3675:
1474:
117:
was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof. He used this proof in his 1852 lectures, which were published only in 1904. Later
3405:
911:
1189:
3027:
2842:
2295:
2231:
3911:
649:
is compact, take a finite subcover of this cover. This subcover is the finite union of balls of radius 1. Consider all pairs of centers of these (finitely many) balls (of radius 1) and let
3780:
3213:
105:
The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of
3368:
1060:
518:
2196:
607:
1401:
456:
2488:
2531:
2009:
have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
1112:
2786:
2746:
2260:
1288:
3181:
753:
3708:
2713:
2087:
2959:
2933:
2680:
544:
2903:
2592:
2462:
2042:
1972:
1838:
1750:
760:
721:
694:
483:
959:
3728:
3322:
3302:
3282:
3262:
3242:
3161:
3141:
3117:
3097:
2871:
2636:
2616:
2560:
2508:
2430:
2152:
2124:
2062:
1996:
1908:
1885:
1862:
1795:
1773:
1718:
933:
667:
647:
627:
564:
427:
3950:
3400:
3670:{\displaystyle d(x,y)=\sum _{k=0}^{\infty }{\frac {1}{2^{k}}}\cdot {\frac {\max _{t\in }|x^{(k)}(t)-y^{(k)}(t)|}{1+\max _{t\in }|x^{(k)}(t)-y^{(k)}(t)|}}}
3975:
2647:
Diedonnné, Jean (1969): Foundations of Modern
Analysis, Volume 1, enlarged and corrected printing. Academic Press, New York, London, p. 58
3917:
1927:, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. These spaces are said to have the
3874:
2005:(or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional
3965:
3828:
134:
in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to
2314:
3893:
114:
3796:
P. Dugac (1989). "Sur la correspondance de Borel et le théorème de
Dirichlet–Heine–Weierstrass–Borel–Schoenflies–Lebesgue".
1013:
3980:
3938:
2964:
2339:
3933:
192:
3928:
3809:
2791:
2265:
2201:
3733:
1133:
3970:
3186:
1238:
2104:
1924:
1364:
itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section
3327:
3910:
Ivan Kenig, Dr. Prof. Hans-Christian Graf v. Botthmer, Dmitrij
Tiessen, Andreas Timm, Viktor Wittman (2004).
488:
2165:
569:
2092:
432:
2467:
2513:
2370:
127:
110:
2751:
2393:
2374:
2366:
2348:
143:
106:
2718:
2236:
3889:
3870:
3824:
3166:
1802:
726:
3680:
2685:
2072:
1084:
3851:
2938:
2912:
2659:
2358:
2159:
2158:
have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional
523:
123:
75:
2876:
2565:
2435:
2015:
1945:
1811:
1723:
699:
672:
461:
2595:
2002:
1521:
1469:{\displaystyle T_{0}\supset T_{1}\supset T_{2}\supset \ldots \supset T_{k}\supset \ldots }
390:
57:
938:
2362:
2337:
Raman-Sundström, Manya (August–September 2015). "A Pedagogical
History of Compactness".
3945:
3817:
3782:) has the Heine–Borel property as a topological vector space but not as a metric space.
3713:
3307:
3287:
3267:
3247:
3227:
3146:
3126:
3102:
3082:
2856:
2621:
2601:
2545:
2493:
2415:
2137:
2109:
2065:
2047:
1981:
1893:
1870:
1847:
1780:
1758:
1703:
918:
652:
632:
612:
549:
412:
131:
43:
3856:
3839:
3373:
2001:
Many metric spaces fail to have the Heine–Borel property, such as the metric space of
3959:
2378:
2298:
2069:
135:
71:
31:
1324:
that does not admit any finite subcover. Through bisection of each of the sides of
3264:
this definition is not equivalent to the definition of the Heine–Borel property of
2302:
2155:
2006:
1940:
1920:
119:
39:
17:
385:
The proof above applies with almost no change to showing that any compact subset
172:
93:
3951:"An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof"
906:{\displaystyle d(p,q)\leq d(p,C_{p})+d(C_{p},C_{q})+d(C_{q},q)\leq 1+M+1=M+2.}
89:
27:
Subset of
Euclidean space is compact if and only if it is closed and bounded
139:
2392:
Sundström, Manya Raman (2010). "A pedagogical history of compactness".
50:
2398:
1392:. Continuing in like manner yields a decreasing sequence of nested
2909:
if it is contained in a ball of a finite radius, i.e. there exists
2464:
is called precompact (or sometimes "totally bounded"), if for any
2353:
723:
are the centers (respectively) of unit balls containing arbitrary
3304:
as a metric space is different from the notion of bounded set in
1342:-boxes, each of which has diameter equal to half the diameter of
2044:
has a Heine–Borel metric which is Cauchy locally identical to
1388:, at least one of which must require an infinite subcover of
362:
discussed previously, and thus cannot be an open subcover of
3746:
3334:
3224:
In the case when the topology of a topological vector space
1919:
The Heine–Borel theorem does not hold as stated for general
3063:
3061:
3059:
669:
be the maximum of the distances between them. Then if
1313:
is not compact. Then there exists an infinite open cover
1191:
that is a finite subcollection of the original collection
113:
on a closed and bounded interval is uniformly continuous.
1697:
In general metric spaces, we have the following theorem:
327:, chosen small enough to not intersect some neighborhood
211:. Indeed, the intersection of the finite family of sets
3324:
as a topological vector space. For instance, the space
1296:> 0. By the lemma above, it is enough to show that
3284:
as a metric space, since the notion of bounded set in
2682:
is said to be a cluster value of an infinite sequence
1200:. It is thus possible to extract from any open cover
3736:
3716:
3683:
3408:
3376:
3330:
3310:
3290:
3270:
3250:
3230:
3189:
3169:
3149:
3129:
3105:
3085:
2967:
2941:
2915:
2879:
2859:
2794:
2754:
2721:
2688:
2662:
2624:
2604:
2568:
2548:
2516:
2496:
2470:
2438:
2418:
2268:
2239:
2204:
2168:
2140:
2112:
2075:
2050:
2018:
1984:
1948:
1896:
1873:
1850:
1814:
1783:
1761:
1726:
1706:
1404:
1241:
1136:
1087:
1016:
941:
921:
763:
729:
702:
675:
655:
635:
615:
572:
552:
526:
491:
464:
435:
415:
296:
is compact but not closed, then it has a limit point
3840:"Construction metrics with the Heine-Borel property"
3022:{\displaystyle B\subseteq \{x\in X:\ d(x,a)\leq r\}}
1217:
If a set is closed and bounded, then it is compact.
965:
Lemma: A closed subset of a compact set is compact.
3816:
3774:
3722:
3702:
3669:
3394:
3362:
3316:
3296:
3276:
3256:
3236:
3207:
3175:
3155:
3135:
3111:
3091:
3021:
2953:
2927:
2897:
2865:
2836:
2780:
2740:
2707:
2674:
2630:
2610:
2586:
2554:
2525:
2502:
2482:
2456:
2424:
2289:
2254:
2225:
2190:
2146:
2118:
2081:
2056:
2036:
1990:
1966:
1902:
1879:
1856:
1832:
1789:
1767:
1744:
1712:
1468:
1282:
1183:
1106:
1054:
953:
927:
905:
747:
715:
688:
661:
641:
621:
601:
558:
538:
512:
477:
450:
421:
3050:
1840:, the following three conditions are equivalent:
3574:
3475:
2802:
3916:. Hannover: Leibniz Universität. Archived from
3038:
2837:{\displaystyle x=\lim _{k\to \infty }x_{n_{k}}}
2290:{\displaystyle \Omega \subset \mathbb {C} ^{n}}
2226:{\displaystyle \Omega \subset \mathbb {R} ^{n}}
1752:, the following two statements are equivalent:
63:, the following two statements are equivalent:
1229:is bounded, then it can be enclosed within an
3775:{\displaystyle x\in {\mathcal {C}}^{\infty }}
1649:, but then the infinite number of members of
1499:tends to infinity. Let us define a sequence (
1184:{\displaystyle C_{K}'=C_{T}'\setminus \{U\},}
520:. Then the set of all such balls centered at
8:
3867:Theorems and Problems in Functional Analysis
3016:
2974:
1175:
1169:
1049:
1043:
155:If a set is compact, then it must be closed.
2332:
2330:
146:(1900) generalized it to arbitrary covers.
3208:{\displaystyle B\subseteq \lambda \cdot U}
2099:In the theory of topological vector spaces
1678:is closed and a subset of the compact set
1349:. Then at least one of the 2 sections of
3855:
3751:
3745:
3744:
3735:
3715:
3688:
3682:
3659:
3638:
3610:
3601:
3577:
3560:
3539:
3511:
3502:
3478:
3471:
3460:
3451:
3445:
3434:
3407:
3375:
3339:
3333:
3332:
3329:
3309:
3289:
3269:
3249:
3229:
3188:
3168:
3148:
3128:
3104:
3084:
2966:
2940:
2914:
2878:
2858:
2826:
2821:
2805:
2793:
2767:
2762:
2753:
2726:
2720:
2696:
2687:
2661:
2623:
2603:
2567:
2547:
2515:
2495:
2469:
2437:
2417:
2397:
2352:
2281:
2277:
2276:
2267:
2238:
2217:
2213:
2212:
2203:
2173:
2167:
2139:
2111:
2074:
2049:
2017:
1983:
1947:
1895:
1872:
1849:
1813:
1782:
1760:
1725:
1705:
1693:Generalization of the Heine-Borel theorem
1454:
1435:
1422:
1409:
1403:
1274:
1246:
1240:
1157:
1141:
1135:
1092:
1086:
1034:
1021:
1015:
940:
920:
858:
836:
823:
801:
762:
728:
707:
701:
680:
674:
654:
634:
614:
593:
577:
571:
551:
525:
504:
499:
490:
469:
463:
442:
437:
434:
414:
3921:(avi • mp4 • mov • swf • streamed video)
3865:Kirillov, A.A.; Gvishiani, A.D. (1982).
3363:{\displaystyle {\mathcal {C}}^{\infty }}
2262:of holomorphic functions on an open set
1381:can be bisected, yielding 2 sections of
405:If a set is compact, then it is bounded.
3067:
2326:
2305:have the Heine–Borel property as well.
1689:is also compact (see the lemma above).
1166:
366:. This contradicts the compactness of
2297:. More generally, any quasi-complete
1306:Assume, by way of contradiction, that
183:of open sets, such that each open set
3815:Jeffreys, H.; Jeffreys, B.S. (1988).
1356:must require an infinite subcover of
1055:{\displaystyle C_{T}=C_{K}\cup \{U\}}
513:{\displaystyle x\in \mathbf {R} ^{n}}
7:
3370:of smooth functions on the interval
2191:{\displaystyle C^{\infty }(\Omega )}
2154:is compact. No infinite-dimensional
1524:, so it must converge to some limit
973:be a closed subset of a compact set
2371:10.4169/amer.math.monthly.122.7.619
2363:10.4169/amer.math.monthly.122.7.619
2198:of smooth functions on an open set
602:{\displaystyle \cup _{x\in S}U_{x}}
311:consisting of an open neighborhood
167:. Observe first the following: if
109:and the theorem stating that every
3838:Williamson, R.; Janos, L. (1987).
3752:
3446:
3340:
2812:
2301:has the Heine–Borel property. All
2269:
2246:
2205:
2182:
2174:
351:, but any finite subcollection of
25:
3929:"Borel-Lebesgue covering theorem"
3857:10.1090/S0002-9939-1987-0891165-X
3123:if for each neighborhood of zero
2162:do have, for instance, the space
1114:that also covers the smaller set
3976:Properties of topological spaces
2748:, if there exists a subsequence
2134:) if each closed bounded set in
1805:, theorem 3.16.1, which states:
1801:The above follows directly from
755:, the triangle inequality says:
500:
451:{\displaystyle \mathbf {R} ^{n}}
438:
3946:Mathworld "Heine-Borel Theorem"
3819:Methods of Mathematical Physics
3730:-th derivative of the function
485:a ball of radius 1 centered at
370:. Hence, every limit point of
3888:. Holt, Rinehart and Winston.
3823:. Cambridge University Press.
3769:
3757:
3695:
3689:
3660:
3656:
3650:
3645:
3639:
3628:
3622:
3617:
3611:
3602:
3596:
3584:
3561:
3557:
3551:
3546:
3540:
3529:
3523:
3518:
3512:
3503:
3497:
3485:
3424:
3412:
3389:
3377:
3357:
3345:
3099:in a topological vector space
3007:
2995:
2892:
2880:
2809:
2775:
2755:
2702:
2689:
2581:
2569:
2490:there is a finite covering of
2483:{\displaystyle \epsilon >0}
2451:
2439:
2249:
2243:
2185:
2179:
2031:
2019:
1978:if each closed bounded set in
1961:
1949:
1935:In the theory of metric spaces
1827:
1815:
1739:
1727:
1548:) is eventually always inside
1271:
1255:
1122:does not contain any point of
870:
851:
842:
816:
807:
788:
779:
767:
138:covers. Pierre Cousin (1895),
115:Peter Gustav Lejeune Dirichlet
1:
3051:Kirillov & Gvishiani 1982
2526:{\displaystyle <\epsilon }
2340:American Mathematical Monthly
1887:has at least a cluster value;
1867:(b) any infinite sequence in
1660:can be replaced by just one:
260:is not covered by the family
179:, then any finite collection
3869:. Springer-Verlag New York.
3244:is generated by some metric
2618:is convergent to a point in
2130:(R.E. Edwards uses the term
1338:can be broken up into 2 sub
546:is clearly an open cover of
3934:Encyclopedia of Mathematics
3039:Williamson & Janos 1987
2781:{\displaystyle (x_{n_{k}})}
2594:is called complete, if any
2315:Bolzano–Weierstrass theorem
1910:is precompact and complete.
1797:is precompact and complete.
3997:
2741:{\displaystyle x_{n}\in X}
2255:{\displaystyle H(\Omega )}
1584:, then it has some member
1283:{\displaystyle T_{0}=^{n}}
3966:Theorems in real analysis
1925:topological vector spaces
1478:where the side length of
304:. Consider a collection
207:, fails to be a cover of
130:used similar techniques.
3176:{\displaystyle \lambda }
2105:topological vector space
1537:is closed, and for each
748:{\displaystyle p,q\in S}
281:and hence disjoint from
3913:The Heine–Borel Theorem
3703:{\displaystyle x^{(k)}}
2708:{\displaystyle (x_{n})}
2132:boundedly compact space
2082:{\displaystyle \sigma }
1374:Likewise, the sides of
1107:{\displaystyle C_{T}',}
3884:Edwards, R.E. (1965).
3776:
3724:
3704:
3671:
3450:
3396:
3364:
3318:
3298:
3278:
3258:
3238:
3209:
3177:
3163:there exists a scalar
3157:
3137:
3113:
3093:
3023:
2955:
2954:{\displaystyle r>0}
2929:
2928:{\displaystyle a\in X}
2899:
2867:
2838:
2782:
2742:
2709:
2676:
2675:{\displaystyle x\in X}
2632:
2612:
2588:
2556:
2527:
2504:
2484:
2458:
2426:
2291:
2256:
2227:
2192:
2148:
2120:
2083:
2058:
2038:
1992:
1968:
1904:
1881:
1858:
1834:
1791:
1769:
1746:
1714:
1495:, which tends to 0 as
1470:
1284:
1185:
1130:is already covered by
1108:
1081:has a finite subcover
1056:
955:
929:
907:
749:
717:
690:
663:
643:
623:
603:
560:
540:
539:{\displaystyle x\in S}
514:
479:
452:
423:
191:is disjoint from some
101:History and motivation
74:, that is, every open
3777:
3725:
3705:
3672:
3430:
3397:
3365:
3319:
3299:
3279:
3259:
3239:
3210:
3178:
3158:
3138:
3114:
3094:
3024:
2956:
2930:
2900:
2898:{\displaystyle (X,d)}
2868:
2839:
2783:
2743:
2710:
2677:
2633:
2613:
2589:
2587:{\displaystyle (X,d)}
2557:
2528:
2505:
2485:
2459:
2457:{\displaystyle (X,d)}
2427:
2292:
2257:
2228:
2193:
2149:
2121:
2084:
2064:if and only if it is
2059:
2039:
2037:{\displaystyle (X,d)}
1993:
1969:
1967:{\displaystyle (X,d)}
1905:
1882:
1859:
1835:
1833:{\displaystyle (X,d)}
1792:
1770:
1747:
1745:{\displaystyle (X,d)}
1715:
1604:is open, there is an
1471:
1285:
1186:
1109:
1057:
956:
930:
908:
750:
718:
716:{\displaystyle C_{q}}
691:
689:{\displaystyle C_{p}}
664:
644:
624:
604:
561:
541:
515:
480:
478:{\displaystyle U_{x}}
453:
424:
244:must contain a point
82:has a finite subcover
3981:Compactness theorems
3810:Heine-Borel Property
3798:Arch. Int. Hist. Sci
3734:
3714:
3681:
3406:
3374:
3328:
3308:
3288:
3268:
3248:
3228:
3187:
3167:
3147:
3127:
3103:
3083:
2965:
2939:
2913:
2877:
2857:
2792:
2752:
2719:
2686:
2660:
2622:
2602:
2566:
2546:
2514:
2510:by sets of diameter
2494:
2468:
2436:
2416:
2266:
2237:
2202:
2166:
2138:
2128:Heine–Borel property
2126:is said to have the
2110:
2073:
2048:
2016:
1982:
1976:Heine–Borel property
1974:is said to have the
1946:
1929:Heine–Borel property
1915:Heine–Borel property
1894:
1871:
1848:
1812:
1781:
1759:
1724:
1704:
1622:. For large enough
1402:
1239:
1134:
1085:
1064:is an open cover of
1014:
990:be an open cover of
939:
919:
761:
727:
700:
673:
653:
633:
613:
570:
550:
524:
489:
462:
433:
429:be a compact set in
413:
347:is an open cover of
236:is a limit point of
3886:Functional analysis
1808:For a metric space
1803:Jean Dieudonné
1674:is compact. Since
1664:, a contradiction.
1520:. This sequence is
1213:a finite subcover.
1165:
1149:
1100:
1008:is an open set and
954:{\displaystyle M+2}
915:So the diameter of
128:Salvatore Pincherle
111:continuous function
36:Heine–Borel theorem
18:Heine-Borel theorem
3772:
3720:
3700:
3667:
3600:
3501:
3392:
3360:
3314:
3294:
3274:
3254:
3234:
3205:
3173:
3153:
3133:
3109:
3089:
3019:
2951:
2925:
2895:
2873:in a metric space
2863:
2834:
2816:
2778:
2738:
2705:
2672:
2628:
2608:
2584:
2562:of a metric space
2552:
2523:
2500:
2480:
2454:
2432:of a metric space
2422:
2287:
2252:
2223:
2188:
2144:
2116:
2079:
2054:
2034:
1988:
1964:
1900:
1877:
1854:
1830:
1787:
1765:
1742:
1720:of a metric space
1710:
1466:
1280:
1181:
1153:
1137:
1104:
1088:
1052:
951:
925:
903:
745:
713:
686:
659:
639:
619:
599:
556:
536:
510:
475:
448:
419:
393:topological space
220:is a neighborhood
107:uniform continuity
3876:978-1-4613-8155-6
3723:{\displaystyle k}
3665:
3573:
3474:
3466:
3317:{\displaystyle X}
3297:{\displaystyle X}
3277:{\displaystyle X}
3257:{\displaystyle d}
3237:{\displaystyle X}
3156:{\displaystyle X}
3136:{\displaystyle U}
3112:{\displaystyle X}
3092:{\displaystyle B}
2991:
2866:{\displaystyle B}
2801:
2631:{\displaystyle S}
2611:{\displaystyle S}
2555:{\displaystyle S}
2503:{\displaystyle S}
2425:{\displaystyle S}
2147:{\displaystyle X}
2119:{\displaystyle X}
2057:{\displaystyle d}
1991:{\displaystyle X}
1903:{\displaystyle X}
1880:{\displaystyle X}
1857:{\displaystyle X}
1790:{\displaystyle S}
1768:{\displaystyle S}
1713:{\displaystyle S}
1506:) such that each
1493:) / 2
1072:is compact, then
928:{\displaystyle S}
662:{\displaystyle M}
642:{\displaystyle S}
622:{\displaystyle S}
559:{\displaystyle S}
422:{\displaystyle S}
285:, which contains
272:is disjoint from
16:(Redirected from
3988:
3971:General topology
3942:
3924:
3922:
3899:
3880:
3861:
3859:
3834:
3822:
3805:
3783:
3781:
3779:
3778:
3773:
3756:
3755:
3750:
3749:
3729:
3727:
3726:
3721:
3709:
3707:
3706:
3701:
3699:
3698:
3676:
3674:
3673:
3668:
3666:
3664:
3663:
3649:
3648:
3621:
3620:
3605:
3599:
3565:
3564:
3550:
3549:
3522:
3521:
3506:
3500:
3472:
3467:
3465:
3464:
3452:
3449:
3444:
3402:with the metric
3401:
3399:
3398:
3395:{\displaystyle }
3393:
3369:
3367:
3366:
3361:
3344:
3343:
3338:
3337:
3323:
3321:
3320:
3315:
3303:
3301:
3300:
3295:
3283:
3281:
3280:
3275:
3263:
3261:
3260:
3255:
3243:
3241:
3240:
3235:
3222:
3216:
3214:
3212:
3211:
3206:
3182:
3180:
3179:
3174:
3162:
3160:
3159:
3154:
3142:
3140:
3139:
3134:
3118:
3116:
3115:
3110:
3098:
3096:
3095:
3090:
3077:
3071:
3065:
3054:
3048:
3042:
3036:
3030:
3028:
3026:
3025:
3020:
2989:
2960:
2958:
2957:
2952:
2934:
2932:
2931:
2926:
2904:
2902:
2901:
2896:
2872:
2870:
2869:
2864:
2851:
2845:
2843:
2841:
2840:
2835:
2833:
2832:
2831:
2830:
2815:
2787:
2785:
2784:
2779:
2774:
2773:
2772:
2771:
2747:
2745:
2744:
2739:
2731:
2730:
2714:
2712:
2711:
2706:
2701:
2700:
2681:
2679:
2678:
2673:
2654:
2648:
2645:
2639:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2593:
2591:
2590:
2585:
2561:
2559:
2558:
2553:
2540:
2534:
2532:
2530:
2529:
2524:
2509:
2507:
2506:
2501:
2489:
2487:
2486:
2481:
2463:
2461:
2460:
2455:
2431:
2429:
2428:
2423:
2410:
2404:
2403:
2401:
2389:
2383:
2382:
2356:
2334:
2296:
2294:
2293:
2288:
2286:
2285:
2280:
2261:
2259:
2258:
2253:
2232:
2230:
2229:
2224:
2222:
2221:
2216:
2197:
2195:
2194:
2189:
2178:
2177:
2153:
2151:
2150:
2145:
2125:
2123:
2122:
2117:
2088:
2086:
2085:
2080:
2063:
2061:
2060:
2055:
2043:
2041:
2040:
2035:
2003:rational numbers
1997:
1995:
1994:
1989:
1973:
1971:
1970:
1965:
1909:
1907:
1906:
1901:
1886:
1884:
1883:
1878:
1863:
1861:
1860:
1855:
1839:
1837:
1836:
1831:
1796:
1794:
1793:
1788:
1774:
1772:
1771:
1766:
1751:
1749:
1748:
1743:
1719:
1717:
1716:
1711:
1653:needed to cover
1648:
1621:
1494:
1475:
1473:
1472:
1467:
1459:
1458:
1440:
1439:
1427:
1426:
1414:
1413:
1289:
1287:
1286:
1281:
1279:
1278:
1251:
1250:
1190:
1188:
1187:
1182:
1161:
1145:
1113:
1111:
1110:
1105:
1096:
1061:
1059:
1058:
1053:
1039:
1038:
1026:
1025:
1007:
960:
958:
957:
952:
934:
932:
931:
926:
912:
910:
909:
904:
863:
862:
841:
840:
828:
827:
806:
805:
754:
752:
751:
746:
722:
720:
719:
714:
712:
711:
695:
693:
692:
687:
685:
684:
668:
666:
665:
660:
648:
646:
645:
640:
628:
626:
625:
620:
609:contains all of
608:
606:
605:
600:
598:
597:
588:
587:
565:
563:
562:
557:
545:
543:
542:
537:
519:
517:
516:
511:
509:
508:
503:
484:
482:
481:
476:
474:
473:
457:
455:
454:
449:
447:
446:
441:
428:
426:
425:
420:
358:has the form of
357:
346:
310:
264:, because every
124:Karl Weierstrass
21:
3996:
3995:
3991:
3990:
3989:
3987:
3986:
3985:
3956:
3955:
3927:
3920:
3909:
3906:
3896:
3883:
3877:
3864:
3837:
3831:
3814:
3795:
3792:
3787:
3786:
3743:
3732:
3731:
3712:
3711:
3684:
3679:
3678:
3634:
3606:
3566:
3535:
3507:
3473:
3456:
3404:
3403:
3372:
3371:
3331:
3326:
3325:
3306:
3305:
3286:
3285:
3266:
3265:
3246:
3245:
3226:
3225:
3223:
3219:
3185:
3184:
3165:
3164:
3145:
3144:
3125:
3124:
3101:
3100:
3081:
3080:
3078:
3074:
3066:
3057:
3049:
3045:
3037:
3033:
2963:
2962:
2937:
2936:
2911:
2910:
2875:
2874:
2855:
2854:
2852:
2848:
2822:
2817:
2790:
2789:
2763:
2758:
2750:
2749:
2722:
2717:
2716:
2715:of elements of
2692:
2684:
2683:
2658:
2657:
2655:
2651:
2646:
2642:
2620:
2619:
2600:
2599:
2596:Cauchy sequence
2564:
2563:
2544:
2543:
2541:
2537:
2512:
2511:
2492:
2491:
2466:
2465:
2434:
2433:
2414:
2413:
2411:
2407:
2391:
2390:
2386:
2336:
2335:
2328:
2323:
2311:
2275:
2264:
2263:
2235:
2234:
2211:
2200:
2199:
2169:
2164:
2163:
2136:
2135:
2108:
2107:
2101:
2093:locally compact
2071:
2070:
2046:
2045:
2014:
2013:
2012:A metric space
1980:
1979:
1944:
1943:
1937:
1917:
1892:
1891:
1869:
1868:
1846:
1845:
1810:
1809:
1779:
1778:
1757:
1756:
1722:
1721:
1702:
1701:
1695:
1684:
1673:
1658:
1635:
1627:
1609:
1583:
1565:
1554:
1547:
1541:the sequence (
1536:
1519:
1512:
1505:
1488:
1486:
1450:
1431:
1418:
1405:
1400:
1399:
1387:
1380:
1370:
1355:
1348:
1337:
1330:
1323:
1312:
1302:
1270:
1242:
1237:
1236:
1208:
1199:
1132:
1131:
1083:
1082:
1080:
1030:
1017:
1012:
1011:
995:
989:
937:
936:
917:
916:
854:
832:
819:
797:
759:
758:
725:
724:
703:
698:
697:
676:
671:
670:
651:
650:
631:
630:
611:
610:
589:
573:
568:
567:
548:
547:
522:
521:
498:
487:
486:
465:
460:
459:
436:
431:
430:
411:
410:
352:
341:
335:
305:
280:
219:
202:
163:be a subset of
152:
103:
58:Euclidean space
28:
23:
22:
15:
12:
11:
5:
3994:
3992:
3984:
3983:
3978:
3973:
3968:
3958:
3957:
3954:
3953:
3948:
3943:
3925:
3923:on 2011-07-19.
3905:
3904:External links
3902:
3901:
3900:
3894:
3881:
3875:
3862:
3850:(3): 567–573.
3835:
3830:978-0521097239
3829:
3812:
3808:BookOfProofs:
3806:
3791:
3788:
3785:
3784:
3771:
3768:
3765:
3762:
3759:
3754:
3748:
3742:
3739:
3719:
3697:
3694:
3691:
3687:
3662:
3658:
3655:
3652:
3647:
3644:
3641:
3637:
3633:
3630:
3627:
3624:
3619:
3616:
3613:
3609:
3604:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3576:
3572:
3569:
3563:
3559:
3556:
3553:
3548:
3545:
3542:
3538:
3534:
3531:
3528:
3525:
3520:
3517:
3514:
3510:
3505:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3477:
3470:
3463:
3459:
3455:
3448:
3443:
3440:
3437:
3433:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3391:
3388:
3385:
3382:
3379:
3359:
3356:
3353:
3350:
3347:
3342:
3336:
3313:
3293:
3273:
3253:
3233:
3217:
3204:
3201:
3198:
3195:
3192:
3172:
3152:
3132:
3119:is said to be
3108:
3088:
3072:
3055:
3043:
3031:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2950:
2947:
2944:
2924:
2921:
2918:
2905:is said to be
2894:
2891:
2888:
2885:
2882:
2862:
2846:
2829:
2825:
2820:
2814:
2811:
2808:
2804:
2800:
2797:
2777:
2770:
2766:
2761:
2757:
2737:
2734:
2729:
2725:
2704:
2699:
2695:
2691:
2671:
2668:
2665:
2649:
2640:
2627:
2607:
2583:
2580:
2577:
2574:
2571:
2551:
2535:
2522:
2519:
2499:
2479:
2476:
2473:
2453:
2450:
2447:
2444:
2441:
2421:
2405:
2384:
2347:(7): 619–635.
2325:
2324:
2322:
2319:
2318:
2317:
2310:
2307:
2284:
2279:
2274:
2271:
2251:
2248:
2245:
2242:
2233:and the space
2220:
2215:
2210:
2207:
2187:
2184:
2181:
2176:
2172:
2160:Fréchet spaces
2143:
2115:
2100:
2097:
2078:
2053:
2033:
2030:
2027:
2024:
2021:
1987:
1963:
1960:
1957:
1954:
1951:
1936:
1933:
1916:
1913:
1912:
1911:
1899:
1888:
1876:
1865:
1853:
1829:
1826:
1823:
1820:
1817:
1799:
1798:
1786:
1776:
1764:
1741:
1738:
1735:
1732:
1729:
1709:
1694:
1691:
1682:
1671:
1656:
1631:
1581:
1563:
1555:, we see that
1552:
1545:
1532:
1517:
1510:
1503:
1482:
1465:
1462:
1457:
1453:
1449:
1446:
1443:
1438:
1434:
1430:
1425:
1421:
1417:
1412:
1408:
1385:
1378:
1368:
1353:
1346:
1335:
1328:
1321:
1310:
1300:
1277:
1273:
1269:
1266:
1263:
1260:
1257:
1254:
1249:
1245:
1204:
1195:
1180:
1177:
1174:
1171:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1103:
1099:
1095:
1091:
1076:
1051:
1048:
1045:
1042:
1037:
1033:
1029:
1024:
1020:
985:
950:
947:
944:
935:is bounded by
924:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
861:
857:
853:
850:
847:
844:
839:
835:
831:
826:
822:
818:
815:
812:
809:
804:
800:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
744:
741:
738:
735:
732:
710:
706:
683:
679:
658:
638:
618:
596:
592:
586:
583:
580:
576:
555:
535:
532:
529:
507:
502:
497:
494:
472:
468:
445:
440:
418:
331:
276:
215:
198:
151:
148:
102:
99:
98:
97:
83:
38:, named after
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3993:
3982:
3979:
3977:
3974:
3972:
3969:
3967:
3964:
3963:
3961:
3952:
3949:
3947:
3944:
3940:
3936:
3935:
3930:
3926:
3919:
3915:
3914:
3908:
3907:
3903:
3897:
3891:
3887:
3882:
3878:
3872:
3868:
3863:
3858:
3853:
3849:
3845:
3841:
3836:
3832:
3826:
3821:
3820:
3813:
3811:
3807:
3803:
3799:
3794:
3793:
3789:
3766:
3763:
3760:
3740:
3737:
3717:
3692:
3685:
3653:
3642:
3635:
3631:
3625:
3614:
3607:
3593:
3590:
3587:
3581:
3578:
3570:
3567:
3554:
3543:
3536:
3532:
3526:
3515:
3508:
3494:
3491:
3488:
3482:
3479:
3468:
3461:
3457:
3453:
3441:
3438:
3435:
3431:
3427:
3421:
3418:
3415:
3409:
3386:
3383:
3380:
3354:
3351:
3348:
3311:
3291:
3271:
3251:
3231:
3221:
3218:
3202:
3199:
3196:
3193:
3190:
3170:
3150:
3130:
3122:
3106:
3086:
3076:
3073:
3069:
3064:
3062:
3060:
3056:
3053:, Theorem 28.
3052:
3047:
3044:
3040:
3035:
3032:
3013:
3010:
3004:
3001:
2998:
2992:
2986:
2983:
2980:
2977:
2971:
2968:
2948:
2945:
2942:
2922:
2919:
2916:
2908:
2889:
2886:
2883:
2860:
2850:
2847:
2827:
2823:
2818:
2806:
2798:
2795:
2768:
2764:
2759:
2735:
2732:
2727:
2723:
2697:
2693:
2669:
2666:
2663:
2653:
2650:
2644:
2641:
2625:
2605:
2597:
2578:
2575:
2572:
2549:
2539:
2536:
2520:
2517:
2497:
2477:
2474:
2471:
2448:
2445:
2442:
2419:
2409:
2406:
2400:
2395:
2388:
2385:
2380:
2376:
2372:
2368:
2364:
2360:
2355:
2350:
2346:
2342:
2341:
2333:
2331:
2327:
2320:
2316:
2313:
2312:
2308:
2306:
2304:
2303:Montel spaces
2300:
2299:nuclear space
2282:
2272:
2240:
2218:
2208:
2170:
2161:
2157:
2156:Banach spaces
2141:
2133:
2129:
2113:
2106:
2098:
2096:
2094:
2090:
2076:
2067:
2051:
2028:
2025:
2022:
2010:
2008:
2007:Banach spaces
2004:
1999:
1985:
1977:
1958:
1955:
1952:
1942:
1934:
1932:
1930:
1926:
1922:
1914:
1897:
1889:
1874:
1866:
1851:
1843:
1842:
1841:
1824:
1821:
1818:
1806:
1804:
1784:
1777:
1762:
1755:
1754:
1753:
1736:
1733:
1730:
1707:
1700:For a subset
1698:
1692:
1690:
1688:
1681:
1677:
1670:
1665:
1663:
1659:
1652:
1647:
1643:
1639:
1634:
1630:
1625:
1620:
1616:
1612:
1607:
1603:
1599:
1595:
1591:
1587:
1580:
1576:
1571:
1569:
1562:
1559: ∈
1558:
1551:
1544:
1540:
1535:
1531:
1528:. Since each
1527:
1523:
1516:
1509:
1502:
1498:
1492:
1485:
1481:
1476:
1463:
1460:
1455:
1451:
1447:
1444:
1441:
1436:
1432:
1428:
1423:
1419:
1415:
1410:
1406:
1397:
1395:
1391:
1384:
1377:
1372:
1367:
1363:
1359:
1352:
1345:
1341:
1334:
1327:
1320:
1316:
1309:
1304:
1299:
1295:
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681:
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584:
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553:
533:
530:
527:
505:
495:
492:
470:
466:
443:
416:
407:
406:
402:
400:
397:is closed in
396:
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59:
55:
52:
47:
45:
41:
37:
33:
32:real analysis
19:
3932:
3918:the original
3912:
3885:
3866:
3847:
3843:
3818:
3801:
3797:
3220:
3120:
3075:
3068:Edwards 1965
3046:
3034:
2906:
2849:
2652:
2643:
2538:
2408:
2387:
2344:
2338:
2131:
2127:
2102:
2011:
2000:
1998:is compact.
1975:
1941:metric space
1938:
1928:
1918:
1807:
1800:
1699:
1696:
1686:
1679:
1675:
1668:
1666:
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1496:
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1483:
1479:
1477:
1398:
1393:
1389:
1382:
1375:
1373:
1365:
1361:
1360:, otherwise
1357:
1350:
1343:
1339:
1332:
1325:
1318:
1314:
1307:
1305:
1303:is compact.
1297:
1293:
1291:
1235:
1230:
1226:
1222:
1220:
1216:
1215:
1210:
1205:
1201:
1196:
1192:
1127:
1123:
1119:
1115:
1077:
1073:
1069:
1065:
1063:
1010:
1004:
1000:
996:
991:
986:
982:
978:
974:
970:
968:
964:
963:
914:
757:
408:
404:
403:
398:
394:
386:
384:
379:
375:
371:
367:
363:
359:
353:
348:
342:
337:
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328:
324:
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316:
312:
306:
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297:
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286:
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277:
273:
269:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
225:
221:
216:
212:
208:
204:
199:
195:
193:neighborhood
188:
184:
180:
176:
168:
164:
160:
158:
154:
153:
120:Eduard Heine
104:
85:
79:
67:
60:
53:
48:
40:Eduard Heine
35:
29:
2399:1006.4131v1
1864:is compact;
1775:is compact,
382:is closed.
319:) for each
173:limit point
144:Schoenflies
142:(1898) and
132:Émile Borel
44:Émile Borel
3960:Categories
3895:0030505356
3790:References
3183:such that
2961:such that
2788:such that
1626:, one has
1592:such that
1331:, the box
1126:, the set
46:, states:
3939:EMS Press
3844:Proc. AMS
3804:: 69–110.
3753:∞
3741:∈
3632:−
3582:∈
3533:−
3483:∈
3469:⋅
3447:∞
3432:∑
3341:∞
3200:⋅
3197:λ
3194:⊆
3171:λ
3011:≤
2981:∈
2972:⊆
2920:∈
2813:∞
2810:→
2733:∈
2667:∈
2521:ϵ
2472:ϵ
2379:119936587
2354:1006.4131
2273:⊂
2270:Ω
2247:Ω
2209:⊂
2206:Ω
2183:Ω
2175:∞
2077:σ
1600:. Since
1566:for each
1489:(2
1464:…
1461:⊃
1448:⊃
1445:…
1442:⊃
1429:⊃
1416:⊃
1259:−
1221:If a set
1167:∖
1118:. Since
1068:. Since
1041:∪
874:≤
783:≤
740:∈
582:∈
575:∪
531:∈
496:∈
391:Hausdorff
136:countable
3070:, 8.4.7.
2656:A point
2309:See also
2089:-compact
2066:complete
1596: ∈
1588: ∈
1396:-boxes:
1163:′
1147:′
1098:′
994:. Then
981:and let
629:. Since
566:, since
356: ′
345: ′
340:. Then
309: ′
232:. Since
140:Lebesgue
3941:, 2001
3710:is the
3121:bounded
2907:bounded
1685:, then
1577:covers
300:not in
252:. This
94:bounded
72:compact
3892:
3873:
3827:
3677:(here
3079:A set
2990:
2853:A set
2542:A set
2412:A set
2377:
2369:
2091:, and
1921:metric
1667:Thus,
1608:-ball
1573:Since
1522:Cauchy
1513:is in
1292:where
458:, and
374:is in
90:closed
51:subset
49:For a
2394:arXiv
2375:S2CID
2367:JSTOR
2349:arXiv
2321:Notes
1233:-box
389:of a
378:, so
171:is a
150:Proof
76:cover
3890:ISBN
3871:ISBN
3825:ISBN
2946:>
2935:and
2518:<
2475:>
1923:and
1890:(c)
1844:(a)
1644:) ⊆
1617:) ⊆
969:Let
696:and
409:Let
159:Let
126:and
92:and
42:and
34:the
3852:doi
3848:100
3575:max
3476:max
3143:in
2803:lim
2598:in
2359:doi
2345:122
1487:is
1317:of
1225:in
1209:of
977:in
336:of
292:If
268:in
248:in
228:in
224:of
203:of
175:of
88:is
78:of
70:is
56:of
30:In
3962::
3937:,
3931:,
3846:.
3842:.
3802:39
3800:.
3058:^
2373:.
2365:.
2357:.
2343:.
2329:^
2103:A
2095:.
2068:,
1939:A
1931:.
1636:⊆
1570:.
1371:.
1003:\
999:=
961:.
901:2.
401:.
323:∈
289:.
256:∈
240:,
187:∈
122:,
3898:.
3879:.
3860:.
3854::
3833:.
3770:]
3767:1
3764:,
3761:0
3758:[
3747:C
3738:x
3718:k
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3693:k
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3651:(
3646:)
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3041:.
3029:.
3017:}
3014:r
3008:)
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2396::
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2020:(
1986:X
1962:)
1959:d
1956:,
1953:X
1950:(
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1386:1
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1379:1
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1301:0
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1276:n
1272:]
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1256:[
1253:=
1248:0
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1179:,
1176:}
1173:U
1170:{
1159:T
1155:C
1151:=
1143:K
1139:C
1128:K
1124:K
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1090:C
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1074:C
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1050:}
1047:U
1044:{
1036:K
1032:C
1028:=
1023:T
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1005:K
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992:K
987:K
983:C
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975:T
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949:2
946:+
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898:+
895:M
892:=
889:1
886:+
883:M
880:+
877:1
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868:q
865:,
860:q
856:C
852:(
849:d
846:+
843:)
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834:C
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825:p
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817:(
814:d
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808:)
803:p
799:C
795:,
792:p
789:(
786:d
780:)
777:q
774:,
771:p
768:(
765:d
743:S
737:q
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709:q
705:C
682:p
678:C
657:M
637:S
617:S
595:x
591:U
585:S
579:x
554:S
534:S
528:x
506:n
501:R
493:x
471:x
467:U
444:n
439:R
417:S
399:X
395:X
387:S
380:S
376:S
372:S
368:S
364:S
360:C
354:C
349:S
343:C
338:a
333:x
329:V
325:S
321:x
317:x
315:(
313:N
307:C
302:S
298:a
294:S
287:x
283:W
278:U
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270:C
266:U
262:C
258:S
254:x
250:S
246:x
242:W
238:S
234:a
230:R
226:a
222:W
217:U
213:V
209:S
205:a
200:U
196:V
189:C
185:U
181:C
177:S
169:a
165:R
161:S
96:.
86:S
80:S
68:S
61:R
54:S
20:)
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