120:. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the
2956:
Since the closure of a connected subset is connected, the
Alexandroff extension of a noncompact connected space is connected. However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of a finite number
1708:
of equivalence classes of compactifications, any minimal element is equivalent to the
Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.
3603:
716:
and hence not dense. Thus a space can only admit a
Hausdorff one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one-point compactification the image of a neighborhood basis for
3743: – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βX
558:
208:
3508:
3455:
3669:
3235:
944:
3323:
2674:
686:
863:
472:
1816:
3402:
1499:
631:
340:
1524:
Recall from the above discussion that any
Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification. In particular, if
1706:
1662:
591:
252:
1251:
3095:
2815:
2793:
2758:
1989:
978:
410:
1090:
2951:
2917:
2716:
2573:
1747:
1064:
3349:
3255:
2878:
2486:
2389:
2315:
2154:
2134:
779:
759:
492:
378:
3913:
1140:
2443:
2416:
2342:
2231:
2181:
2114:
2087:
2067:
2040:
1930:
1883:
1436:
1394:
1360:
1322:
1292:
1201:
1167:
890:
3123:
278:
1021:
3170:
3150:
3069:
3049:
3029:
2997:
2975:
2736:
2633:
2613:
2593:
2506:
2463:
2362:
2295:
2275:
2251:
2201:
2013:
1950:
1903:
1856:
1836:
1767:
1630:
1610:
1586:
1562:
1542:
1110:
1041:
998:
815:
3513:
2827:
are defined as topological spaces that are the continuous image of the power of a one-point compactification of a discrete, locally compact
Hausdorff space.
129:
152:
gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the
Euclidean plane. The inverse stereographic projection
497:
3871:
17:
1749:
be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily
Hausdorff) of
155:
4074:
2508:; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.
3972:
3921:
3774:
3460:
3407:
3608:
3175:
3740:
899:
741:), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of
121:
4035:
4005:
3802:
3264:
2638:
1885:
is the same as the original topology. The last compatibility condition on the topology automatically implies that
3939:
3004:
The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is the
644:
3695:
3369:
97:
820:
3949:
1776:
415:
3375:
3866:
1465:
604:
3944:
283:
696:) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage
1678:
1635:
567:
210:
is an open, dense embedding into a compact
Hausdorff space obtained by adjoining the additional point
4069:
3710:
3126:
2883:
1952:
is not compact, so it cannot be closed in a compact space. Also, it is a fact that the inclusion map
213:
3749:
3734:
1214:
145:
3074:
2798:
2763:
2741:
1955:
957:
383:
2368:
1069:
2679:
2545:
1720:
1046:
3830:
3680:
3328:
3240:
2863:
2471:
2374:
2300:
2139:
2119:
764:
744:
477:
345:
1115:
4045:
3982:
3931:
3372:
with proper continuous maps as morphisms to the category whose objects are continuous maps
2421:
2394:
2320:
2209:
2159:
2092:
2072:
2045:
2018:
1908:
1861:
1414:
1372:
1338:
1300:
1270:
1179:
1145:
868:
70:
4053:
4015:
3892:
3851:
8:
3100:
2488:
one has to pick a suitable subfamily of the complements of all closed compact subsets of
257:
1003:
144:
A geometrically appealing example of one-point compactification is given by the inverse
3855:
3598:{\displaystyle f_{X}\colon X_{1}\rightarrow X_{2},\ f_{Y}\colon Y_{1}\rightarrow Y_{2}}
3155:
3135:
3054:
3034:
3014:
2982:
2978:
2960:
2721:
2618:
2598:
2578:
2491:
2448:
2347:
2280:
2260:
2236:
2186:
1998:
1935:
1888:
1841:
1821:
1752:
1615:
1595:
1571:
1547:
1527:
1095:
1026:
983:
800:
3906:
4031:
4001:
3968:
3917:
3901:
3859:
3821:
3798:
2818:
125:
51:
43:
3008:. This is different from the wedge of countably many circles, which is not compact.
4049:
4011:
3888:
3880:
3847:
3839:
3790:
3005:
2468:
Any topology intermediate between the two topologies above. For neighborhoods of
4041:
3978:
3964:
3927:
1672:
1405:
1397:
133:
109:
105:
3704: – in topology, the connected components of the “ideal boundary” of a space
4027:
3956:
3728:
3701:
2824:
1589:
3884:
3671:. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
2277:
defined above. Here we take the complements of all closed compact subsets of
4063:
3997:
3989:
3689:
3258:
2524:
1509:
is
Hausdorff, noncompact and locally compact. In this case it is called the
47:
3698: – Embedding a topological space into a compact space as a dense subset
3716:
67:
3722:
1565:
31:
3843:
1773:
in this context. So one wants to determine all possible ways to give
1211:(Willard, 19A). Sometimes the same name is used for the inclusion map
593:. This example already contains the key concepts of the general case.
3869:(1973), "Sequentially proper maps and a sequential compactification",
46:
by adjoining a single point in such a way that the resulting space is
3352:
3731: – Model of the extended complex plane plus a point at infinity
3826:"Über die Metrisation der im Kleinen kompakten topologischen Räume"
1992:
1439:
553:{\displaystyle S^{-1}(\mathbb {R} ^{2}\setminus K)\cup \{\infty \}}
35:
2523:
The one-point compactification of the set of positive integers is
3683: – compact Hausdorff group associated to a topological group
3365:
3825:
3775:"General topology – Non-Hausdorff one-point compactifications"
3737: – Particular mapping that projects a sphere onto a plane
1769:
obtained by adding a single point, which could also be called
1712:
203:{\displaystyle S^{-1}:\mathbb {R} ^{2}\hookrightarrow S^{2}}
139:
92:* consists of a single point, typically denoted ∞. The map
112:. For such spaces the Alexandroff extension is called the
58:
be a topological space. Then the
Alexandroff extension of
254:. Under the stereographic projection latitudinal circles
2831:
2517:
1255:
The properties below follow from the above discussion:
2880:
copies of the half-closed interval [0,1), that is, of
418:
412:
corresponds to the complements of closed planar disks
286:
3611:
3516:
3463:
3410:
3378:
3331:
3267:
3243:
3178:
3158:
3138:
3103:
3077:
3057:
3037:
3017:
2985:
2963:
2920:
2886:
2866:
2801:
2766:
2744:
2724:
2682:
2641:
2621:
2601:
2581:
2548:
2494:
2474:
2451:
2424:
2397:
2377:
2350:
2323:
2303:
2283:
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2239:
2212:
2189:
2162:
2142:
2122:
2095:
2075:
2048:
2021:
2001:
1958:
1938:
1911:
1891:
1864:
1844:
1824:
1779:
1755:
1723:
1681:
1638:
1618:
1598:
1574:
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1468:
1417:
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1217:
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1118:
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1006:
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960:
902:
871:
823:
803:
767:
747:
647:
607:
570:
500:
480:
386:
348:
342:. It follows that the deleted neighborhood basis of
260:
216:
158:
3745:
Pages displaying wikidata descriptions as a fallback
3706:
Pages displaying wikidata descriptions as a fallback
3685:
Pages displaying wikidata descriptions as a fallback
3261:. Recall that the definition of the smash product:
2853:. As above, the map can be given explicitly as an
3905:
3663:
3597:
3503:{\displaystyle c_{2}\colon X_{2}\rightarrow Y_{2}}
3502:
3450:{\displaystyle c_{1}\colon X_{1}\rightarrow Y_{1}}
3449:
3396:
3343:
3317:
3249:
3229:
3164:
3144:
3117:
3089:
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2991:
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2787:
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2175:
2148:
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2108:
2081:
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2007:
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1493:
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1388:
1354:
1316:
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1195:
1161:
1134:
1104:
1084:
1058:
1035:
1015:
992:
972:
938:
884:
857:
809:
773:
753:
680:
625:
585:
552:
486:
466:
404:
372:
334:
272:
246:
202:
3664:{\displaystyle f_{Y}\circ c_{1}=c_{2}\circ f_{X}}
3230:{\displaystyle (X\times Y)^{*}=X^{*}\wedge Y^{*}}
2860:The one-point compactification of the product of
1675:. Under the natural partial ordering on the set
1457:
700:is also locally compact Hausdorff. Moreover, if
4061:
939:{\displaystyle V=(X\setminus C)\cup \{\infty \}}
3318:{\displaystyle A\wedge B=(A\times B)/(A\vee B)}
2539:is a positive integer} with the order topology.
474:. More qualitatively, a neighborhood basis at
50:. It is named after the Russian mathematician
2857:-dimensional inverse stereographic projection.
3364:The Alexandroff extension can be viewed as a
2669:{\displaystyle f\colon \mathbb {N} ^{*}\to X}
27:Way to extend a non-compact topological space
2562:
2549:
2418:. This is the smallest topology that makes
1838:is dense in it and the subspace topology on
1805:
1799:
1651:
1645:
1079:
1073:
933:
927:
892:by taking as open sets all the open sets in
849:
843:
792:
654:
648:
547:
541:
148:. Recall that the stereographic projection
3820:
2317:. This is the largest topology that makes
681:{\displaystyle \{\infty \}=Y\setminus c(X)}
641:, with dense image and one-point remainder
3872:Journal of the London Mathematical Society
3355:, and again, / denotes the quotient space.
3937:
3900:
3713: – Real numbers with +∞ and −∞ added
2803:
2746:
2650:
1713:Non-Hausdorff one-point compactifications
1462:In particular, the Alexandroff extension
1092:will contain all except a compact subset
637:to a compact Hausdorff topological space
633:be an embedding from a topological space
573:
519:
177:
140:Example: inverse stereographic projection
3786:
3784:
2371:. Here we add a single neighborhood of
858:{\displaystyle X^{*}=X\cup \{\infty \},}
467:{\textstyle r\geq {\sqrt {(1+c)/(1-c)}}}
4021:
3988:
2116:is determined by the neighbourhoods of
1811:{\displaystyle X^{*}=X\cup \{\infty \}}
1632:is the Alexandroff compactification of
761:must be all sets obtained by adjoining
14:
4062:
3955:
3397:{\displaystyle c\colon X\rightarrow Y}
3125:, where the forward slash denotes the
2832:Compactifications of continuous spaces
1170:
564:ranges through the compact subsets of
380:given by the punctured spherical caps
3865:
3795:An Introduction to Algebraic Topology
3781:
2977:of copies of the interval (0,1) is a
3172:are locally compact Hausdorff, then
3071:, the one-point compactification of
2518:Compactifications of discrete spaces
1494:{\displaystyle c:X\rightarrow X^{*}}
626:{\displaystyle c:X\hookrightarrow Y}
335:{\textstyle r={\sqrt {(1+c)/(1-c)}}}
3725: – Basic concept in set theory
2512:
2233:that make it a compactification of
1501:is a Hausdorff compactification of
896:together with all sets of the form
24:
3692: – Type of mathematical space
2837:The one-point compactification of
2773:
2475:
2378:
2304:
2143:
2123:
1802:
1684:
1263:is continuous and open: it embeds
1076:
1050:
930:
846:
768:
748:
651:
544:
481:
217:
25:
4086:
3719: – Type of topological space
3404:and for which the morphisms from
3081:
2156:is necessarily the complement in
1701:{\displaystyle {\mathcal {C}}(X)}
1642:
1544:is a compact Hausdorff space and
964:
915:
663:
529:
2445:a one-point compactification of
2344:a one-point compactification of
1657:{\displaystyle X\setminus \{p\}}
586:{\displaystyle \mathbb {R} ^{2}}
42:is a way to extend a noncompact
3752: – A compactification of T
3359:
2912:{\displaystyle [0,1)^{\kappa }}
725:gives a neighborhood basis for
247:{\displaystyle \infty =(0,0,1)}
4075:Compactification (mathematics)
3940:"Aleksandrov compactification"
3767:
3696:Compactification (mathematics)
3582:
3540:
3487:
3434:
3388:
3370:category of topological spaces
3312:
3300:
3292:
3280:
3192:
3179:
2934:
2921:
2900:
2887:
2776:
2770:
2692:
2686:
2660:
2183:of a closed compact subset of
1968:
1736:
1724:
1695:
1689:
1478:
1458:The one-point compactification
1227:
921:
909:
675:
669:
617:
535:
514:
459:
447:
439:
427:
367:
349:
327:
315:
307:
295:
241:
223:
187:
84:* such that the complement of
13:
1:
3814:
3510:are pairs of continuous maps
2841:-dimensional Euclidean space
2257:The Alexandroff extension of
2069:must contain every member of
1818:a compact topology such that
1246:{\displaystyle c:X\to X^{*}.}
817:be a topological space. Put
596:
280:get mapped to planar circles
3090:{\displaystyle X\setminus C}
2810:{\displaystyle \mathbb {N} }
2788:{\displaystyle f(\infty )=a}
2753:{\displaystyle \mathbb {N} }
1984:{\displaystyle c:X\to X^{*}}
1515:Alexandroff compactification
1066:and thus any open cover of
973:{\displaystyle X\setminus C}
405:{\displaystyle c\leq z<1}
118:Alexandroff compactification
7:
3945:Encyclopedia of Mathematics
3806:(See Chapter 11 for proof.)
3741:Stone–Čech compactification
3674:
2527:to the space consisting of
2203:, as previously discussed.
1771:one-point compactifications
1085:{\displaystyle \{\infty \}}
1043:is an open neighborhood of
122:Stone–Čech compactification
62:is a certain compact space
10:
4091:
2946:{\displaystyle ^{\kappa }}
2711:{\displaystyle f(n)=a_{n}}
1511:one-point compactification
980:denotes the complement of
114:one-point compactification
18:One-point compactification
4022:Willard, Stephen (1970),
3938:Fedorchuk, V.V. (2001) ,
3711:Extended real number line
2635:, if and only if the map
2568:{\displaystyle \{a_{n}\}}
2391:, namely the whole space
1742:{\displaystyle (X,\tau )}
950:is closed and compact in
793:The Alexandroff extension
789:with compact complement.
494:is furnished by the sets
3760:
3750:Wallman compactification
3735:Stereographic projection
1059:{\displaystyle \infty ,}
146:stereographic projection
3914:Helderman Verlag Berlin
3885:10.1112/jlms/s2-7.3.515
3797:(1988) Springer-Verlag
3344:{\displaystyle A\vee B}
3250:{\displaystyle \wedge }
2873:{\displaystyle \kappa }
2845:is homeomorphic to the
2575:in a topological space
2481:{\displaystyle \infty }
2384:{\displaystyle \infty }
2369:open extension topology
2310:{\displaystyle \infty }
2149:{\displaystyle \infty }
2136:. Any neighborhood of
2129:{\displaystyle \infty }
774:{\displaystyle \infty }
754:{\displaystyle \infty }
487:{\displaystyle \infty }
373:{\displaystyle (0,0,1)}
3665:
3599:
3504:
3451:
3398:
3345:
3319:
3251:
3231:
3166:
3146:
3119:
3091:
3065:
3045:
3031:compact Hausdorff and
3025:
2993:
2971:
2947:
2913:
2874:
2811:
2789:
2754:
2732:
2712:
2670:
2629:
2609:
2589:
2569:
2502:
2482:
2459:
2439:
2412:
2385:
2358:
2338:
2311:
2291:
2271:
2247:
2227:
2197:
2177:
2150:
2130:
2110:
2083:
2063:
2036:
2009:
1985:
1946:
1926:
1899:
1879:
1852:
1832:
1812:
1763:
1743:
1702:
1658:
1626:
1606:
1582:
1558:
1538:
1495:
1432:
1390:
1356:
1318:
1288:
1247:
1197:
1163:
1136:
1135:{\displaystyle X^{*},}
1106:
1086:
1060:
1037:
1017:
994:
974:
940:
886:
859:
811:
775:
755:
682:
627:
587:
554:
488:
468:
406:
374:
336:
274:
248:
204:
54:. More precisely, let
3831:Mathematische Annalen
3822:Alexandroff, Pavel S.
3681:Bohr compactification
3666:
3600:
3505:
3452:
3399:
3346:
3320:
3252:
3232:
3167:
3147:
3120:
3092:
3066:
3051:any closed subset of
3046:
3026:
2994:
2972:
2948:
2914:
2875:
2812:
2790:
2755:
2733:
2713:
2671:
2630:
2610:
2595:converges to a point
2590:
2570:
2503:
2483:
2460:
2440:
2438:{\displaystyle X^{*}}
2413:
2411:{\displaystyle X^{*}}
2386:
2359:
2339:
2337:{\displaystyle X^{*}}
2312:
2292:
2272:
2248:
2228:
2226:{\displaystyle X^{*}}
2198:
2178:
2176:{\displaystyle X^{*}}
2151:
2131:
2111:
2109:{\displaystyle X^{*}}
2089:. So the topology on
2084:
2082:{\displaystyle \tau }
2064:
2062:{\displaystyle X^{*}}
2037:
2035:{\displaystyle X^{*}}
2010:
1986:
1947:
1927:
1925:{\displaystyle X^{*}}
1900:
1880:
1878:{\displaystyle X^{*}}
1853:
1833:
1813:
1764:
1744:
1703:
1659:
1627:
1607:
1583:
1559:
1539:
1496:
1433:
1431:{\displaystyle X^{*}}
1391:
1389:{\displaystyle X^{*}}
1357:
1355:{\displaystyle X^{*}}
1319:
1317:{\displaystyle X^{*}}
1289:
1287:{\displaystyle X^{*}}
1267:as an open subset of
1248:
1205:Alexandroff extension
1198:
1196:{\displaystyle X^{*}}
1164:
1162:{\displaystyle X^{*}}
1137:
1107:
1087:
1061:
1038:
1018:
995:
975:
941:
887:
885:{\displaystyle X^{*}}
860:
812:
776:
756:
712:) would be closed in
683:
628:
588:
555:
489:
469:
407:
375:
337:
275:
249:
205:
130:provides an embedding
124:which exists for any
40:Alexandroff extension
3963:, Berlin, New York:
3609:
3514:
3461:
3408:
3376:
3329:
3265:
3241:
3176:
3156:
3136:
3101:
3075:
3055:
3035:
3015:
2983:
2961:
2918:
2884:
2864:
2799:
2795:is continuous. Here
2764:
2742:
2722:
2680:
2639:
2619:
2599:
2579:
2546:
2492:
2472:
2449:
2422:
2395:
2375:
2348:
2321:
2301:
2297:as neighborhoods of
2281:
2261:
2237:
2210:
2187:
2160:
2140:
2120:
2093:
2073:
2046:
2042:and the topology on
2019:
1999:
1995:embedding, that is,
1956:
1936:
1909:
1889:
1862:
1842:
1822:
1777:
1753:
1721:
1679:
1636:
1616:
1596:
1572:
1548:
1528:
1466:
1415:
1373:
1339:
1301:
1271:
1215:
1180:
1146:
1116:
1096:
1070:
1047:
1027:
1004:
984:
958:
900:
869:
821:
801:
765:
745:
645:
605:
568:
498:
478:
416:
384:
346:
284:
258:
214:
156:
3118:{\displaystyle X/C}
781:to the image under
273:{\displaystyle z=c}
66:* together with an
3902:Engelking, Ryszard
3844:10.1007/BF01448011
3756:topological spaces
3661:
3595:
3500:
3447:
3394:
3341:
3315:
3247:
3227:
3162:
3142:
3115:
3087:
3061:
3041:
3021:
2989:
2967:
2943:
2909:
2870:
2807:
2785:
2750:
2728:
2708:
2666:
2625:
2605:
2585:
2565:
2498:
2478:
2455:
2435:
2408:
2381:
2354:
2334:
2307:
2287:
2267:
2243:
2223:
2206:The topologies on
2193:
2173:
2146:
2126:
2106:
2079:
2059:
2032:
2005:
1991:is necessarily an
1981:
1942:
1922:
1895:
1875:
1848:
1828:
1808:
1759:
1739:
1698:
1671:be any noncompact
1654:
1622:
1602:
1578:
1554:
1534:
1491:
1428:
1386:
1352:
1314:
1284:
1243:
1193:
1159:
1132:
1102:
1082:
1056:
1033:
1016:{\displaystyle X.}
1013:
990:
970:
936:
882:
855:
807:
771:
751:
704:were compact then
678:
623:
583:
550:
484:
464:
402:
370:
332:
270:
244:
200:
3974:978-0-387-90125-1
3923:978-0-201-08707-9
3558:
3165:{\displaystyle Y}
3145:{\displaystyle X}
3064:{\displaystyle X}
3044:{\displaystyle C}
3024:{\displaystyle X}
2992:{\displaystyle n}
2970:{\displaystyle n}
2819:discrete topology
2731:{\displaystyle n}
2628:{\displaystyle X}
2608:{\displaystyle a}
2588:{\displaystyle X}
2501:{\displaystyle X}
2458:{\displaystyle X}
2357:{\displaystyle X}
2290:{\displaystyle X}
2270:{\displaystyle X}
2246:{\displaystyle X}
2196:{\displaystyle X}
2008:{\displaystyle X}
1945:{\displaystyle X}
1898:{\displaystyle X}
1851:{\displaystyle X}
1831:{\displaystyle X}
1762:{\displaystyle X}
1625:{\displaystyle X}
1605:{\displaystyle X}
1581:{\displaystyle X}
1557:{\displaystyle p}
1537:{\displaystyle X}
1404:is Hausdorff and
1105:{\displaystyle C}
1036:{\displaystyle V}
993:{\displaystyle C}
810:{\displaystyle X}
462:
330:
126:topological space
52:Pavel Alexandroff
44:topological space
16:(Redirected from
4082:
4070:General topology
4056:
4024:General Topology
4018:
3996:(2nd ed.),
3985:
3961:General Topology
3952:
3934:
3911:
3908:General Topology
3895:
3862:
3838:(3–4): 294–301,
3808:
3791:Joseph J. Rotman
3788:
3779:
3778:
3771:
3746:
3707:
3686:
3670:
3668:
3667:
3662:
3660:
3659:
3647:
3646:
3634:
3633:
3621:
3620:
3604:
3602:
3601:
3596:
3594:
3593:
3581:
3580:
3568:
3567:
3556:
3552:
3551:
3539:
3538:
3526:
3525:
3509:
3507:
3506:
3501:
3499:
3498:
3486:
3485:
3473:
3472:
3456:
3454:
3453:
3448:
3446:
3445:
3433:
3432:
3420:
3419:
3403:
3401:
3400:
3395:
3350:
3348:
3347:
3342:
3324:
3322:
3321:
3316:
3299:
3256:
3254:
3253:
3248:
3236:
3234:
3233:
3228:
3226:
3225:
3213:
3212:
3200:
3199:
3171:
3169:
3168:
3163:
3151:
3149:
3148:
3143:
3124:
3122:
3121:
3116:
3111:
3096:
3094:
3093:
3088:
3070:
3068:
3067:
3062:
3050:
3048:
3047:
3042:
3030:
3028:
3027:
3022:
3006:Hawaiian earring
2998:
2996:
2995:
2990:
2976:
2974:
2973:
2968:
2952:
2950:
2949:
2944:
2942:
2941:
2916:
2915:
2910:
2908:
2907:
2879:
2877:
2876:
2871:
2816:
2814:
2813:
2808:
2806:
2794:
2792:
2791:
2786:
2759:
2757:
2756:
2751:
2749:
2737:
2735:
2734:
2729:
2717:
2715:
2714:
2709:
2707:
2706:
2675:
2673:
2672:
2667:
2659:
2658:
2653:
2634:
2632:
2631:
2626:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2574:
2572:
2571:
2566:
2561:
2560:
2513:Further examples
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2464:
2462:
2461:
2456:
2444:
2442:
2441:
2436:
2434:
2433:
2417:
2415:
2414:
2409:
2407:
2406:
2390:
2388:
2387:
2382:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2333:
2332:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2253:are as follows:
2252:
2250:
2249:
2244:
2232:
2230:
2229:
2224:
2222:
2221:
2202:
2200:
2199:
2194:
2182:
2180:
2179:
2174:
2172:
2171:
2155:
2153:
2152:
2147:
2135:
2133:
2132:
2127:
2115:
2113:
2112:
2107:
2105:
2104:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2058:
2057:
2041:
2039:
2038:
2033:
2031:
2030:
2015:must be open in
2014:
2012:
2011:
2006:
1990:
1988:
1987:
1982:
1980:
1979:
1951:
1949:
1948:
1943:
1931:
1929:
1928:
1923:
1921:
1920:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1874:
1873:
1857:
1855:
1854:
1849:
1837:
1835:
1834:
1829:
1817:
1815:
1814:
1809:
1789:
1788:
1768:
1766:
1765:
1760:
1748:
1746:
1745:
1740:
1707:
1705:
1704:
1699:
1688:
1687:
1663:
1661:
1660:
1655:
1631:
1629:
1628:
1623:
1611:
1609:
1608:
1603:
1587:
1585:
1584:
1579:
1563:
1561:
1560:
1555:
1543:
1541:
1540:
1535:
1500:
1498:
1497:
1492:
1490:
1489:
1437:
1435:
1434:
1429:
1427:
1426:
1395:
1393:
1392:
1387:
1385:
1384:
1361:
1359:
1358:
1353:
1351:
1350:
1323:
1321:
1320:
1315:
1313:
1312:
1293:
1291:
1290:
1285:
1283:
1282:
1252:
1250:
1249:
1244:
1239:
1238:
1202:
1200:
1199:
1194:
1192:
1191:
1173:, p. 150).
1168:
1166:
1165:
1160:
1158:
1157:
1141:
1139:
1138:
1133:
1128:
1127:
1111:
1109:
1108:
1103:
1091:
1089:
1088:
1083:
1065:
1063:
1062:
1057:
1042:
1040:
1039:
1034:
1022:
1020:
1019:
1014:
999:
997:
996:
991:
979:
977:
976:
971:
945:
943:
942:
937:
891:
889:
888:
883:
881:
880:
864:
862:
861:
856:
833:
832:
816:
814:
813:
808:
780:
778:
777:
772:
760:
758:
757:
752:
687:
685:
684:
679:
632:
630:
629:
624:
592:
590:
589:
584:
582:
581:
576:
559:
557:
556:
551:
528:
527:
522:
513:
512:
493:
491:
490:
485:
473:
471:
470:
465:
463:
446:
426:
411:
409:
408:
403:
379:
377:
376:
371:
341:
339:
338:
333:
331:
314:
294:
279:
277:
276:
271:
253:
251:
250:
245:
209:
207:
206:
201:
199:
198:
186:
185:
180:
171:
170:
134:Tychonoff spaces
98:compactification
21:
4090:
4089:
4085:
4084:
4083:
4081:
4080:
4079:
4060:
4059:
4038:
4008:
3975:
3965:Springer-Verlag
3957:Kelley, John L.
3924:
3817:
3812:
3811:
3789:
3782:
3773:
3772:
3768:
3763:
3755:
3744:
3705:
3684:
3677:
3655:
3651:
3642:
3638:
3629:
3625:
3616:
3612:
3610:
3607:
3606:
3589:
3585:
3576:
3572:
3563:
3559:
3547:
3543:
3534:
3530:
3521:
3517:
3515:
3512:
3511:
3494:
3490:
3481:
3477:
3468:
3464:
3462:
3459:
3458:
3441:
3437:
3428:
3424:
3415:
3411:
3409:
3406:
3405:
3377:
3374:
3373:
3362:
3330:
3327:
3326:
3295:
3266:
3263:
3262:
3242:
3239:
3238:
3221:
3217:
3208:
3204:
3195:
3191:
3177:
3174:
3173:
3157:
3154:
3153:
3137:
3134:
3133:
3107:
3102:
3099:
3098:
3076:
3073:
3072:
3056:
3053:
3052:
3036:
3033:
3032:
3016:
3013:
3012:
2984:
2981:
2980:
2962:
2959:
2958:
2937:
2933:
2919:
2903:
2899:
2885:
2882:
2881:
2865:
2862:
2861:
2834:
2825:Polyadic spaces
2802:
2800:
2797:
2796:
2765:
2762:
2761:
2745:
2743:
2740:
2739:
2723:
2720:
2719:
2702:
2698:
2681:
2678:
2677:
2654:
2649:
2648:
2640:
2637:
2636:
2620:
2617:
2616:
2600:
2597:
2596:
2580:
2577:
2576:
2556:
2552:
2547:
2544:
2543:
2520:
2515:
2493:
2490:
2489:
2473:
2470:
2469:
2450:
2447:
2446:
2429:
2425:
2423:
2420:
2419:
2402:
2398:
2396:
2393:
2392:
2376:
2373:
2372:
2349:
2346:
2345:
2328:
2324:
2322:
2319:
2318:
2302:
2299:
2298:
2282:
2279:
2278:
2262:
2259:
2258:
2238:
2235:
2234:
2217:
2213:
2211:
2208:
2207:
2188:
2185:
2184:
2167:
2163:
2161:
2158:
2157:
2141:
2138:
2137:
2121:
2118:
2117:
2100:
2096:
2094:
2091:
2090:
2074:
2071:
2070:
2053:
2049:
2047:
2044:
2043:
2026:
2022:
2020:
2017:
2016:
2000:
1997:
1996:
1975:
1971:
1957:
1954:
1953:
1937:
1934:
1933:
1916:
1912:
1910:
1907:
1906:
1890:
1887:
1886:
1869:
1865:
1863:
1860:
1859:
1843:
1840:
1839:
1823:
1820:
1819:
1784:
1780:
1778:
1775:
1774:
1754:
1751:
1750:
1722:
1719:
1718:
1715:
1683:
1682:
1680:
1677:
1676:
1673:Tychonoff space
1637:
1634:
1633:
1617:
1614:
1613:
1597:
1594:
1593:
1573:
1570:
1569:
1549:
1546:
1545:
1529:
1526:
1525:
1505:if and only if
1485:
1481:
1467:
1464:
1463:
1460:
1452:
1445:if and only if
1443:
1422:
1418:
1416:
1413:
1412:
1406:locally compact
1400:if and only if
1380:
1376:
1374:
1371:
1370:
1346:
1342:
1340:
1337:
1336:
1308:
1304:
1302:
1299:
1298:
1278:
1274:
1272:
1269:
1268:
1234:
1230:
1216:
1213:
1212:
1187:
1183:
1181:
1178:
1177:
1153:
1149:
1147:
1144:
1143:
1123:
1119:
1117:
1114:
1113:
1097:
1094:
1093:
1071:
1068:
1067:
1048:
1045:
1044:
1028:
1025:
1024:
1005:
1002:
1001:
985:
982:
981:
959:
956:
955:
901:
898:
897:
876:
872:
870:
867:
866:
865:and topologize
828:
824:
822:
819:
818:
802:
799:
798:
795:
785:of a subset of
766:
763:
762:
746:
743:
742:
646:
643:
642:
606:
603:
602:
599:
577:
572:
571:
569:
566:
565:
523:
518:
517:
505:
501:
499:
496:
495:
479:
476:
475:
442:
425:
417:
414:
413:
385:
382:
381:
347:
344:
343:
310:
293:
285:
282:
281:
259:
256:
255:
215:
212:
211:
194:
190:
181:
176:
175:
163:
159:
157:
154:
153:
142:
110:Hausdorff space
106:locally compact
100:if and only if
96:is a Hausdorff
28:
23:
22:
15:
12:
11:
5:
4088:
4078:
4077:
4072:
4058:
4057:
4036:
4028:Addison-Wesley
4019:
4006:
3990:Munkres, James
3986:
3973:
3953:
3935:
3922:
3897:
3896:
3879:(3): 515–522,
3863:
3816:
3813:
3810:
3809:
3780:
3765:
3764:
3762:
3759:
3758:
3757:
3753:
3747:
3738:
3732:
3729:Riemann sphere
3726:
3720:
3714:
3708:
3702:End (topology)
3699:
3693:
3687:
3676:
3673:
3658:
3654:
3650:
3645:
3641:
3637:
3632:
3628:
3624:
3619:
3615:
3592:
3588:
3584:
3579:
3575:
3571:
3566:
3562:
3555:
3550:
3546:
3542:
3537:
3533:
3529:
3524:
3520:
3497:
3493:
3489:
3484:
3480:
3476:
3471:
3467:
3444:
3440:
3436:
3431:
3427:
3423:
3418:
3414:
3393:
3390:
3387:
3384:
3381:
3361:
3358:
3357:
3356:
3340:
3337:
3334:
3314:
3311:
3308:
3305:
3302:
3298:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3246:
3224:
3220:
3216:
3211:
3207:
3203:
3198:
3194:
3190:
3187:
3184:
3181:
3161:
3141:
3130:
3127:quotient space
3114:
3110:
3106:
3086:
3083:
3080:
3060:
3040:
3020:
3009:
3002:
2988:
2966:
2954:
2940:
2936:
2932:
2929:
2926:
2923:
2906:
2902:
2898:
2895:
2892:
2889:
2869:
2858:
2833:
2830:
2829:
2828:
2822:
2805:
2784:
2781:
2778:
2775:
2772:
2769:
2748:
2727:
2705:
2701:
2697:
2694:
2691:
2688:
2685:
2665:
2662:
2657:
2652:
2647:
2644:
2624:
2604:
2584:
2564:
2559:
2555:
2551:
2540:
2519:
2516:
2514:
2511:
2510:
2509:
2497:
2477:
2466:
2454:
2432:
2428:
2405:
2401:
2380:
2365:
2353:
2331:
2327:
2306:
2286:
2266:
2242:
2220:
2216:
2192:
2170:
2166:
2145:
2125:
2103:
2099:
2078:
2056:
2052:
2029:
2025:
2004:
1978:
1974:
1970:
1967:
1964:
1961:
1941:
1919:
1915:
1894:
1872:
1868:
1847:
1827:
1807:
1804:
1801:
1798:
1795:
1792:
1787:
1783:
1758:
1738:
1735:
1732:
1729:
1726:
1714:
1711:
1697:
1694:
1691:
1686:
1653:
1650:
1647:
1644:
1641:
1621:
1601:
1590:isolated point
1577:
1553:
1533:
1488:
1484:
1480:
1477:
1474:
1471:
1459:
1456:
1455:
1454:
1450:
1441:
1425:
1421:
1409:
1383:
1379:
1367:
1366:is noncompact.
1349:
1345:
1335:) is dense in
1325:
1311:
1307:
1295:
1281:
1277:
1242:
1237:
1233:
1229:
1226:
1223:
1220:
1203:is called the
1190:
1186:
1156:
1152:
1142:implying that
1131:
1126:
1122:
1101:
1081:
1078:
1075:
1055:
1052:
1032:
1012:
1009:
989:
969:
966:
963:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
879:
875:
854:
851:
848:
845:
842:
839:
836:
831:
827:
806:
794:
791:
770:
750:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
622:
619:
616:
613:
610:
598:
595:
580:
575:
549:
546:
543:
540:
537:
534:
531:
526:
521:
516:
511:
508:
504:
483:
461:
458:
455:
452:
449:
445:
441:
438:
435:
432:
429:
424:
421:
401:
398:
395:
392:
389:
369:
366:
363:
360:
357:
354:
351:
329:
326:
323:
320:
317:
313:
309:
306:
303:
300:
297:
292:
289:
269:
266:
263:
243:
240:
237:
234:
231:
228:
225:
222:
219:
197:
193:
189:
184:
179:
174:
169:
166:
162:
141:
138:
26:
9:
6:
4:
3:
2:
4087:
4076:
4073:
4071:
4068:
4067:
4065:
4055:
4051:
4047:
4043:
4039:
4037:3-88538-006-4
4033:
4029:
4025:
4020:
4017:
4013:
4009:
4007:0-13-181629-2
4003:
3999:
3998:Prentice Hall
3995:
3991:
3987:
3984:
3980:
3976:
3970:
3966:
3962:
3958:
3954:
3951:
3947:
3946:
3941:
3936:
3933:
3929:
3925:
3919:
3915:
3910:
3909:
3903:
3899:
3898:
3894:
3890:
3886:
3882:
3878:
3874:
3873:
3868:
3867:Brown, Ronald
3864:
3861:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3832:
3827:
3823:
3819:
3818:
3807:
3804:
3803:0-387-96678-1
3800:
3796:
3792:
3787:
3785:
3776:
3770:
3766:
3751:
3748:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3703:
3700:
3697:
3694:
3691:
3690:Compact space
3688:
3682:
3679:
3678:
3672:
3656:
3652:
3648:
3643:
3639:
3635:
3630:
3626:
3622:
3617:
3613:
3590:
3586:
3577:
3573:
3569:
3564:
3560:
3553:
3548:
3544:
3535:
3531:
3527:
3522:
3518:
3495:
3491:
3482:
3478:
3474:
3469:
3465:
3442:
3438:
3429:
3425:
3421:
3416:
3412:
3391:
3385:
3382:
3379:
3371:
3367:
3354:
3338:
3335:
3332:
3309:
3306:
3303:
3296:
3289:
3286:
3283:
3277:
3274:
3271:
3268:
3260:
3259:smash product
3244:
3222:
3218:
3214:
3209:
3205:
3201:
3196:
3188:
3185:
3182:
3159:
3139:
3131:
3128:
3112:
3108:
3104:
3084:
3078:
3058:
3038:
3018:
3010:
3007:
3003:
3000:
2986:
2964:
2955:
2938:
2930:
2927:
2924:
2904:
2896:
2893:
2890:
2867:
2859:
2856:
2852:
2848:
2844:
2840:
2836:
2835:
2826:
2823:
2820:
2782:
2779:
2767:
2725:
2703:
2699:
2695:
2689:
2683:
2663:
2655:
2645:
2642:
2622:
2602:
2582:
2557:
2553:
2541:
2538:
2534:
2530:
2526:
2522:
2521:
2495:
2467:
2452:
2430:
2426:
2403:
2399:
2370:
2366:
2351:
2329:
2325:
2284:
2264:
2256:
2255:
2254:
2240:
2218:
2214:
2204:
2190:
2168:
2164:
2101:
2097:
2076:
2054:
2050:
2027:
2023:
2002:
1994:
1976:
1972:
1965:
1962:
1959:
1939:
1917:
1913:
1892:
1870:
1866:
1858:induced from
1845:
1825:
1796:
1793:
1790:
1785:
1781:
1772:
1756:
1733:
1730:
1727:
1710:
1692:
1674:
1670:
1665:
1648:
1639:
1619:
1599:
1591:
1588:(i.e. not an
1575:
1567:
1551:
1531:
1522:
1520:
1516:
1512:
1508:
1504:
1486:
1482:
1475:
1472:
1469:
1448:
1444:
1423:
1419:
1410:
1407:
1403:
1399:
1381:
1377:
1368:
1365:
1347:
1343:
1334:
1330:
1326:
1309:
1305:
1296:
1279:
1275:
1266:
1262:
1258:
1257:
1256:
1253:
1240:
1235:
1231:
1224:
1221:
1218:
1210:
1206:
1188:
1184:
1174:
1172:
1154:
1150:
1129:
1124:
1120:
1099:
1053:
1030:
1010:
1007:
987:
967:
961:
953:
949:
924:
918:
912:
906:
903:
895:
877:
873:
852:
840:
837:
834:
829:
825:
804:
790:
788:
784:
740:
736:
732:
728:
724:
720:
715:
711:
707:
703:
699:
695:
691:
672:
666:
660:
657:
640:
636:
620:
614:
611:
608:
594:
578:
563:
538:
532:
524:
509:
506:
502:
456:
453:
450:
443:
436:
433:
430:
422:
419:
399:
396:
393:
390:
387:
364:
361:
358:
355:
352:
324:
321:
318:
311:
304:
301:
298:
290:
287:
267:
264:
261:
238:
235:
232:
229:
226:
220:
195:
191:
182:
172:
167:
164:
160:
151:
147:
137:
135:
131:
127:
123:
119:
115:
111:
108:, noncompact
107:
103:
99:
95:
91:
87:
83:
80: →
79:
76: :
75:
72:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4023:
3993:
3960:
3943:
3907:
3876:
3875:, Series 2,
3870:
3835:
3829:
3805:
3794:
3769:
3717:Normal space
3363:
3360:As a functor
2854:
2850:
2846:
2842:
2838:
2536:
2532:
2528:
2525:homeomorphic
2205:
1905:is dense in
1770:
1716:
1668:
1666:
1523:
1518:
1514:
1510:
1506:
1502:
1461:
1446:
1401:
1363:
1332:
1328:
1264:
1260:
1254:
1208:
1204:
1175:
1169:is compact (
951:
947:
893:
796:
786:
782:
738:
734:
730:
726:
722:
718:
713:
709:
705:
701:
697:
693:
689:
638:
634:
600:
561:
149:
143:
132:exactly for
117:
113:
101:
93:
89:
85:
81:
77:
73:
63:
59:
55:
39:
32:mathematical
29:
3723:Pointed set
2542:A sequence
2531:= {0} U {1/
1566:limit point
1324:is compact.
1171:Kelley 1975
4064:Categories
4054:0205.26601
4016:0951.54001
3893:0269.54015
3852:50.0128.04
3815:References
3605:such that
1932:, because
1411:The space
1369:The space
1327:The image
1297:The space
1176:The space
1023:Note that
597:Motivation
3950:EMS Press
3860:121699713
3649:∘
3623:∘
3583:→
3570::
3541:→
3528::
3488:→
3475::
3435:→
3422::
3389:→
3383::
3368:from the
3353:wedge sum
3336:∨
3307:∨
3287:×
3272:∧
3245:∧
3223:∗
3215:∧
3210:∗
3197:∗
3186:×
3082:∖
2979:wedge of
2939:κ
2905:κ
2868:κ
2774:∞
2676:given by
2661:→
2656:∗
2646::
2476:∞
2431:∗
2404:∗
2379:∞
2330:∗
2305:∞
2219:∗
2169:∗
2144:∞
2124:∞
2102:∗
2077:τ
2055:∗
2028:∗
1977:∗
1969:→
1918:∗
1871:∗
1803:∞
1797:∪
1786:∗
1734:τ
1643:∖
1487:∗
1479:→
1424:∗
1398:Hausdorff
1382:∗
1348:∗
1310:∗
1280:∗
1236:∗
1228:→
1189:∗
1155:∗
1125:∗
1077:∞
1051:∞
965:∖
954:. Here,
931:∞
925:∪
916:∖
878:∗
847:∞
841:∪
830:∗
769:∞
749:∞
664:∖
652:∞
618:↪
545:∞
539:∪
530:∖
507:−
482:∞
454:−
423:≥
391:≤
322:−
218:∞
188:↪
165:−
71:embedding
34:field of
3994:Topology
3992:(1999),
3959:(1975),
3904:(1989),
3824:(1924),
3675:See also
2849:-sphere
2817:has the
1259:The map
688:. Then
36:topology
4046:0264581
3983:0370454
3932:1039321
3366:functor
3351:is the
3257:is the
2999:circles
48:compact
30:In the
4052:
4044:
4034:
4014:
4004:
3981:
3971:
3930:
3920:
3891:
3858:
3850:
3801:
3557:
3325:where
3237:where
3011:Given
946:where
38:, the
3856:S2CID
3761:Notes
1564:is a
1362:, if
733:) in
128:(but
104:is a
4032:ISBN
4002:ISBN
3969:ISBN
3918:ISBN
3799:ISBN
3152:and
2760:and
2718:for
2367:The
1993:open
1717:Let
1667:Let
1449:is T
797:Let
601:Let
397:<
68:open
4050:Zbl
4012:Zbl
3889:Zbl
3881:doi
3848:JFM
3840:doi
3457:to
3132:If
3097:is
2738:in
2615:in
1612:),
1592:of
1568:of
1517:of
1513:or
1438:is
1396:is
1207:of
1112:of
1000:in
721:in
560:as
136:).
116:or
88:in
4066::
4048:,
4042:MR
4040:,
4030:,
4026:,
4010:,
4000:,
3979:MR
3977:,
3967:,
3948:,
3942:,
3928:MR
3926:,
3916:,
3912:,
3887:,
3854:,
3846:,
3836:92
3834:,
3828:,
3793:,
3783:^
2535:|
1664:.
1521:.
3883::
3877:7
3842::
3777:.
3754:1
3657:X
3653:f
3644:2
3640:c
3636:=
3631:1
3627:c
3618:Y
3614:f
3591:2
3587:Y
3578:1
3574:Y
3565:Y
3561:f
3554:,
3549:2
3545:X
3536:1
3532:X
3523:X
3519:f
3496:2
3492:Y
3483:2
3479:X
3470:2
3466:c
3443:1
3439:Y
3430:1
3426:X
3417:1
3413:c
3392:Y
3386:X
3380:c
3339:B
3333:A
3313:)
3310:B
3304:A
3301:(
3297:/
3293:)
3290:B
3284:A
3281:(
3278:=
3275:B
3269:A
3219:Y
3206:X
3202:=
3193:)
3189:Y
3183:X
3180:(
3160:Y
3140:X
3129:.
3113:C
3109:/
3105:X
3085:C
3079:X
3059:X
3039:C
3019:X
3001:.
2987:n
2965:n
2953:.
2935:]
2931:1
2928:,
2925:0
2922:[
2901:)
2897:1
2894:,
2891:0
2888:[
2855:n
2851:S
2847:n
2843:R
2839:n
2821:.
2804:N
2783:a
2780:=
2777:)
2771:(
2768:f
2747:N
2726:n
2704:n
2700:a
2696:=
2693:)
2690:n
2687:(
2684:f
2664:X
2651:N
2643:f
2623:X
2603:a
2583:X
2563:}
2558:n
2554:a
2550:{
2537:n
2533:n
2529:K
2496:X
2465:.
2453:X
2427:X
2400:X
2364:.
2352:X
2326:X
2285:X
2265:X
2241:X
2215:X
2191:X
2165:X
2098:X
2051:X
2024:X
2003:X
1973:X
1966:X
1963::
1960:c
1940:X
1914:X
1893:X
1867:X
1846:X
1826:X
1806:}
1800:{
1794:X
1791:=
1782:X
1757:X
1737:)
1731:,
1728:X
1725:(
1696:)
1693:X
1690:(
1685:C
1669:X
1652:}
1649:p
1646:{
1640:X
1620:X
1600:X
1576:X
1552:p
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1473::
1470:c
1453:.
1451:1
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1442:1
1440:T
1420:X
1408:.
1402:X
1378:X
1364:X
1344:X
1333:X
1331:(
1329:c
1306:X
1294:.
1276:X
1265:X
1261:c
1241:.
1232:X
1225:X
1222::
1219:c
1209:X
1185:X
1151:X
1130:,
1121:X
1100:C
1080:}
1074:{
1054:,
1031:V
1011:.
1008:X
988:C
968:C
962:X
952:X
948:C
934:}
928:{
922:)
919:C
913:X
910:(
907:=
904:V
894:X
874:X
853:,
850:}
844:{
838:X
835:=
826:X
805:X
787:X
783:c
739:X
737:(
735:c
731:x
729:(
727:c
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719:x
714:Y
710:X
708:(
706:c
702:X
698:X
694:X
692:(
690:c
676:)
673:X
670:(
667:c
661:Y
658:=
655:}
649:{
639:Y
635:X
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615:X
612::
609:c
579:2
574:R
562:K
548:}
542:{
536:)
533:K
525:2
520:R
515:(
510:1
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451:1
448:(
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437:c
434:+
431:1
428:(
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365:1
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359:0
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353:0
350:(
328:)
325:c
319:1
316:(
312:/
308:)
305:c
302:+
299:1
296:(
291:=
288:r
268:c
265:=
262:z
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239:1
236:,
233:0
230:,
227:0
224:(
221:=
196:2
192:S
183:2
178:R
173::
168:1
161:S
150:S
102:X
94:c
90:X
86:X
82:X
78:X
74:c
64:X
60:X
56:X
20:)
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