:
3316:
3315:
3299:
3296:
3295:
3294:via the metric
3279:
3276:
3275:
3259:
3256:
3255:
3239:
3236:
3235:
3220:
3214:
3158:
3149:and not of the
3143:
3119:
3099:
3096:
3095:
3001:
2997:
2988:
2984:
2983:
2979:
2970:
2952:
2948:
2939:
2935:
2934:
2930:
2925:
2922:
2921:
2896:
2892:
2888:
2879:
2875:
2873:
2870:
2869:
2848:
2844:
2840:
2831:
2827:
2825:
2822:
2821:
2714:
2712:
2709:
2708:
2697:
2692:
2666:
2662:
2655:
2648:
2643:
2639:
2613:
2609:
2596:
2589:
2584:
2580:
2572:
2569:
2568:
2552:
2549:
2548:
2527:
2523:
2521:
2518:
2517:
2490:
2486:
2479:
2472:
2467:
2463:
2437:
2433:
2420:
2413:
2408:
2404:
2396:
2393:
2392:
2376:
2373:
2372:
2355:
2351:
2349:
2346:
2345:
2323:
2320:
2319:
2296:
2292:
2283:
2279:
2277:
2274:
2273:
2257:
2254:
2253:
2249:
2246:
2203:. That is, the
2162:
2159:
2158:
2130:
2127:
2126:
2090:
2086:
2084:
2081:
2080:
2064:
2061:
2060:
2040:
2037:
2036:
1943:
1940:
1939:
1920:
1917:
1916:
1885:
1882:
1881:
1865:
1862:
1861:
1857:
1853:
1822:
1819:
1818:
1802:
1799:
1798:
1778:
1775:
1774:
1755:
1752:
1751:
1729:
1726:
1725:
1697:
1694:
1693:
1677:
1674:
1673:
1651:
1648:
1647:
1619:
1616:
1615:
1606:
1590:
1587:
1586:
1579:totally bounded
1567:
1530:
1527:
1526:
1495:
1492:
1491:
1474:
1470:
1468:
1465:
1464:
1443:
1439:
1437:
1434:
1433:
1417:
1414:
1413:
1390:
1388:
1385:
1384:
1363:
1359:
1355:
1353:
1350:
1349:
1328:
1324:
1320:
1318:
1315:
1314:
1298:
1295:
1294:
1288:
1272:
1269:
1268:
1230:
1227:
1226:
1214:
1180:
1156:
1144:
1124:
1103:Euclidean space
1099:complex numbers
1082:
1079:closed interval
1055:
1046:
1042:
1040:
1037:
1036:
1032:
1008:
1006:
1003:
1002:
969:
965:
963:
960:
959:
938:
925:
917:
914:
913:
894:
891:
890:
868:
864:
859:
845:
841:
839:
824:
820:
818:
815:
814:
791:
787:
785:
782:
781:
762:
735:
731:
729:
726:
725:
703:
700:
699:
669:
665:
661:
653:
650:
649:
630:
627:
626:
609:
605:
590:
586:
584:
581:
580:
563:
559:
557:
554:
553:
526:
523:
522:
496:
493:
492:
476:
473:
472:
456:
453:
452:
431:
428:
427:
411:
408:
407:
369:
366:
365:
364:A metric space
329:
325:
316:
312:
304:
301:
300:
270:
267:
266:
250:
247:
246:
221:
218:
217:
182:
179:
178:
152:
148:
139:
135:
126:
122:
120:
117:
116:
108:Cauchy sequence
105:
76:
74:
71:
70:
58:
50:
47:Cauchy sequence
34:
24:
17:
16:Metric geometry
12:
11:
5:
3757:
3747:
3746:
3744:Uniform spaces
3741:
3736:
3722:
3721:
3715:
3702:
3689:
3672:
3666:
3648:
3645:
3642:
3641:
3632:
3617:
3599:
3592:
3570:
3569:
3567:
3564:
3563:
3562:
3556:
3547:
3541:
3532:
3523:
3515:
3512:
3495:
3475:
3472:
3425:
3422:
3419:
3416:
3413:
3410:
3390:
3370:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3303:
3283:
3263:
3243:
3216:Main article:
3213:
3210:
3168:one considers
3142:
3139:
3106:
3103:
3072:total ordering
3048:converging to
3010:
3004:
3000:
2996:
2991:
2987:
2982:
2978:
2973:
2969:
2965:
2961:
2955:
2951:
2947:
2942:
2938:
2933:
2929:
2904:
2899:
2895:
2891:
2887:
2882:
2878:
2856:
2851:
2847:
2843:
2839:
2834:
2830:
2786:is determined
2740:dense subspace
2721:
2718:
2696:
2693:
2691:
2690:
2679:
2675:
2669:
2665:
2658:
2654:
2651:
2647:
2642:
2638:
2635:
2632:
2629:
2626:
2622:
2616:
2612:
2608:
2605:
2599:
2595:
2592:
2588:
2583:
2579:
2576:
2556:
2530:
2526:
2514:
2503:
2499:
2493:
2489:
2482:
2478:
2475:
2471:
2466:
2462:
2459:
2456:
2453:
2450:
2446:
2440:
2436:
2432:
2429:
2423:
2419:
2416:
2412:
2407:
2403:
2400:
2380:
2358:
2354:
2330:
2327:
2307:
2304:
2299:
2295:
2291:
2286:
2282:
2261:
2240:
2222:states that a
2181:
2178:
2175:
2172:
2169:
2166:
2146:
2143:
2140:
2137:
2134:
2110:
2107:
2104:
2101:
2098:
2093:
2089:
2068:
2044:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1924:
1904:
1901:
1898:
1895:
1892:
1889:
1869:
1841:
1838:
1835:
1832:
1829:
1826:
1806:
1782:
1759:
1739:
1736:
1733:
1713:
1710:
1707:
1704:
1701:
1681:
1661:
1658:
1655:
1635:
1632:
1629:
1626:
1623:
1594:
1575:if and only if
1566:
1563:
1537:
1534:
1524:discrete space
1499:
1477:
1473:
1446:
1442:
1421:
1398:
1395:
1371:
1366:
1362:
1358:
1336:
1331:
1327:
1323:
1302:
1276:
1237:
1234:
1210:
1123:. The space C
1110:usual distance
1063:
1060:
1054:
1049:
1045:
1012:
983:
980:
977:
972:
968:
945:
942:
937:
932:
929:
924:
921:
901:
898:
878:
871:
867:
863:
858:
853:
848:
844:
838:
833:
830:
827:
823:
802:
799:
794:
790:
774:absolute value
761:
758:
757:
756:
755:
754:
743:
738:
734:
713:
710:
707:
687:
684:
681:
677:
672:
668:
664:
660:
657:
637:
634:
612:
608:
604:
599:
596:
593:
589:
566:
562:
533:
530:
519:closed subsets
512:
500:
480:
460:
449:
438:
435:
415:
385:
382:
379:
376:
373:
360:Complete space
346:
343:
340:
337:
332:
328:
324:
319:
315:
311:
308:
289:
286:
283:
280:
277:
274:
254:
231:
228:
225:
198:
195:
192:
189:
186:
163:
160:
155:
151:
147:
142:
138:
134:
129:
125:
104:
101:
80:
15:
9:
6:
4:
3:
2:
3756:
3745:
3742:
3740:
3737:
3735:
3732:
3731:
3729:
3718:
3716:0-19-851485-9
3712:
3708:
3703:
3701:
3700:0-387-94001-4
3697:
3693:
3690:
3688:
3687:0-471-03729-X
3684:
3680:
3676:
3673:
3669:
3667:0-387-90125-6
3663:
3659:
3655:
3651:
3650:
3636:
3628:
3624:
3620:
3618:981-238-067-1
3614:
3610:
3603:
3595:
3589:
3585:
3581:
3575:
3571:
3560:
3557:
3551:
3548:
3545:
3542:
3536:
3533:
3527:
3524:
3521:
3518:
3517:
3511:
3509:
3508:Cauchy spaces
3493:
3473:
3470:
3462:
3458:
3457:
3452:
3447:
3445:
3441:
3440:uniform space
3436:
3423:
3420:
3417:
3414:
3411:
3408:
3388:
3368:
3361:
3345:
3342:
3339:
3333:
3330:
3327:
3321:
3301:
3281:
3261:
3241:
3233:
3229:
3225:
3219:
3209:
3207:
3203:
3198:
3196:
3192:
3188:
3184:
3179:
3177:
3173:
3172:
3167:
3162:
3156:
3152:
3148:
3138:
3136:
3135:Hilbert space
3132:
3127:
3125:
3124:-adic numbers
3122:
3104:
3101:
3092:
3090:
3085:
3081:
3077:
3073:
3069:
3065:
3061:
3057:
3055:
3051:
3047:
3043:
3039:
3035:
3031:
3027:
3022:
3008:
3002:
2998:
2994:
2989:
2985:
2980:
2976:
2971:
2963:
2959:
2953:
2949:
2945:
2940:
2936:
2931:
2927:
2919:
2902:
2897:
2893:
2889:
2885:
2880:
2876:
2854:
2849:
2845:
2841:
2837:
2832:
2828:
2819:
2815:
2811:
2806:
2804:
2800:
2796:
2792:
2789:
2785:
2782:. The space
2781:
2778:that extends
2777:
2773:
2769:
2765:
2761:
2757:
2753:
2749:
2745:
2741:
2737:
2716:
2706:
2702:
2677:
2673:
2667:
2663:
2652:
2649:
2645:
2640:
2636:
2633:
2630:
2627:
2624:
2620:
2614:
2610:
2606:
2603:
2593:
2590:
2586:
2581:
2577:
2574:
2554:
2546:
2528:
2524:
2515:
2501:
2497:
2491:
2487:
2476:
2473:
2469:
2464:
2460:
2457:
2454:
2451:
2448:
2444:
2438:
2434:
2430:
2427:
2417:
2414:
2410:
2405:
2401:
2398:
2378:
2371:is closed in
2356:
2352:
2343:
2342:
2341:
2328:
2325:
2305:
2302:
2297:
2293:
2289:
2284:
2280:
2259:
2239:
2237:
2233:
2229:
2225:
2221:
2216:
2214:
2210:
2209:nowhere dense
2206:
2202:
2198:
2193:
2176:
2173:
2170:
2164:
2144:
2138:
2135:
2132:
2124:
2105:
2102:
2099:
2091:
2087:
2066:
2058:
2042:
2033:
2017:
2014:
2011:
2008:
1999:
1993:
1990:
1984:
1978:
1972:
1963:
1957:
1954:
1951:
1945:
1938:
1937:supremum norm
1922:
1899:
1896:
1893:
1887:
1867:
1836:
1833:
1830:
1824:
1804:
1796:
1780:
1771:
1757:
1737:
1734:
1731:
1708:
1705:
1702:
1679:
1659:
1656:
1653:
1630:
1627:
1624:
1612:
1609:
1592:
1584:
1580:
1576:
1572:
1565:Some theorems
1562:
1560:
1556:
1552:
1548:
1535:
1532:
1525:
1521:
1517:
1513:
1497:
1475:
1471:
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1440:
1419:
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1369:
1364:
1360:
1356:
1334:
1329:
1325:
1321:
1300:
1291:
1274:
1265:
1263:
1259:
1255:
1251:
1235:
1232:
1225:
1221:
1220:-adic numbers
1219:
1213:
1209:
1204:
1202:
1198:
1194:
1193:Fréchet space
1188:
1184:
1178:
1174:
1170:
1164:
1160:
1152:
1148:
1142:
1141:supremum norm
1138:
1132:
1128:
1122:
1121:Banach spaces
1118:
1115:
1111:
1107:
1104:
1100:
1096:
1092:
1087:
1085:
1083:[0,1]
1080:
1061:
1058:
1052:
1047:
1043:
1031:
1030:open interval
1026:
1010:
1001:
997:
981:
978:
975:
970:
966:
943:
940:
935:
930:
927:
922:
919:
899:
896:
876:
869:
865:
861:
856:
851:
846:
842:
836:
831:
828:
825:
821:
800:
797:
792:
788:
779:
775:
772:given by the
771:
767:
741:
736:
732:
711:
708:
705:
685:
682:
675:
670:
666:
662:
658:
655:
635:
632:
610:
606:
602:
597:
594:
591:
587:
564:
560:
551:
547:
531:
528:
520:
517:
513:
498:
478:
471:converges in
458:
450:
436:
433:
413:
405:
404:
403:
402:
401:
399:
380:
377:
374:
362:
361:
357:
344:
341:
338:
330:
326:
322:
317:
313:
306:
287:
284:
281:
278:
275:
272:
252:
245:
229:
226:
223:
216:
212:
193:
190:
187:
177:
161:
158:
153:
149:
145:
140:
136:
132:
127:
123:
115:
110:
109:
100:
98:
93:
78:
67:
62:
56:
49:of points in
48:
44:
40:
33:
29:
22:
3706:
3678:
3660:. Springer.
3657:
3635:
3608:
3602:
3583:
3574:
3520:Cauchy space
3454:
3450:
3448:
3437:
3221:
3206:Polish space
3199:
3190:
3182:
3180:
3169:
3163:
3155:homeomorphic
3150:
3146:
3144:
3128:
3120:
3094:For a prime
3093:
3079:
3058:
3053:
3049:
3045:
3041:
3037:
3033:
3026:pseudometric
3023:
2917:
2817:
2809:
2807:
2802:
2798:
2794:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2751:
2747:
2735:
2704:
2700:
2698:
2247:(C. Ursescu)
2241:
2217:
2194:
2034:
1772:
1613:
1607:
1568:
1549:
1512:homeomorphic
1289:
1266:
1261:
1257:
1253:
1249:
1224:prime number
1217:
1211:
1207:
1205:
1186:
1182:
1162:
1158:
1150:
1146:
1130:
1126:
1105:
1094:
1090:
1088:
1027:
996:real numbers
958:necessarily
765:
763:
550:intersection
397:
363:
359:
358:
210:
176:metric space
111:
107:
106:
96:
63:
43:Cauchy space
42:
38:
32:metric space
25:
3692:Lang, Serge
3442:, where an
3076:isomorphism
2228:fixed point
2201:Baire space
1108:, with the
215:real number
45:) if every
3728:Categories
3647:References
3459:or Cauchy
3358:but by an
3078:). It is
2799:completion
2695:Completion
2123:continuous
1260:completes
1206:The space
1089:The space
778:difference
764:The space
625:for every
209:is called
103:Definition
97:completion
37:is called
3627:285163112
3451:sequences
3444:entourage
3418:∈
3412:−
3343:ε
3282:ε
3202:separable
2954:∙
2941:∙
2881:∙
2833:∙
2720:¯
2653:∈
2646:⋂
2637:
2631:
2607:
2594:∈
2587:⋂
2578:
2477:∈
2470:⋃
2461:
2455:
2431:
2418:∈
2411:⋃
2402:
2306:…
2142:→
2015:∈
1964:≡
1935:with the
1735:⊆
1657:⊆
1520:countable
1252:with the
709:∈
680:→
659:
603:⊆
546:diameters
516:non-empty
162:…
3739:Topology
3656:(1975).
3582:(1975).
3514:See also
3166:topology
3151:topology
3068:subfield
2791:isometry
2516:If each
2344:If each
2213:interior
1461:distinct
1203:metric.
1173:topology
760:Examples
398:complete
114:sequence
39:complete
3461:filters
3080:defined
2754:is any
2243:Theorem
1516:product
1514:to the
776:of the
244:integer
3713:
3698:
3685:
3664:
3625:
3615:
3590:
3222:Since
3147:metric
3060:Cantor
2245:
1569:Every
1412:where
1383:to be
770:metric
211:Cauchy
53:has a
41:(or a
3566:Notes
3486:then
3159:(0,1)
3064:field
2788:up to
2770:from
2758:from
2746:: if
2738:as a
2567:then
2391:then
2232:prove
2205:union
2055:is a
1856:from
1793:is a
1518:of a
1463:from
1133:]
1125:[
1033:(0,1)
552:: if
544:with
174:in a
55:limit
3711:ISBN
3696:ISBN
3683:ISBN
3662:ISBN
3623:OCLC
3613:ISBN
3588:ISBN
3456:nets
3340:<
3254:and
3117:the
2868:and
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2252:Let
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2195:The
2059:and
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1028:The
813:and
656:diam
648:and
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1967:sup
1860:to
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