3192:. Moreover, a locally compact ANR has the homotopy type of a locally finite CW complex; and, by West, a compact ANR has the homotopy type of a finite CW complex. In this sense, ANRs avoid all the homotopy-theoretic pathologies of arbitrary topological spaces. For example, the
662:
3554:
that is strictly locally contractible but is not homotopy equivalent to a CW complex. It is not known whether a compact (or locally compact) metrizable space that is strictly locally contractible must be an
2671:
2633:
2388:
2147:
3200:(for all choices of base point) is a homotopy equivalence. Since ANRs include topological manifolds, Hilbert cube manifolds, Banach manifolds, and so on, these results apply to a large class of spaces.
3337:
312:
3276:
924:
2887:
is an ANR but not an AR (because it is not contractible). In infinite dimensions, Hanner's theorem implies that every
Hilbert cube manifold as well as the (rather different, for example not
1010:
256:
1867:
Any topological space that deformation retracts to a point is contractible and vice versa. However, there exist contractible spaces that do not strongly deformation retract to a point.
1394:
520:
1444:
1666:
1162:
1053:
3063:
1346:
2765:
2591:
2563:
2496:
2432:
2347:
2259:
2193:
2106:
2052:
1820:
151:
3115:
1785:
4056:
1701:
2828:
1554:
805:
1852:
1514:
1093:
1621:
1580:
451:
1753:
1485:
1202:
1129:
750:
400:
2798:
213:
2884:
873:
3552:
3532:
3512:
3492:
3472:
3444:
3419:
3392:
3372:
3179:
3159:
3139:
3037:
3017:
2997:
2977:
2957:
2937:
2725:
2701:
2536:
2516:
2472:
2452:
2408:
2319:
2299:
2279:
2235:
2214:
2168:
2077:
2025:
2005:
1985:
1965:
1941:
1921:
2906:
is an ANR. An arbitrary CW complex need not be metrizable, but every CW complex has the homotopy type of an ANR (which is metrizable, by definition).
559:
347:
if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (any constant map yields a retraction). If
2597:
has been found to give the most satisfactory theory. For that reason, the notations AR and ANR by themselves are used in this article to mean
2638:
4143:
4034:
3996:
3940:
3905:
2600:
3591:
2352:
3988:
2111:
1018:
3284:
752:
is a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on
275:
3587:
3181:.) Borsuk also found a compact subset of the Hilbert cube that is locally contractible (as defined above) but not an ANR.
4026:
1892:
1247:
3222:
932:
879:
694:
A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space
229:
3932:
3897:
1860:
Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are homotopy equivalent
1267:
3398:
2681:
1284:
Among all closed inclusions, cofibrations can be characterized as follows. The inclusion of a closed subspace
3862:
3826:
3771:
475:
1857:
If a subspace is a retract of a space, then the inclusion induces an injection between fundamental groups.
1351:
161:
1399:
4094:
3423:
3340:
2888:
2732:
1626:
4172:
2054:
be a class of topological spaces, closed under homeomorphisms and passage to closed subsets. Following
756:. In this formulation, a deformation retraction carries with it a homotopy between the identity map on
2858:
for metrizable spaces.) It follows that every topological manifold is an ANR. For example, the sphere
1138:
1029:
3073:
metrizable space is an ANR if and only if it is locally contractible in this sense. For example, the
2915:
1307:
1270:
688:
82:
3042:
2572:
2544:
2477:
2413:
2328:
2240:
2174:
2087:
2033:
124:
2738:
2728:
1877:
1790:
3976:
3091:
1758:
4119:
4073:
3185:
3070:
1132:
711:
43:
4017:
1671:
4189:
4139:
4030:
3992:
3936:
3901:
3193:
3082:
1518:
769:
173:
51:
47:
2806:
1490:
1058:
4103:
4065:
3871:
3835:
3790:
3780:
3279:
2892:
2594:
1825:
1585:
1559:
4153:
4115:
4085:
4044:
4006:
3969:
3950:
3915:
3885:
3849:
3813:
1175:
1102:
421:
4149:
4111:
4081:
4040:
4002:
3965:
3946:
3911:
3881:
3845:
3809:
3794:
3447:
2896:
1726:
1449:
1251:
1237:
723:
373:
352:
54:
that preserves the position of all points in that subspace. The subspace is then called a
2773:
183:
3344:
3197:
2862:
2854:
2677:
1861:
851:
703:
3926:
720:
An equivalent definition of deformation retraction is the following. A continuous map
4183:
4123:
3922:
3078:
1259:
1099:
a positive integer, together with the closed line segment connecting the origin with
835:
360:
262:
86:
4013:
3801:
3762:
3537:
3517:
3497:
3477:
3457:
3429:
3404:
3377:
3357:
3164:
3144:
3124:
3066:
3022:
3002:
2982:
2962:
2942:
2922:
2834:
2801:
2710:
2686:
2566:
2521:
2501:
2457:
2437:
2393:
2304:
2284:
2264:
2220:
2199:
2153:
2062:
2055:
2010:
1990:
1970:
1950:
1926:
1906:
78:
3534:
that is not homotopy equivalent to a CW complex. Thus there is a metrizable space
17:
4051:
3957:
3494:
is contractible and not an AR, it is also not an ANR. By Cauty's theorem above,
3189:
2845:
1242:
35:
27:
Continuous, position-preserving mapping from a topological space into a subspace
763:
If, in the definition of a deformation retraction, we add the requirement that
4168:
3980:
3348:
3074:
2903:
2849:
2704:
407:
90:
2676:
A metrizable space is an AR if and only if it is contractible and an ANR. By
1706:
For example, the inclusion of a subcomplex in a CW complex is a cofibration.
3785:
2768:
1864:
they are both homeomorphic to deformation retracts of a single larger space.
657:{\displaystyle F(x,0)=x,\quad F(x,1)\in A,\quad {\mbox{and}}\quad F(a,1)=a.}
3876:
3840:
4135:
4092:
Puppe, Dieter (1967), "Bemerkungen über die
Erweiterung von Homotopien",
3454:(that is, a complete metric linear space). (By Dugundji's theorem above,
842:
668:
458:
31:
4107:
4077:
3451:
1893:
Brouwer fixed-point theorem § A proof using homology or cohomology
1055:
consisting of closed line segments connecting the origin and the point
1881:
4069:
830:. In other words, a strong deformation retraction leaves points in
3117:
that is an ANR but not strictly locally contractible. (A space is
3081:
subset of the real line that is not an ANR, since it is not even
1015:
Note that the condition of being a strong deformation retract is
4167:
This article incorporates material from
Neighborhood retract on
329:
4130:
West, James (2004), "Absolute retracts", in Hart, K. P. (ed.),
2666:{\displaystyle \operatorname {ANR} \left({\mathcal {M}}\right)}
4054:(1959), "On spaces having the homotopy type of a CW-complex",
3979:(1999), "Absolute neighborhood retracts and shape theory", in
3858:"Un espace métrique linéaire qui n'est pas un rétracte absolu"
3857:
3821:
3766:
2628:{\displaystyle \operatorname {AR} \left({\mathcal {M}}\right)}
1250:
for maps to any space. This is one of the central concepts of
3196:
holds for ANRs: a map of ANRs that induces an isomorphism on
2383:{\textstyle \operatorname {ANR} \left({\mathcal {C}}\right),}
2654:
2616:
2578:
2550:
2483:
2419:
2368:
2334:
2246:
2180:
2142:{\textstyle \operatorname {AR} \left({\mathcal {C}}\right),}
2127:
2093:
2039:
901:
1172:
consisting of the line segment connecting the origin with
838:, take this as the definition of deformation retraction.)
3332:{\textstyle \left(X,B\right)\rightarrow \left(Y,A\right)}
3822:"Une caractérisation des rétractes absolus de voisinage"
3351:
of any CW complex has the homotopy type of a CW complex.
926:
as strong deformation retraction one can choose the map
3215:
be any compact metrizable space with a closed subspace
2569:
have been considered in this definition, but the class
3540:
3520:
3500:
3480:
3460:
3432:
3407:
3380:
3360:
3287:
3225:
3167:
3147:
3127:
3094:
3045:
3025:
3005:
2985:
2965:
2945:
2925:
2865:
2809:
2776:
2741:
2713:
2689:
2524:
2504:
2460:
2440:
2396:
2355:
2307:
2287:
2267:
2223:
2202:
2156:
2114:
2065:
2013:
1993:
1973:
1953:
1929:
1909:
1828:
1793:
1761:
1729:
1629:
1452:
1402:
1354:
1131:. Let X have the subspace topology inherited from the
882:
854:
834:
fixed throughout the homotopy. (Some authors, such as
726:
620:
424:
376:
307:{\displaystyle r\circ \iota =\operatorname {id} _{A},}
186:
2641:
2603:
2575:
2547:
2480:
2416:
2331:
2243:
2177:
2090:
2036:
1674:
1588:
1562:
1521:
1493:
1310:
1178:
1141:
1105:
1061:
1032:
935:
772:
562:
478:
278:
232:
127:
3184:
Every ANR has the homotopy type of a CW complex, by
1022:
than being a deformation retract. For instance, let
2735:or not) is an AR. More concretely, Euclidean space
3546:
3526:
3506:
3486:
3466:
3438:
3413:
3386:
3366:
3331:
3270:
3173:
3153:
3133:
3109:
3088:Counterexamples: Borsuk found a compact subset of
3057:
3031:
3011:
2991:
2971:
2951:
2931:
2878:
2837:" topological spaces. Among their properties are:
2822:
2792:
2759:
2719:
2695:
2665:
2627:
2585:
2557:
2530:
2510:
2490:
2466:
2446:
2426:
2402:
2382:
2341:
2313:
2293:
2273:
2253:
2229:
2208:
2187:
2162:
2141:
2100:
2071:
2046:
2019:
1999:
1979:
1959:
1935:
1915:
1846:
1814:
1779:
1747:
1695:
1660:
1615:
1574:
1548:
1508:
1479:
1438:
1388:
1340:
1196:
1156:
1123:
1087:
1047:
1004:
918:
867:
799:
744:
687:. A deformation retraction is a special case of a
656:
514:
465:Deformation retract and strong deformation retract
445:
418:. Conversely, given any idempotent continuous map
394:
306:
250:
207:
145:
4057:Transactions of the American Mathematical Society
3203:Many mapping spaces are ANRs. In particular, let
1374:
4173:Creative Commons Attribution/Share-Alike License
3649:Hu (1965), Corollary II.14.2 and Theorem II.3.1.
3271:{\textstyle \left(Y,A\right)^{\left(X,B\right)}}
1220:Cofibration and neighborhood deformation retract
919:{\textstyle \mathbb {R} ^{n+1}\backslash \{0\};}
3347:) is an ANR. It follows, for example, that the
2852:by ANRs is an ANR. (That is, being an ANR is a
1891:−1)-sphere, is not a retract of the ball. (See
1005:{\displaystyle F(x,t)=(1-t)x+t{x \over \|x\|}.}
251:{\displaystyle \iota \colon A\hookrightarrow X}
3730:Hu (1965), Theorem VII.3.1 and Remark VII.2.3.
3712:Fritsch & Piccinini (1990), Theorem 5.2.1.
3676:Fritsch & Piccinini (1990), Theorem 5.2.1.
3374:is an ANR if and only if every open subset of
2919:in the sense that for every open neighborhood
667:In other words, a deformation retraction is a
325:. Note that, by definition, a retraction maps
81:type of topological space. For example, every
3161:contains a contractible open neighborhood of
671:between a retraction and the identity map on
8:
993:
987:
910:
904:
3422:(meaning a topological vector space with a
3892:Fritsch, Rudolf; Piccinini, Renzo (1990),
3875:
3839:
3808:, Warsaw: Państwowe Wydawnictwo Naukowe,
3784:
3539:
3519:
3499:
3479:
3459:
3431:
3406:
3379:
3359:
3286:
3246:
3224:
3166:
3146:
3126:
3101:
3097:
3096:
3093:
3044:
3024:
3004:
2984:
2964:
2944:
2924:
2870:
2864:
2814:
2808:
2781:
2775:
2748:
2744:
2743:
2740:
2712:
2703:is an AR; more generally, every nonempty
2688:
2653:
2652:
2640:
2615:
2614:
2602:
2577:
2576:
2574:
2549:
2548:
2546:
2523:
2503:
2482:
2481:
2479:
2459:
2439:
2418:
2417:
2415:
2395:
2367:
2366:
2354:
2333:
2332:
2330:
2306:
2286:
2266:
2245:
2244:
2242:
2222:
2201:
2179:
2178:
2176:
2155:
2126:
2125:
2113:
2092:
2091:
2089:
2064:
2038:
2037:
2035:
2012:
1992:
1972:
1952:
1928:
1908:
1827:
1792:
1760:
1728:
1673:
1628:
1587:
1561:
1520:
1492:
1451:
1401:
1365:
1353:
1309:
1304:, meaning that there is a continuous map
1177:
1148:
1144:
1143:
1140:
1104:
1068:
1060:
1039:
1035:
1034:
1031:
981:
934:
889:
885:
884:
881:
859:
853:
771:
725:
619:
561:
477:
453:we obtain a retraction onto the image of
423:
375:
295:
277:
231:
185:
126:
3426:metric) that is not an AR. One can take
1212:but not a strong deformation retract of
402:is a retraction, then the composition ι∘
3565:
62:is a mapping that captures the idea of
4019:A Concise Course in Algebraic Topology
3394:has the homotopy type of a CW complex.
2841:Every open subset of an ANR is an ANR.
321:with the inclusion is the identity of
89:of a very simple topological space, a
3748:Cauty (1994), Fund. Math. 146: 85–99.
3739:Cauty (1994), Fund. Math. 144: 11–22.
1389:{\textstyle A=u^{-1}\!\left(0\right)}
515:{\displaystyle F\colon X\times \to X}
7:
1987:is a retract of some open subset of
1439:{\textstyle H:X\times \rightarrow X}
1899:Absolute neighborhood retract (ANR)
1661:{\textstyle H\left(x,1\right)\in A}
1273:), then the image of a cofibration
876:is a strong deformation retract of
265:, a retraction is a continuous map
3584:Fundamentals of Algebraic Topology
2680:, every locally convex metrizable
25:
3604:Hatcher (2002), Proposition 4H.1.
3474:cannot be locally convex.) Since
3207:be an ANR with a closed subspace
2848:, a metrizable space that has an
2833:ANRs form a remarkable class of "
4132:Encyclopedia of General Topology
3964:, Wayne State University Press,
2979:, there is an open neighborhood
1715:One basic property of a retract
1298:neighborhood deformation retract
1292:is a cofibration if and only if
1157:{\displaystyle \mathbb {R} ^{2}}
1048:{\displaystyle \mathbb {R} ^{2}}
3894:Cellular Structures in Topology
3058:{\textstyle V\hookrightarrow U}
2217:is a closed subset of a space
1755:) is that every continuous map
1341:{\displaystyle u:X\rightarrow }
1258:is always injective, in fact a
626:
618:
590:
4171:, which is licensed under the
3703:Borsuk (1967), Theorem V.11.1.
3640:Hu (1965), Proposition II.7.2.
3307:
3049:
2760:{\textstyle \mathbb {R} ^{n},}
2586:{\displaystyle {\mathcal {M}}}
2558:{\displaystyle {\mathcal {C}}}
2491:{\displaystyle {\mathcal {C}}}
2454:is a closed subset of a space
2427:{\displaystyle {\mathcal {C}}}
2342:{\displaystyle {\mathcal {C}}}
2254:{\displaystyle {\mathcal {C}}}
2188:{\displaystyle {\mathcal {C}}}
2101:{\displaystyle {\mathcal {C}}}
2047:{\displaystyle {\mathcal {C}}}
1815:{\textstyle g:X\rightarrow Y,}
1803:
1771:
1739:
1684:
1678:
1607:
1595:
1537:
1525:
1468:
1456:
1430:
1427:
1415:
1335:
1323:
1320:
1191:
1179:
1118:
1106:
1082:
1062:
969:
957:
951:
939:
788:
776:
736:
642:
630:
606:
594:
578:
566:
506:
503:
491:
434:
386:
242:
196:
190:
146:{\displaystyle r\colon X\to A}
137:
1:
3622:Hatcher (2002), Exercise 0.6.
3588:Graduate Texts in Mathematics
3354:By Cauty, a metrizable space
3119:strictly locally contractible
3110:{\textstyle \mathbb {R} ^{3}}
2518:is a neighborhood retract of
2323:absolute neighborhood retract
1780:{\textstyle f:A\rightarrow Y}
828:strong deformation retraction
85:is an ANR. Every ANR has the
71:absolute neighborhood retract
3694:Borsuk (1967), section IV.4.
2058:(starting in 1931), a space
1236:of topological spaces is a (
1208:is a deformation retract of
317:that is, the composition of
223:. Equivalently, denoting by
4027:University of Chicago Press
3658:Hu (1965), Theorem III.8.1.
3121:if every open neighborhood
2727:is an AR. For example, any
1787:has at least one extension
1248:homotopy extension property
110:be a topological space and
4206:
3933:Cambridge University Press
3898:Cambridge University Press
3685:Hu (1965), Theorem V.7.1.
1696:{\displaystyle u(x)<1}
66:a space into a subspace.
58:of the original space. A
3667:Mardešiċ (1999), p. 245.
3631:Mardešiċ (1999), p. 242.
3211:that is an ANR, and let
3039:such that the inclusion
2823:{\textstyle I^{\omega }}
2682:topological vector space
1549:{\displaystyle H(a,t)=a}
800:{\displaystyle F(a,t)=a}
118:. Then a continuous map
3863:Fundamenta Mathematicae
3827:Fundamenta Mathematicae
3786:10.4064/fm-17-1-152-170
3772:Fundamenta Mathematicae
2707:of such a vector space
1923:of a topological space
1847:{\textstyle g=f\circ r}
1509:{\displaystyle x\in X,}
1088:{\displaystyle (1/n,1)}
3877:10.4064/fm-146-1-85-99
3856:Cauty, Robert (1994),
3841:10.4064/fm-144-1-11-22
3820:Cauty, Robert (1994),
3548:
3528:
3508:
3488:
3468:
3440:
3415:
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3333:
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3033:
3013:
2993:
2973:
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2933:
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2667:
2629:
2587:
2559:
2532:
2512:
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2448:
2428:
2404:
2384:
2343:
2315:
2295:
2275:
2255:
2231:
2210:
2189:
2164:
2143:
2102:
2073:
2048:
2021:
2001:
1981:
1961:
1937:
1917:
1848:
1816:
1781:
1749:
1697:
1662:
1617:
1616:{\displaystyle t\in ,}
1576:
1575:{\displaystyle a\in A}
1550:
1510:
1481:
1440:
1390:
1342:
1198:
1158:
1125:
1089:
1049:
1006:
920:
869:
801:
746:
658:
527:deformation retraction
516:
447:
446:{\textstyle s:X\to X,}
396:
308:
252:
209:
147:
64:continuously shrinking
60:deformation retraction
4095:Archiv der Mathematik
3613:Puppe (1967), Satz 1.
3582:Weintraub, Steven H.
3549:
3529:
3509:
3489:
3469:
3441:
3424:translation-invariant
3416:
3397:By Cauty, there is a
3389:
3369:
3341:compact-open topology
3334:
3273:
3176:
3156:
3136:
3112:
3060:
3034:
3014:
2994:
2974:
2954:
2934:
2902:Every locally finite
2881:
2825:
2795:
2762:
2722:
2698:
2668:
2630:
2588:
2560:
2533:
2513:
2493:
2469:
2449:
2429:
2405:
2385:
2344:
2316:
2296:
2276:
2256:
2232:
2211:
2190:
2165:
2144:
2103:
2074:
2049:
2022:
2002:
1982:
1962:
1938:
1918:
1872:No-retraction theorem
1849:
1817:
1782:
1750:
1748:{\textstyle r:X\to A}
1698:
1663:
1618:
1577:
1551:
1511:
1482:
1480:{\textstyle H(x,0)=x}
1441:
1391:
1343:
1199:
1197:{\displaystyle (0,1)}
1159:
1126:
1124:{\displaystyle (0,1)}
1090:
1050:
1007:
921:
870:
802:
747:
745:{\textstyle r:X\to A}
659:
517:
448:
397:
395:{\textstyle r:X\to A}
309:
253:
210:
148:
3991:, pp. 241–269,
3721:West (2004), p. 119.
3538:
3518:
3498:
3478:
3458:
3430:
3405:
3378:
3358:
3285:
3223:
3165:
3145:
3125:
3092:
3043:
3023:
3003:
2983:
2963:
2943:
2923:
2916:locally contractible
2863:
2807:
2774:
2739:
2711:
2687:
2639:
2601:
2573:
2545:
2522:
2502:
2478:
2458:
2438:
2414:
2394:
2353:
2329:
2305:
2285:
2265:
2241:
2221:
2200:
2175:
2154:
2112:
2088:
2063:
2034:
2011:
1991:
1971:
1951:
1945:neighborhood retract
1927:
1907:
1826:
1791:
1759:
1727:
1672:
1627:
1586:
1560:
1519:
1491:
1450:
1400:
1352:
1308:
1271:weak Hausdorff space
1176:
1139:
1103:
1059:
1030:
933:
880:
852:
770:
724:
689:homotopy equivalence
560:
476:
422:
410:continuous map from
374:
276:
230:
184:
125:
83:topological manifold
77:) is a particularly
3985:History of Topology
3767:"Sur les rétractes"
3514:has an open subset
3399:metric linear space
2793:{\textstyle I^{n},}
2729:normed vector space
1268:compactly generated
1266:is Hausdorff (or a
1168:be the subspace of
1026:be the subspace of
841:As an example, the
681:deformation retract
457:by restricting the
208:{\textstyle r(a)=a}
4108:10.1007/BF01899475
3962:Theory of Retracts
3928:Algebraic Topology
3806:Theory of Retracts
3544:
3524:
3504:
3484:
3464:
3436:
3411:
3384:
3364:
3329:
3268:
3171:
3151:
3131:
3107:
3071:finite-dimensional
3065:is homotopic to a
3055:
3029:
3009:
2989:
2969:
2949:
2929:
2879:{\textstyle S^{n}}
2876:
2820:
2790:
2757:
2717:
2693:
2663:
2625:
2583:
2555:
2528:
2508:
2488:
2464:
2444:
2424:
2400:
2380:
2339:
2311:
2291:
2271:
2251:
2227:
2206:
2185:
2160:
2139:
2098:
2069:
2044:
2017:
1997:
1977:
1957:
1933:
1913:
1844:
1812:
1777:
1745:
1693:
1658:
1613:
1572:
1546:
1506:
1477:
1436:
1386:
1338:
1194:
1154:
1133:Euclidean topology
1121:
1085:
1045:
1002:
916:
868:{\textstyle S^{n}}
865:
797:
742:
706:(and in fact that
654:
624:
512:
443:
392:
304:
248:
205:
143:
44:continuous mapping
18:Retract (topology)
3590:. Vol. 270.
3219:. Then the space
3194:Whitehead theorem
3083:locally connected
2893:Hilbert manifolds
2595:metrizable spaces
1885:-dimensional ball
1723:(with retraction
1262:to its image. If
997:
698:would imply that
623:
469:A continuous map
48:topological space
16:(Redirected from
4197:
4156:
4126:
4088:
4047:
4024:
4009:
3972:
3953:
3918:
3888:
3879:
3852:
3843:
3816:
3797:
3788:
3749:
3746:
3740:
3737:
3731:
3728:
3722:
3719:
3713:
3710:
3704:
3701:
3695:
3692:
3686:
3683:
3677:
3674:
3668:
3665:
3659:
3656:
3650:
3647:
3641:
3638:
3632:
3629:
3623:
3620:
3614:
3611:
3605:
3602:
3596:
3595:
3579:
3573:
3570:
3553:
3551:
3550:
3545:
3533:
3531:
3530:
3525:
3513:
3511:
3510:
3505:
3493:
3491:
3490:
3485:
3473:
3471:
3470:
3465:
3445:
3443:
3442:
3437:
3420:
3418:
3417:
3412:
3393:
3391:
3390:
3385:
3373:
3371:
3370:
3365:
3338:
3336:
3335:
3330:
3328:
3324:
3306:
3302:
3277:
3275:
3274:
3269:
3267:
3266:
3265:
3261:
3245:
3241:
3180:
3178:
3177:
3172:
3160:
3158:
3157:
3152:
3140:
3138:
3137:
3132:
3116:
3114:
3113:
3108:
3106:
3105:
3100:
3064:
3062:
3061:
3056:
3038:
3036:
3035:
3030:
3018:
3016:
3015:
3010:
2998:
2996:
2995:
2990:
2978:
2976:
2975:
2970:
2958:
2956:
2955:
2950:
2938:
2936:
2935:
2930:
2897:Banach manifolds
2885:
2883:
2882:
2877:
2875:
2874:
2829:
2827:
2826:
2821:
2819:
2818:
2799:
2797:
2796:
2791:
2786:
2785:
2766:
2764:
2763:
2758:
2753:
2752:
2747:
2726:
2724:
2723:
2718:
2702:
2700:
2699:
2694:
2672:
2670:
2669:
2664:
2662:
2658:
2657:
2634:
2632:
2631:
2626:
2624:
2620:
2619:
2592:
2590:
2589:
2584:
2582:
2581:
2564:
2562:
2561:
2556:
2554:
2553:
2541:Various classes
2537:
2535:
2534:
2529:
2517:
2515:
2514:
2509:
2497:
2495:
2494:
2489:
2487:
2486:
2473:
2471:
2470:
2465:
2453:
2451:
2450:
2445:
2433:
2431:
2430:
2425:
2423:
2422:
2409:
2407:
2406:
2401:
2389:
2387:
2386:
2381:
2376:
2372:
2371:
2348:
2346:
2345:
2340:
2338:
2337:
2320:
2318:
2317:
2312:
2300:
2298:
2297:
2292:
2281:is a retract of
2280:
2278:
2277:
2272:
2260:
2258:
2257:
2252:
2250:
2249:
2236:
2234:
2233:
2228:
2215:
2213:
2212:
2207:
2194:
2192:
2191:
2186:
2184:
2183:
2169:
2167:
2166:
2161:
2148:
2146:
2145:
2140:
2135:
2131:
2130:
2107:
2105:
2104:
2099:
2097:
2096:
2082:absolute retract
2078:
2076:
2075:
2070:
2053:
2051:
2050:
2045:
2043:
2042:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1986:
1984:
1983:
1978:
1966:
1964:
1963:
1958:
1942:
1940:
1939:
1934:
1922:
1920:
1919:
1914:
1903:A closed subset
1887:, that is, the (
1853:
1851:
1850:
1845:
1821:
1819:
1818:
1813:
1786:
1784:
1783:
1778:
1754:
1752:
1751:
1746:
1702:
1700:
1699:
1694:
1667:
1665:
1664:
1659:
1651:
1647:
1622:
1620:
1619:
1614:
1581:
1579:
1578:
1573:
1555:
1553:
1552:
1547:
1515:
1513:
1512:
1507:
1486:
1484:
1483:
1478:
1445:
1443:
1442:
1437:
1395:
1393:
1392:
1387:
1385:
1373:
1372:
1347:
1345:
1344:
1339:
1254:. A cofibration
1203:
1201:
1200:
1195:
1163:
1161:
1160:
1155:
1153:
1152:
1147:
1130:
1128:
1127:
1122:
1094:
1092:
1091:
1086:
1072:
1054:
1052:
1051:
1046:
1044:
1043:
1038:
1011:
1009:
1008:
1003:
998:
996:
982:
925:
923:
922:
917:
900:
899:
888:
874:
872:
871:
866:
864:
863:
806:
804:
803:
798:
751:
749:
748:
743:
663:
661:
660:
655:
625:
621:
533:onto a subspace
521:
519:
518:
513:
452:
450:
449:
444:
401:
399:
398:
393:
313:
311:
310:
305:
300:
299:
257:
255:
254:
249:
214:
212:
211:
206:
152:
150:
149:
144:
21:
4205:
4204:
4200:
4199:
4198:
4196:
4195:
4194:
4180:
4179:
4163:
4146:
4129:
4091:
4070:10.2307/1993204
4050:
4037:
4022:
4012:
3999:
3975:
3956:
3943:
3921:
3908:
3891:
3855:
3819:
3800:
3761:
3758:
3753:
3752:
3747:
3743:
3738:
3734:
3729:
3725:
3720:
3716:
3711:
3707:
3702:
3698:
3693:
3689:
3684:
3680:
3675:
3671:
3666:
3662:
3657:
3653:
3648:
3644:
3639:
3635:
3630:
3626:
3621:
3617:
3612:
3608:
3603:
3599:
3581:
3580:
3576:
3571:
3567:
3562:
3536:
3535:
3516:
3515:
3496:
3495:
3476:
3475:
3456:
3455:
3428:
3427:
3403:
3402:
3376:
3375:
3356:
3355:
3314:
3310:
3292:
3288:
3283:
3282:
3251:
3247:
3231:
3227:
3226:
3221:
3220:
3198:homotopy groups
3163:
3162:
3143:
3142:
3123:
3122:
3095:
3090:
3089:
3041:
3040:
3021:
3020:
3001:
3000:
2981:
2980:
2961:
2960:
2941:
2940:
2921:
2920:
2889:locally compact
2866:
2861:
2860:
2810:
2805:
2804:
2777:
2772:
2771:
2742:
2737:
2736:
2709:
2708:
2685:
2684:
2648:
2637:
2636:
2610:
2599:
2598:
2571:
2570:
2543:
2542:
2520:
2519:
2500:
2499:
2476:
2475:
2456:
2455:
2436:
2435:
2412:
2411:
2392:
2391:
2362:
2351:
2350:
2327:
2326:
2303:
2302:
2283:
2282:
2263:
2262:
2239:
2238:
2219:
2218:
2198:
2197:
2173:
2172:
2152:
2151:
2121:
2110:
2109:
2086:
2085:
2061:
2060:
2032:
2031:
2009:
2008:
1989:
1988:
1969:
1968:
1949:
1948:
1925:
1924:
1905:
1904:
1901:
1874:
1824:
1823:
1789:
1788:
1757:
1756:
1725:
1724:
1712:
1670:
1669:
1637:
1633:
1625:
1624:
1584:
1583:
1558:
1557:
1517:
1516:
1489:
1488:
1448:
1447:
1398:
1397:
1396:and a homotopy
1375:
1361:
1350:
1349:
1306:
1305:
1252:homotopy theory
1222:
1174:
1173:
1142:
1137:
1136:
1101:
1100:
1057:
1056:
1033:
1028:
1027:
986:
931:
930:
883:
878:
877:
855:
850:
849:
768:
767:
722:
721:
675:. The subspace
558:
557:
474:
473:
467:
420:
419:
372:
371:
291:
274:
273:
228:
227:
182:
181:
123:
122:
104:
99:
28:
23:
22:
15:
12:
11:
5:
4203:
4201:
4193:
4192:
4182:
4181:
4178:
4177:
4162:
4161:External links
4159:
4158:
4157:
4144:
4127:
4089:
4064:(2): 272–280,
4048:
4035:
4010:
3997:
3977:Mardešić, Sibe
3973:
3954:
3941:
3923:Hatcher, Allen
3919:
3906:
3889:
3853:
3817:
3798:
3757:
3754:
3751:
3750:
3741:
3732:
3723:
3714:
3705:
3696:
3687:
3678:
3669:
3660:
3651:
3642:
3633:
3624:
3615:
3606:
3597:
3574:
3572:Borsuk (1931).
3564:
3563:
3561:
3558:
3557:
3556:
3547:{\textstyle U}
3543:
3527:{\textstyle U}
3523:
3507:{\textstyle V}
3503:
3487:{\textstyle V}
3483:
3467:{\textstyle V}
3463:
3439:{\textstyle V}
3435:
3414:{\textstyle V}
3410:
3395:
3387:{\textstyle X}
3383:
3367:{\textstyle X}
3363:
3352:
3327:
3323:
3320:
3317:
3313:
3309:
3305:
3301:
3298:
3295:
3291:
3264:
3260:
3257:
3254:
3250:
3244:
3240:
3237:
3234:
3230:
3201:
3182:
3174:{\textstyle x}
3170:
3154:{\textstyle x}
3150:
3141:of each point
3134:{\textstyle U}
3130:
3104:
3099:
3086:
3054:
3051:
3048:
3032:{\textstyle U}
3028:
3012:{\textstyle x}
3008:
2992:{\textstyle V}
2988:
2972:{\textstyle X}
2968:
2952:{\textstyle x}
2948:
2932:{\textstyle U}
2928:
2907:
2900:
2873:
2869:
2855:local property
2842:
2817:
2813:
2789:
2784:
2780:
2756:
2751:
2746:
2720:{\textstyle V}
2716:
2696:{\textstyle V}
2692:
2661:
2656:
2651:
2647:
2644:
2623:
2618:
2613:
2609:
2606:
2580:
2552:
2531:{\textstyle Y}
2527:
2511:{\textstyle X}
2507:
2485:
2467:{\textstyle Y}
2463:
2447:{\textstyle X}
2443:
2421:
2403:{\textstyle X}
2399:
2379:
2375:
2370:
2365:
2361:
2358:
2336:
2325:for the class
2314:{\textstyle X}
2310:
2294:{\textstyle Y}
2290:
2274:{\textstyle X}
2270:
2248:
2230:{\textstyle Y}
2226:
2209:{\textstyle X}
2205:
2182:
2163:{\textstyle X}
2159:
2138:
2134:
2129:
2124:
2120:
2117:
2095:
2084:for the class
2072:{\textstyle X}
2068:
2041:
2020:{\textstyle X}
2016:
2007:that contains
2000:{\textstyle Y}
1996:
1980:{\textstyle X}
1976:
1960:{\textstyle Y}
1956:
1936:{\textstyle Y}
1932:
1916:{\textstyle X}
1912:
1900:
1897:
1873:
1870:
1869:
1868:
1865:
1862:if and only if
1858:
1855:
1843:
1840:
1837:
1834:
1831:
1811:
1808:
1805:
1802:
1799:
1796:
1776:
1773:
1770:
1767:
1764:
1744:
1741:
1738:
1735:
1732:
1711:
1708:
1692:
1689:
1686:
1683:
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1657:
1654:
1650:
1646:
1643:
1640:
1636:
1632:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1571:
1568:
1565:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1505:
1502:
1499:
1496:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1384:
1381:
1378:
1371:
1368:
1364:
1360:
1357:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1246:if it has the
1221:
1218:
1193:
1190:
1187:
1184:
1181:
1151:
1146:
1120:
1117:
1114:
1111:
1108:
1084:
1081:
1078:
1075:
1071:
1067:
1064:
1042:
1037:
1013:
1012:
1001:
995:
992:
989:
985:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
915:
912:
909:
906:
903:
898:
895:
892:
887:
862:
858:
808:
807:
796:
793:
790:
787:
784:
781:
778:
775:
741:
738:
735:
732:
729:
704:path connected
665:
664:
653:
650:
647:
644:
641:
638:
635:
632:
629:
617:
614:
611:
608:
605:
602:
599:
596:
593:
589:
586:
583:
580:
577:
574:
571:
568:
565:
537:if, for every
523:
522:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
466:
463:
442:
439:
436:
433:
430:
427:
391:
388:
385:
382:
379:
315:
314:
303:
298:
294:
290:
287:
284:
281:
259:
258:
247:
244:
241:
238:
235:
204:
201:
198:
195:
192:
189:
154:
153:
142:
139:
136:
133:
130:
114:a subspace of
103:
100:
98:
95:
34:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4202:
4191:
4188:
4187:
4185:
4176:
4174:
4170:
4165:
4164:
4160:
4155:
4151:
4147:
4145:0-444-50355-2
4141:
4137:
4134:, Amsterdam:
4133:
4128:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4096:
4090:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4058:
4053:
4049:
4046:
4042:
4038:
4036:0-226-51182-0
4032:
4028:
4021:
4020:
4015:
4014:May, J. Peter
4011:
4008:
4004:
4000:
3998:0-444-82375-1
3994:
3990:
3989:North-Holland
3987:, Amsterdam:
3986:
3982:
3978:
3974:
3971:
3967:
3963:
3959:
3955:
3952:
3948:
3944:
3942:0-521-79540-0
3938:
3934:
3930:
3929:
3924:
3920:
3917:
3913:
3909:
3907:0-521-32784-9
3903:
3899:
3895:
3890:
3887:
3883:
3878:
3873:
3869:
3865:
3864:
3859:
3854:
3851:
3847:
3842:
3837:
3833:
3829:
3828:
3823:
3818:
3815:
3811:
3807:
3803:
3802:Borsuk, Karol
3799:
3796:
3792:
3787:
3782:
3778:
3774:
3773:
3768:
3764:
3763:Borsuk, Karol
3760:
3759:
3755:
3745:
3742:
3736:
3733:
3727:
3724:
3718:
3715:
3709:
3706:
3700:
3697:
3691:
3688:
3682:
3679:
3673:
3670:
3664:
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3655:
3652:
3646:
3643:
3637:
3634:
3628:
3625:
3619:
3616:
3610:
3607:
3601:
3598:
3594:. p. 20.
3593:
3589:
3585:
3578:
3575:
3569:
3566:
3559:
3541:
3521:
3501:
3481:
3461:
3453:
3449:
3433:
3425:
3421:
3408:
3400:
3396:
3381:
3361:
3353:
3350:
3346:
3345:mapping space
3342:
3325:
3321:
3318:
3315:
3311:
3303:
3299:
3296:
3293:
3289:
3281:
3262:
3258:
3255:
3252:
3248:
3242:
3238:
3235:
3232:
3228:
3218:
3214:
3210:
3206:
3202:
3199:
3195:
3191:
3187:
3183:
3168:
3148:
3128:
3120:
3102:
3087:
3084:
3080:
3076:
3072:
3068:
3052:
3046:
3026:
3019:contained in
3006:
2986:
2966:
2946:
2926:
2918:
2917:
2912:
2908:
2905:
2901:
2898:
2894:
2890:
2886:
2871:
2867:
2857:
2856:
2851:
2847:
2843:
2840:
2839:
2838:
2836:
2831:
2815:
2811:
2803:
2787:
2782:
2778:
2770:
2754:
2749:
2734:
2730:
2714:
2706:
2705:convex subset
2690:
2683:
2679:
2674:
2659:
2649:
2645:
2642:
2621:
2611:
2607:
2604:
2596:
2568:
2567:normal spaces
2539:
2525:
2505:
2461:
2441:
2434:and whenever
2397:
2377:
2373:
2363:
2359:
2356:
2324:
2308:
2288:
2268:
2224:
2216:
2203:
2195:and whenever
2170:
2157:
2136:
2132:
2122:
2118:
2115:
2083:
2080:is called an
2079:
2066:
2057:
2028:
2014:
1994:
1974:
1954:
1946:
1930:
1910:
1898:
1896:
1894:
1890:
1886:
1884:
1879:
1871:
1866:
1863:
1859:
1856:
1841:
1838:
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1809:
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1800:
1797:
1794:
1774:
1768:
1765:
1762:
1742:
1736:
1733:
1730:
1722:
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1714:
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1690:
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1317:
1314:
1311:
1303:
1299:
1295:
1291:
1287:
1282:
1280:
1277:is closed in
1276:
1272:
1269:
1265:
1261:
1260:homeomorphism
1257:
1253:
1249:
1245:
1244:
1239:
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1231:
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1219:
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1211:
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794:
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785:
782:
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773:
766:
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739:
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727:
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713:
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536:
532:
528:
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472:
471:
470:
464:
462:
460:
456:
440:
437:
431:
428:
425:
417:
413:
409:
405:
389:
383:
380:
377:
368:
366:
362:
361:closed subset
358:
354:
350:
346:
342:
338:
335:. A subspace
334:
331:
328:
324:
320:
301:
296:
292:
288:
285:
282:
279:
272:
271:
270:
268:
264:
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140:
134:
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128:
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120:
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109:
101:
96:
94:
92:
88:
87:homotopy type
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4166:
4131:
4099:
4093:
4061:
4055:
4052:Milnor, John
4018:
3984:
3981:James, I. M.
3961:
3958:Hu, Sze-Tsen
3927:
3893:
3867:
3861:
3831:
3825:
3805:
3776:
3770:
3744:
3735:
3726:
3717:
3708:
3699:
3690:
3681:
3672:
3663:
3654:
3645:
3636:
3627:
3618:
3609:
3600:
3583:
3577:
3568:
3401:
3216:
3212:
3208:
3204:
3118:
3067:constant map
2914:
2910:
2859:
2853:
2835:well-behaved
2832:
2802:Hilbert cube
2675:
2540:
2322:
2196:
2150:
2081:
2059:
2029:
1944:
1943:is called a
1902:
1888:
1882:
1875:
1720:
1716:
1705:
1301:
1297:
1293:
1289:
1285:
1283:
1278:
1274:
1263:
1255:
1241:
1233:
1229:
1225:
1223:
1213:
1209:
1205:
1169:
1165:
1096:
1023:
1016:
1014:
848:
843:
840:
831:
827:
826:is called a
823:
819:
815:
811:
809:
762:
760:and itself.
757:
753:
717:
716:
712:contractible
707:
699:
695:
693:
684:
680:
679:is called a
676:
672:
666:
550:
546:
542:
538:
534:
530:
526:
524:
468:
454:
415:
411:
403:
369:
364:
356:
348:
344:
340:
339:is called a
336:
332:
326:
322:
318:
316:
266:
260:
220:
216:
177:
174:identity map
169:
165:
157:
155:
115:
111:
107:
105:
79:well-behaved
74:
70:
68:
63:
59:
55:
39:
29:
3779:: 152–170,
3278:of maps of
2939:of a point
1288:in a space
1243:cofibration
529:of a space
180:; that is,
162:restriction
97:Definitions
36:mathematics
4169:PlanetMath
3795:0003.02701
3756:References
3349:loop space
3339:(with the
3075:Cantor set
2909:Every ANR
2904:CW complex
2850:open cover
2349:, written
2301:. A space
2108:, written
1710:Properties
1446:such that
1164:. Now let
408:idempotent
359:must be a
269:such that
158:retraction
91:CW complex
40:retraction
4124:120021003
4102:: 81–88,
3870:: 85–99,
3834:: 11–22,
3448:separable
3308:→
3186:Whitehead
3050:↪
2899:are ANRs.
2830:are ARs.
2816:ω
2769:unit cube
2646:
2608:
2360:
2119:
1839:∘
1804:→
1772:→
1740:→
1653:∈
1593:∈
1567:∈
1498:∈
1431:→
1413:×
1367:−
1321:→
1017:strictly
994:‖
988:‖
964:−
902:∖
737:→
610:∈
507:→
489:×
483::
435:→
387:→
353:Hausdorff
286:ι
283:∘
263:inclusion
243:↪
237::
234:ι
138:→
132::
4190:Topology
4184:Category
4136:Elsevier
4016:(1999),
3960:(1965),
3925:(2002),
3804:(1967),
3765:(1931),
3592:Springer
2800:and the
2733:complete
2678:Dugundji
2565:such as
1878:boundary
1556:for all
1487:for all
1238:Hurewicz
1019:stronger
814:in and
810:for all
669:homotopy
459:codomain
215:for all
52:subspace
32:topology
4154:2049453
4116:0206954
4086:0100267
4078:1993204
4045:1702278
4007:1674915
3983:(ed.),
3970:0181977
3951:1867354
3916:1074175
3886:1305261
3850:1271475
3814:0216473
3452:F-space
3450:and an
3343:on the
3079:compact
1880:of the
1822:namely
1204:. Then
846:-sphere
836:Hatcher
822:, then
355:, then
341:retract
172:is the
160:if the
102:Retract
56:retract
50:into a
46:from a
4152:
4142:
4122:
4114:
4084:
4076:
4043:
4033:
4005:
3995:
3968:
3949:
3939:
3914:
3904:
3884:
3848:
3812:
3793:
3446:to be
3190:Milnor
2846:Hanner
2410:is in
2321:is an
2171:is in
2056:Borsuk
1224:A map
406:is an
4120:S2CID
4074:JSTOR
4023:(PDF)
3560:Notes
3280:pairs
3077:is a
1348:with
1296:is a
718:Note:
525:is a
156:is a
42:is a
4140:ISBN
4031:ISBN
3993:ISBN
3937:ISBN
3902:ISBN
3555:ANR.
3188:and
3069:. A
2895:and
2767:the
2635:and
2030:Let
1876:The
1688:<
1623:and
1582:and
1095:for
545:and
330:onto
261:the
106:Let
38:, a
4104:doi
4066:doi
3872:doi
3868:146
3836:doi
3832:144
3791:Zbl
3781:doi
2999:of
2959:in
2913:is
2844:By
2643:ANR
2593:of
2474:in
2390:if
2357:ANR
2237:in
2149:if
1967:if
1947:of
1895:.)
1719:of
1668:if
1300:of
1135:on
818:in
714:).
710:is
702:is
683:of
622:and
549:in
541:in
414:to
370:If
363:of
351:is
343:of
219:in
176:on
168:to
164:of
75:ANR
69:An
30:In
4186::
4150:MR
4148:,
4138:,
4118:,
4112:MR
4110:,
4100:18
4098:,
4082:MR
4080:,
4072:,
4062:90
4060:,
4041:MR
4039:,
4029:,
4025:,
4003:MR
4001:,
3966:MR
3947:MR
3945:,
3935:,
3931:,
3912:MR
3910:,
3900:,
3896:,
3882:MR
3880:,
3866:,
3860:,
3846:MR
3844:,
3830:,
3824:,
3810:MR
3789:,
3777:17
3775:,
3769:,
3586:.
2891:)
2673:.
2605:AR
2538:.
2498:,
2261:,
2116:AR
2027:.
1703:.
1281:.
1240:)
1232:→
1228::
1216:.
691:.
553:,
461:.
367:.
293:id
93:.
4175:.
4106::
4068::
3874::
3838::
3783::
3542:U
3522:U
3502:V
3482:V
3462:V
3434:V
3409:V
3382:X
3362:X
3326:)
3322:A
3319:,
3316:Y
3312:(
3304:)
3300:B
3297:,
3294:X
3290:(
3263:)
3259:B
3256:,
3253:X
3249:(
3243:)
3239:A
3236:,
3233:Y
3229:(
3217:B
3213:X
3209:A
3205:Y
3169:x
3149:x
3129:U
3103:3
3098:R
3085:.
3053:U
3047:V
3027:U
3007:x
2987:V
2967:X
2947:x
2927:U
2911:X
2872:n
2868:S
2812:I
2788:,
2783:n
2779:I
2755:,
2750:n
2745:R
2731:(
2715:V
2691:V
2660:)
2655:M
2650:(
2622:)
2617:M
2612:(
2579:M
2551:C
2526:Y
2506:X
2484:C
2462:Y
2442:X
2420:C
2398:X
2378:,
2374:)
2369:C
2364:(
2335:C
2309:X
2289:Y
2269:X
2247:C
2225:Y
2204:X
2181:C
2158:X
2137:,
2133:)
2128:C
2123:(
2094:C
2067:X
2040:C
2015:X
1995:Y
1975:X
1955:Y
1931:Y
1911:X
1889:n
1883:n
1854:.
1842:r
1836:f
1833:=
1830:g
1810:,
1807:Y
1801:X
1798::
1795:g
1775:Y
1769:A
1766::
1763:f
1743:A
1737:X
1734::
1731:r
1721:X
1717:A
1691:1
1685:)
1682:x
1679:(
1676:u
1656:A
1649:)
1645:1
1642:,
1639:x
1635:(
1631:H
1611:,
1608:]
1605:1
1602:,
1599:0
1596:[
1590:t
1570:A
1564:a
1544:a
1541:=
1538:)
1535:t
1532:,
1529:a
1526:(
1523:H
1504:,
1501:X
1495:x
1475:x
1472:=
1469:)
1466:0
1463:,
1460:x
1457:(
1454:H
1434:X
1428:]
1425:1
1422:,
1419:0
1416:[
1410:X
1407::
1404:H
1383:)
1380:0
1377:(
1370:1
1363:u
1359:=
1356:A
1336:]
1333:1
1330:,
1327:0
1324:[
1318:X
1315::
1312:u
1302:X
1294:A
1290:X
1286:A
1279:X
1275:f
1264:X
1256:f
1234:X
1230:A
1226:f
1214:X
1210:X
1206:A
1192:)
1189:1
1186:,
1183:0
1180:(
1170:X
1166:A
1150:2
1145:R
1119:)
1116:1
1113:,
1110:0
1107:(
1097:n
1083:)
1080:1
1077:,
1074:n
1070:/
1066:1
1063:(
1041:2
1036:R
1024:X
1000:.
991:x
984:x
979:t
976:+
973:x
970:)
967:t
961:1
958:(
955:=
952:)
949:t
946:,
943:x
940:(
937:F
914:;
911:}
908:0
905:{
897:1
894:+
891:n
886:R
861:n
857:S
844:n
832:A
824:F
820:A
816:a
812:t
795:a
792:=
789:)
786:t
783:,
780:a
777:(
774:F
758:X
754:X
740:A
734:X
731::
728:r
708:X
700:X
696:X
685:X
677:A
673:X
652:.
649:a
646:=
643:)
640:1
637:,
634:a
631:(
628:F
616:,
613:A
607:)
604:1
601:,
598:x
595:(
592:F
588:,
585:x
582:=
579:)
576:0
573:,
570:x
567:(
564:F
551:A
547:a
543:X
539:x
535:A
531:X
510:X
504:]
501:1
498:,
495:0
492:[
486:X
480:F
455:s
441:,
438:X
432:X
429::
426:s
416:X
412:X
404:r
390:A
384:X
381::
378:r
365:X
357:A
349:X
345:X
337:A
333:A
327:X
323:A
319:r
302:,
297:A
289:=
280:r
267:r
246:X
240:A
221:A
217:a
203:a
200:=
197:)
194:a
191:(
188:r
178:A
170:A
166:r
141:A
135:X
129:r
116:X
112:A
108:X
73:(
20:)
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