2773:
3789:
1355:
4170:
is to say that it is equivalent to the statement that the
Cartesian product of a collection of non-empty sets is non-empty. The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a
3994:
470:
4178:
on compact sets is a more complex and subtle example of a statement that requires the axiom of choice and is equivalent to it in its most general formulation, and shows why the product topology may be considered the more useful topology to put on a
Cartesian product.
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In particular, for a finite product (in particular, for the product of two topological spaces), the set of all
Cartesian products between one basis element from each
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with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a
3784:{\displaystyle {\operatorname {Cl} _{X}}{\Bigl (}\prod _{i\in I}S_{i}{\Bigr )}=\prod _{i\in I}{\bigl (}{\operatorname {Cl} _{X_{i}}}S_{i}{\bigr )}.}
1350:{\displaystyle \left\{p_{i}^{-1}\left(U_{i}\right)\mathbin {\big \vert } i\in I{\text{ and }}U_{i}\subseteq X_{i}{\text{ is open in }}X_{i}\right\}}
209:
1924:
4171:
representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.
1995:
3989:{\displaystyle \left\{x=\left(x_{i}\right)_{i\in I}\in X\mathbin {\big \vert } x_{i}=z_{i}{\text{ for all but finitely many }}i\right\}}
4191: – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology
2068:
1625:
4325:
3220:
1801:
465:{\displaystyle {\begin{aligned}p_{i}:\ \prod _{j\in I}X_{j}&\to X_{i},\\(x_{j})_{j\in I}&\mapsto x_{i}.\\\end{aligned}}}
2191:
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be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact
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1421:
1097:
544:
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4374:
2783:
4206: – Coarsest topology making certain functions continuous - Sometimes called the projective limit topology
17:
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660:. The open sets in the product topology are arbitrary unions (finite or infinite) of sets of the form
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3029:. This means that any open subset of the product space remains open when projected down to the
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The axiom of choice occurs again in the study of (topological) product spaces; for example,
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65:. This topology differs from another, perhaps more natural-seeming, topology called the
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81:; in that sense the product topology is the natural topology on the Cartesian product.
3819:(and not the full strength of the axiom of choice) states that any product of compact
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541:(that is, the topology with the fewest open sets) for which all the projections
4197:
4139:
Every product of hereditarily disconnected spaces is hereditarily disconnected.
2819:
2604:
together with the canonical projections, can be characterized by the following
2575:
2502:
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2520:
The set of
Cartesian products between the open sets of the topologies of each
4394:
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4209:
4133:
4074:
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1528:
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is continuous is usually more difficult; one tries to use the fact that the
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66:
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1985:{\displaystyle p_{i}\left(s_{\bullet }\right):=p_{i}\circ s_{\bullet }}
3999:
2459:
2285:
110:
2058:{\displaystyle \left(p_{i}\left(s_{n}\right)\right)_{n=1}^{\infty }}
4052:
4037:
3026:
1545:
38:
4136:(resp. path-connected) spaces is connected (resp. path-connected).
2578:
than the product topology, but for finite products they coincide.
2896:
In many cases it is easier to check that the component functions
1360:
2501:
is homeomorphic to the product of countably many copies of the
1387:
4122:
locally compact (This condition is sufficient and necessary).
2126:{\displaystyle \left(p_{i}\left(s_{a}\right)\right)_{a\in A}}
2508:
Several additional examples are given in the article on the
1681:{\textstyle s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }}
4255:
4253:
3285:{\textstyle \left\{(x,y)\in \mathbb {R} ^{2}:xy=1\right\},}
2979:
In addition to being continuous, the canonical projections
1595:
converges if and only if all its projections to the spaces
264:{\displaystyle X:=\prod X_{\bullet }:=\prod _{i\in I}X_{i}}
4340:. Berlin New York: Springer Science & Business Media.
2786:. It follows from the above universal property that a map
1858:{\displaystyle p_{i}\left(s_{\bullet }\right)\to p_{i}(x)}
2234:{\textstyle \prod _{i\in I}\mathbb {R} =\mathbb {R} ^{I}}
4250:
4200: – Finest topology making some functions continuous
4025:
4232:
Pages displaying short descriptions of redirect targets
3083:
of the product space whose projections down to all the
2505:, where again each copy carries the discrete topology.
2272:
and convergence in the product topology is the same as
69:, which can also be given to a product space and which
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1742:{\textstyle s_{\bullet }=\left(s_{a}\right)_{a\in A}}
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Pages displaying wikidata descriptions as a fallback
30:"Product space" redirects here. For other uses, see
27:
Topology on
Cartesian products of topological spaces
3206:{\textstyle W=\mathbb {R} ^{2}\setminus (0,1)^{2}.}
1410:is open if and only if it is a (possibly infinite)
4014:
3988:
3878:
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4392:
650:endowed with the product topology is called the
77:of its factors, whereas the box topology is too
3879:{\textstyle z=\left(z_{i}\right)_{i\in I}\in X}
4218: – A notion of convergence in mathematics
3213:) The canonical projections are not generally
896:That is, for a finite product, the set of all
590:{\textstyle p_{i}:\prod X_{\bullet }\to X_{i}}
3945:
3773:
3737:
1467:{\displaystyle p_{i}^{-1}\left(U_{i}\right).}
1285:
1143:{\displaystyle p_{i}^{-1}\left(U_{i}\right),}
4286:
4104:Every product of compact spaces is compact (
3313:
3307:
2484:
2472:
2313:then the product topology on the product of
1520:{\displaystyle p_{i}^{-1}\left(U_{i}\right)}
3592:More generally, the closure of the product
3319:{\displaystyle \mathbb {R} \setminus \{0\}}
4369:. Reading, Mass.: Addison-Wesley Pub. Co.
4287:Hocking, John G.; Young, Gail S. (1988) ,
3635:of arbitrary subsets in the product space
2771:
2411:is finite, this is also equivalent to the
850:gives a basis for the product topology of
3655:is equal to the product of the closures:
3372:is a product of arbitrary subsets, where
3300:
3249:
3165:
2424:
2368:
2341:
2295:
2221:
2212:
2154:
4320:
4259:
3811:is a compact space. A specialization of
3515:is a closed subset of the product space
2188:then the Cartesian product is the space
1538:The product topology is also called the
4362:
4212: – Construction in category theory
2923:are continuous. Checking whether a map
2778:This shows that the product space is a
2702:is a continuous map, then there exists
999:is a basis for the product topology of
969:is an element of the (chosen) basis of
14:
4393:
4271:
4224: – Topological space construction
2628:is a topological space, and for every
1791:{\textstyle x\in \prod _{i\in I}X_{i}}
4026:Relation to other topological notions
3975: for all but finitely many
3292:whose projections onto both axes are
3217:(consider for example the closed set
2547:forms a basis for what is called the
3405:{\displaystyle S_{i}\subseteq X_{i}}
643:{\textstyle X:=\prod _{i\in I}X_{i}}
24:
4161:
2860:{\displaystyle f_{i}=p_{i}\circ f}
2161:{\displaystyle X_{i}=\mathbb {R} }
2050:
1673:
1418:of finitely many sets of the form
1038:{\textstyle \prod _{i\in I}X_{i}.}
935:{\textstyle \prod _{i\in I}U_{i},}
889:{\textstyle \prod _{i\in I}X_{i}.}
699:{\textstyle \prod _{i\in I}U_{i},}
25:
4417:
3628:{\textstyle \prod _{i\in I}S_{i}}
3508:{\textstyle \prod _{i\in I}S_{i}}
3365:{\textstyle \prod _{i\in I}S_{i}}
3304:
3175:
2440:{\displaystyle \mathbb {R} ^{n}.}
2384:{\displaystyle \mathbb {R} ^{n}.}
1622:converge. Explicitly, a sequence
1588:{\textstyle \prod _{i\in I}X_{i}}
1541:topology of pointwise convergence
1083:{\textstyle \prod _{i\in I}X_{i}}
530:{\textstyle \prod _{i\in I}X_{i}}
4166:One of many ways to express the
3018:{\displaystyle p_{i}:X\to X_{i}}
2695:{\displaystyle f_{i}:Y\to X_{i}}
2574:In general, the box topology is
4274:Foundations of General Topology
793:{\displaystyle U_{i}\neq X_{i}}
4327:General Topology: Chapters 1–4
4280:
4265:
3241:
3229:
3191:
3178:
3002:
2933:
2802:
2784:category of topological spaces
2722:
2679:
1852:
1846:
1833:
1531:, and their intersections are
574:
442:
423:
409:
389:
13:
1:
4314:
3807:, states that any product of
3803:, which is equivalent to the
3059:The converse is not true: if
2515:
1749:) converges to a given point
84:
4276:, Academic Press, p. 33
3797:is again a Hausdorff space.
2976:are continuous in some way.
2348:{\displaystyle \mathbb {R} }
2302:{\displaystyle \mathbb {R} }
7:
4272:Pervin, William J. (1964),
4182:
3823:spaces is a compact space.
2279:
10:
4422:
4230: – Inherited topology
29:
4363:Willard, Stephen (1970).
4239: – Mathematical term
4222:Quotient space (topology)
4189:Disjoint union (topology)
2355:is equal to the ordinary
1233:In other words, the sets
4406:Operations on structures
4338:Éléments de mathématique
4243:
2811:{\displaystyle f:Y\to X}
2731:{\displaystyle f:Y\to X}
1047:The product topology on
601:. The Cartesian product
3434:{\displaystyle i\in I.}
3150:(consider for instance
2889:{\displaystyle i\in I.}
2650:{\displaystyle i\in I,}
2490:{\displaystyle \{0,1\}}
2065:(respectively, denotes
1914:{\displaystyle i\in I,}
800:for only finitely many
484:, sometimes called the
295:{\displaystyle i\in I,}
135:{\displaystyle i\in I,}
109:will be some non-empty
4149:Countable products of
4113:locally compact spaces
4016:
3990:
3886:is fixed then the set
3880:
3785:
3649:
3629:
3586:
3585:{\displaystyle X_{i}.}
3562:is a closed subset of
3556:
3529:
3509:
3462:
3435:
3406:
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3320:
3286:
3207:
3144:
3124:
3104:
3073:
3053:
3052:{\displaystyle X_{i}.}
3019:
2970:
2943:
2942:{\displaystyle X\to Y}
2917:
2890:
2867:is continuous for all
2861:
2812:
2764:the following diagram
2758:
2757:{\displaystyle i\in I}
2732:
2696:
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1986:
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1589:
1548:(or more generally, a
1521:
1468:
1404:
1380:
1351:
1329: is open in
1227:
1226:{\displaystyle X_{i}.}
1197:
1170:
1169:{\displaystyle i\in I}
1144:
1084:
1039:
993:
992:{\displaystyle X_{i},}
963:
936:
890:
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700:
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591:
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466:
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296:
265:
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163:
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4216:Pointwise convergence
4017:
4002:of the product space
3991:
3881:
3817:the ultrafilter lemma
3786:
3650:
3630:
3587:
3557:
3555:{\displaystyle S_{i}}
3535:if and only if every
3530:
3510:
3463:
3461:{\displaystyle S_{i}}
3436:
3407:
3367:
3321:
3287:
3208:
3145:
3125:
3105:
3103:{\displaystyle X_{i}}
3074:
3054:
3020:
2971:
2969:{\displaystyle p_{i}}
2944:
2918:
2916:{\displaystyle f_{i}}
2891:
2862:
2813:
2759:
2733:
2697:
2652:
2623:
2599:
2569:
2542:
2540:{\displaystyle X_{i}}
2492:
2442:
2406:
2386:
2350:
2328:
2304:
2274:pointwise convergence
2267:
2236:
2183:
2163:
2133:). In particular, if
2128:
2060:
1987:
1916:
1887:
1885:{\displaystyle X_{i}}
1860:
1793:
1744:
1688:(respectively, a net
1683:
1617:
1615:{\displaystyle X_{i}}
1590:
1527:are sometimes called
1522:
1469:
1405:
1381:
1352:
1228:
1203:is an open subset of
1198:
1196:{\displaystyle U_{i}}
1171:
1145:
1085:
1040:
994:
964:
962:{\displaystyle U_{i}}
937:
891:
845:
843:{\displaystyle X_{i}}
818:
795:
755:
753:{\displaystyle X_{i}}
728:
726:{\displaystyle U_{i}}
701:
645:
592:
537:is defined to be the
532:
467:
317:
297:
266:
199:
197:{\displaystyle X_{i}}
164:
162:{\displaystyle X_{i}}
137:
104:
41:and related areas of
4006:
3892:
3830:
3661:
3639:
3596:
3566:
3539:
3519:
3476:
3445:
3416:
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3333:
3296:
3221:
3154:
3134:
3130:need not be open in
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3087:
3063:
3033:
2983:
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2871:
2825:
2790:
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2632:
2612:
2585:
2555:
2524:
2469:
2419:
2395:
2363:
2337:
2317:
2309:is endowed with its
2291:
2253:
2192:
2172:
2137:
2069:
1996:
1925:
1896:
1869:
1802:
1753:
1692:
1626:
1599:
1556:
1478:
1422:
1394:
1367:
1363:for the topology on
1239:
1207:
1180:
1154:
1098:
1094:by sets of the form
1051:
1003:
973:
946:
900:
854:
827:
804:
764:
737:
710:
664:
605:
545:
498:
336:
326:canonical projection
306:
277:
273:and for every index
210:
181:
146:
117:
113:and for every index
93:
4228:Subspace (topology)
4176:Tychonoff's theorem
4106:Tychonoff's theorem
3815:that requires only
3813:Tychonoff's theorem
3801:Tychonoff's theorem
2738:such that for each
2054:
1677:
1498:
1442:
1264:
1118:
75:categorical product
4333:Topologie Générale
4012:
3986:
3876:
3781:
3734:
3698:
3645:
3625:
3614:
3582:
3552:
3525:
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3203:
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2754:
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2692:
2647:
2618:
2606:universal property
2597:{\displaystyle X,}
2594:
2581:The product space
2567:{\displaystyle X.}
2564:
2537:
2499:irrational numbers
2487:
2458:to the product of
2437:
2401:
2381:
2357:Euclidean topology
2345:
2323:
2299:
2265:{\displaystyle I,}
2262:
2231:
2210:
2178:
2158:
2123:
2055:
1999:
1982:
1911:
1882:
1855:
1788:
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1678:
1642:
1612:
1585:
1574:
1517:
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1425:
1400:
1379:{\displaystyle X.}
1376:
1347:
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1223:
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1166:
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1101:
1080:
1069:
1035:
1021:
989:
959:
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886:
872:
840:
816:{\displaystyle i.}
813:
790:
750:
723:
696:
682:
640:
629:
587:
527:
516:
489:Tychonoff topology
462:
460:
374:
312:
292:
261:
250:
194:
159:
132:
99:
55:topological spaces
4347:978-3-540-64241-1
4322:Bourbaki, Nicolas
4304:978-0-486-65676-2
4293:, Dover, p.
4262:, pp. 43–50.
4155:metrizable spaces
4132:Every product of
4080:Every product of
4073:Every product of
4066:Every product of
4051:Every product of
4036:Every product of
4015:{\displaystyle X}
3976:
3719:
3683:
3648:{\displaystyle X}
3599:
3528:{\displaystyle X}
3479:
3336:
3143:{\displaystyle X}
3123:{\displaystyle W}
3072:{\displaystyle W}
2621:{\displaystyle Y}
2497:and the space of
2404:{\displaystyle n}
2326:{\displaystyle n}
2311:standard topology
2195:
2181:{\displaystyle i}
1762:
1559:
1403:{\displaystyle X}
1330:
1302:
1054:
1006:
903:
857:
667:
614:
539:coarsest topology
501:
359:
358:
315:{\displaystyle i}
235:
175:Cartesian product
171:topological space
102:{\displaystyle I}
51:Cartesian product
32:The Product Space
16:(Redirected from
4413:
4401:General topology
4387:
4385:
4383:
4366:General Topology
4359:
4308:
4307:
4284:
4278:
4277:
4269:
4263:
4257:
4233:
4204:Initial topology
4194:
4082:Tychonoff spaces
4068:Hausdorff spaces
4021:
4019:
4018:
4013:
3995:
3993:
3992:
3987:
3985:
3981:
3977:
3974:
3972:
3971:
3959:
3958:
3949:
3948:
3936:
3935:
3924:
3920:
3919:
3885:
3883:
3882:
3877:
3869:
3868:
3857:
3853:
3852:
3795:Hausdorff spaces
3790:
3788:
3787:
3782:
3777:
3776:
3770:
3769:
3760:
3759:
3758:
3757:
3756:
3741:
3740:
3733:
3715:
3714:
3708:
3707:
3697:
3682:
3681:
3675:
3674:
3673:
3654:
3652:
3651:
3646:
3634:
3632:
3631:
3626:
3624:
3623:
3613:
3591:
3589:
3588:
3583:
3578:
3577:
3561:
3559:
3558:
3553:
3551:
3550:
3534:
3532:
3531:
3526:
3514:
3512:
3511:
3506:
3504:
3503:
3493:
3467:
3465:
3464:
3459:
3457:
3456:
3440:
3438:
3437:
3432:
3411:
3409:
3408:
3403:
3401:
3400:
3388:
3387:
3371:
3369:
3368:
3363:
3361:
3360:
3350:
3325:
3323:
3322:
3317:
3303:
3291:
3289:
3288:
3283:
3278:
3274:
3258:
3257:
3252:
3212:
3210:
3209:
3204:
3199:
3198:
3174:
3173:
3168:
3149:
3147:
3146:
3141:
3129:
3127:
3126:
3121:
3109:
3107:
3106:
3101:
3099:
3098:
3078:
3076:
3075:
3070:
3058:
3056:
3055:
3050:
3045:
3044:
3024:
3022:
3021:
3016:
3014:
3013:
2995:
2994:
2975:
2973:
2972:
2967:
2965:
2964:
2948:
2946:
2945:
2940:
2922:
2920:
2919:
2914:
2912:
2911:
2895:
2893:
2892:
2887:
2866:
2864:
2863:
2858:
2850:
2849:
2837:
2836:
2817:
2815:
2814:
2809:
2775:
2763:
2761:
2760:
2755:
2737:
2735:
2734:
2729:
2701:
2699:
2698:
2693:
2691:
2690:
2672:
2671:
2656:
2654:
2653:
2648:
2627:
2625:
2624:
2619:
2603:
2601:
2600:
2595:
2573:
2571:
2570:
2565:
2546:
2544:
2543:
2538:
2536:
2535:
2510:initial topology
2496:
2494:
2493:
2488:
2446:
2444:
2443:
2438:
2433:
2432:
2427:
2410:
2408:
2407:
2402:
2390:
2388:
2387:
2382:
2377:
2376:
2371:
2354:
2352:
2351:
2346:
2344:
2332:
2330:
2329:
2324:
2308:
2306:
2305:
2300:
2298:
2271:
2269:
2268:
2263:
2240:
2238:
2237:
2232:
2230:
2229:
2224:
2215:
2209:
2187:
2185:
2184:
2179:
2168:is used for all
2167:
2165:
2164:
2159:
2157:
2149:
2148:
2132:
2130:
2129:
2124:
2122:
2121:
2110:
2106:
2105:
2101:
2100:
2087:
2086:
2064:
2062:
2061:
2056:
2053:
2048:
2037:
2033:
2032:
2028:
2027:
2014:
2013:
1991:
1989:
1988:
1983:
1981:
1980:
1968:
1967:
1955:
1951:
1950:
1937:
1936:
1920:
1918:
1917:
1912:
1892:for every index
1891:
1889:
1888:
1883:
1881:
1880:
1864:
1862:
1861:
1856:
1845:
1844:
1832:
1828:
1827:
1814:
1813:
1797:
1795:
1794:
1789:
1787:
1786:
1776:
1748:
1746:
1745:
1740:
1738:
1737:
1726:
1722:
1721:
1704:
1703:
1687:
1685:
1684:
1679:
1676:
1671:
1660:
1656:
1655:
1638:
1637:
1621:
1619:
1618:
1613:
1611:
1610:
1594:
1592:
1591:
1586:
1584:
1583:
1573:
1526:
1524:
1523:
1518:
1516:
1512:
1511:
1497:
1489:
1473:
1471:
1470:
1465:
1460:
1456:
1455:
1441:
1433:
1409:
1407:
1406:
1401:
1385:
1383:
1382:
1377:
1356:
1354:
1353:
1348:
1346:
1342:
1341:
1340:
1331:
1328:
1326:
1325:
1313:
1312:
1303:
1300:
1289:
1288:
1282:
1278:
1277:
1263:
1255:
1232:
1230:
1229:
1224:
1219:
1218:
1202:
1200:
1199:
1194:
1192:
1191:
1175:
1173:
1172:
1167:
1149:
1147:
1146:
1141:
1136:
1132:
1131:
1117:
1109:
1090:is the topology
1089:
1087:
1086:
1081:
1079:
1078:
1068:
1044:
1042:
1041:
1036:
1031:
1030:
1020:
998:
996:
995:
990:
985:
984:
968:
966:
965:
960:
958:
957:
941:
939:
938:
933:
928:
927:
917:
895:
893:
892:
887:
882:
881:
871:
849:
847:
846:
841:
839:
838:
822:
820:
819:
814:
799:
797:
796:
791:
789:
788:
776:
775:
759:
757:
756:
751:
749:
748:
732:
730:
729:
724:
722:
721:
705:
703:
702:
697:
692:
691:
681:
657:
656:
649:
647:
646:
641:
639:
638:
628:
596:
594:
593:
588:
586:
585:
573:
572:
557:
556:
536:
534:
533:
528:
526:
525:
515:
491:
490:
481:
480:
479:product topology
471:
469:
468:
463:
461:
454:
453:
437:
436:
421:
420:
401:
400:
384:
383:
373:
356:
352:
351:
321:
319:
318:
313:
301:
299:
298:
293:
270:
268:
267:
262:
260:
259:
249:
231:
230:
203:
201:
200:
195:
193:
192:
168:
166:
165:
160:
158:
157:
141:
139:
138:
133:
108:
106:
105:
100:
63:product topology
59:natural topology
57:equipped with a
21:
4421:
4420:
4416:
4415:
4414:
4412:
4411:
4410:
4391:
4390:
4381:
4379:
4377:
4348:
4317:
4312:
4311:
4305:
4285:
4281:
4270:
4266:
4258:
4251:
4246:
4231:
4192:
4185:
4168:axiom of choice
4164:
4162:Axiom of choice
4062:
4056:
4047:
4041:
4028:
4007:
4004:
4003:
3973:
3967:
3963:
3954:
3950:
3944:
3943:
3925:
3915:
3911:
3907:
3906:
3899:
3895:
3893:
3890:
3889:
3858:
3848:
3844:
3840:
3839:
3831:
3828:
3827:
3805:axiom of choice
3793:Any product of
3772:
3771:
3765:
3761:
3752:
3748:
3747:
3743:
3742:
3736:
3735:
3723:
3710:
3709:
3703:
3699:
3687:
3677:
3676:
3669:
3665:
3664:
3662:
3659:
3658:
3640:
3637:
3636:
3619:
3615:
3603:
3597:
3594:
3593:
3573:
3569:
3567:
3564:
3563:
3546:
3542:
3540:
3537:
3536:
3520:
3517:
3516:
3499:
3495:
3483:
3477:
3474:
3473:
3452:
3448:
3446:
3443:
3442:
3417:
3414:
3413:
3396:
3392:
3383:
3379:
3377:
3374:
3373:
3356:
3352:
3340:
3334:
3331:
3330:
3299:
3297:
3294:
3293:
3253:
3248:
3247:
3228:
3224:
3222:
3219:
3218:
3194:
3190:
3169:
3164:
3163:
3155:
3152:
3151:
3135:
3132:
3131:
3115:
3112:
3111:
3110:are open, then
3094:
3090:
3088:
3085:
3084:
3064:
3061:
3060:
3040:
3036:
3034:
3031:
3030:
3009:
3005:
2990:
2986:
2984:
2981:
2980:
2960:
2956:
2954:
2951:
2950:
2928:
2925:
2924:
2907:
2903:
2901:
2898:
2897:
2872:
2869:
2868:
2845:
2841:
2832:
2828:
2826:
2823:
2822:
2791:
2788:
2787:
2776:
2743:
2740:
2739:
2711:
2708:
2707:
2706:continuous map
2686:
2682:
2667:
2663:
2661:
2658:
2657:
2633:
2630:
2629:
2613:
2610:
2609:
2586:
2583:
2582:
2556:
2553:
2552:
2531:
2527:
2525:
2522:
2521:
2518:
2503:natural numbers
2470:
2467:
2466:
2428:
2423:
2422:
2420:
2417:
2416:
2396:
2393:
2392:
2372:
2367:
2366:
2364:
2361:
2360:
2340:
2338:
2335:
2334:
2318:
2315:
2314:
2294:
2292:
2289:
2288:
2282:
2254:
2251:
2250:
2225:
2220:
2219:
2211:
2199:
2193:
2190:
2189:
2173:
2170:
2169:
2153:
2144:
2140:
2138:
2135:
2134:
2111:
2096:
2092:
2088:
2082:
2078:
2077:
2073:
2072:
2070:
2067:
2066:
2049:
2038:
2023:
2019:
2015:
2009:
2005:
2004:
2000:
1997:
1994:
1993:
1976:
1972:
1963:
1959:
1946:
1942:
1938:
1932:
1928:
1926:
1923:
1922:
1897:
1894:
1893:
1876:
1872:
1870:
1867:
1866:
1840:
1836:
1823:
1819:
1815:
1809:
1805:
1803:
1800:
1799:
1798:if and only if
1782:
1778:
1766:
1754:
1751:
1750:
1727:
1717:
1713:
1709:
1708:
1699:
1695:
1693:
1690:
1689:
1672:
1661:
1651:
1647:
1643:
1633:
1629:
1627:
1624:
1623:
1606:
1602:
1600:
1597:
1596:
1579:
1575:
1563:
1557:
1554:
1553:
1507:
1503:
1499:
1490:
1485:
1479:
1476:
1475:
1451:
1447:
1443:
1434:
1429:
1423:
1420:
1419:
1395:
1392:
1391:
1368:
1365:
1364:
1336:
1332:
1327:
1321:
1317:
1308:
1304:
1301: and
1299:
1284:
1283:
1273:
1269:
1265:
1256:
1251:
1246:
1242:
1240:
1237:
1236:
1214:
1210:
1208:
1205:
1204:
1187:
1183:
1181:
1178:
1177:
1155:
1152:
1151:
1127:
1123:
1119:
1110:
1105:
1099:
1096:
1095:
1074:
1070:
1058:
1052:
1049:
1048:
1026:
1022:
1010:
1004:
1001:
1000:
980:
976:
974:
971:
970:
953:
949:
947:
944:
943:
923:
919:
907:
901:
898:
897:
877:
873:
861:
855:
852:
851:
834:
830:
828:
825:
824:
805:
802:
801:
784:
780:
771:
767:
765:
762:
761:
744:
740:
738:
735:
734:
717:
713:
711:
708:
707:
687:
683:
671:
665:
662:
661:
654:
653:
634:
630:
618:
606:
603:
602:
581:
577:
568:
564:
552:
548:
546:
543:
542:
521:
517:
505:
499:
496:
495:
488:
487:
478:
477:
459:
458:
449:
445:
438:
426:
422:
416:
412:
406:
405:
396:
392:
385:
379:
375:
363:
347:
343:
339:
337:
334:
333:
307:
304:
303:
278:
275:
274:
255:
251:
239:
226:
222:
211:
208:
207:
188:
184:
182:
179:
178:
153:
149:
147:
144:
143:
118:
115:
114:
94:
91:
90:
87:
53:of a family of
35:
28:
23:
22:
15:
12:
11:
5:
4419:
4409:
4408:
4403:
4389:
4388:
4375:
4360:
4346:
4316:
4313:
4310:
4309:
4303:
4279:
4264:
4248:
4247:
4245:
4242:
4241:
4240:
4234:
4225:
4219:
4213:
4207:
4201:
4198:Final topology
4195:
4184:
4181:
4163:
4160:
4159:
4158:
4141:
4140:
4137:
4124:
4123:
4121:
4117:
4109:
4096:
4095:
4093:
4085:
4078:
4075:regular spaces
4071:
4064:
4060:
4054:
4049:
4045:
4039:
4027:
4024:
4011:
3984:
3980:
3970:
3966:
3962:
3957:
3953:
3947:
3942:
3939:
3934:
3931:
3928:
3923:
3918:
3914:
3910:
3905:
3902:
3898:
3875:
3872:
3867:
3864:
3861:
3856:
3851:
3847:
3843:
3838:
3835:
3809:compact spaces
3780:
3775:
3768:
3764:
3755:
3751:
3746:
3739:
3732:
3729:
3726:
3722:
3718:
3713:
3706:
3702:
3696:
3693:
3690:
3686:
3680:
3672:
3668:
3644:
3622:
3618:
3612:
3609:
3606:
3602:
3581:
3576:
3572:
3549:
3545:
3524:
3502:
3498:
3492:
3489:
3486:
3482:
3471:
3455:
3451:
3430:
3427:
3424:
3421:
3399:
3395:
3391:
3386:
3382:
3359:
3355:
3349:
3346:
3343:
3339:
3315:
3312:
3309:
3306:
3302:
3281:
3277:
3273:
3270:
3267:
3264:
3261:
3256:
3251:
3246:
3243:
3240:
3237:
3234:
3231:
3227:
3202:
3197:
3193:
3189:
3186:
3183:
3180:
3177:
3172:
3167:
3162:
3159:
3139:
3119:
3097:
3093:
3068:
3048:
3043:
3039:
3012:
3008:
3004:
3001:
2998:
2993:
2989:
2963:
2959:
2938:
2935:
2932:
2910:
2906:
2885:
2882:
2879:
2876:
2856:
2853:
2848:
2844:
2840:
2835:
2831:
2820:if and only if
2818:is continuous
2807:
2804:
2801:
2798:
2795:
2770:
2753:
2750:
2747:
2727:
2724:
2721:
2718:
2715:
2705:
2689:
2685:
2681:
2678:
2675:
2670:
2666:
2646:
2643:
2640:
2637:
2617:
2593:
2590:
2563:
2560:
2534:
2530:
2517:
2514:
2486:
2483:
2480:
2477:
2474:
2464:discrete space
2462:copies of the
2460:countably many
2436:
2431:
2426:
2400:
2380:
2375:
2370:
2343:
2322:
2297:
2281:
2278:
2276:of functions.
2261:
2258:
2228:
2223:
2218:
2214:
2208:
2205:
2202:
2198:
2177:
2156:
2152:
2147:
2143:
2120:
2117:
2114:
2109:
2104:
2099:
2095:
2091:
2085:
2081:
2076:
2052:
2047:
2044:
2041:
2036:
2031:
2026:
2022:
2018:
2012:
2008:
2003:
1979:
1975:
1971:
1966:
1962:
1958:
1954:
1949:
1945:
1941:
1935:
1931:
1910:
1907:
1904:
1901:
1879:
1875:
1854:
1851:
1848:
1843:
1839:
1835:
1831:
1826:
1822:
1818:
1812:
1808:
1785:
1781:
1775:
1772:
1769:
1765:
1761:
1758:
1736:
1733:
1730:
1725:
1720:
1716:
1712:
1707:
1702:
1698:
1675:
1670:
1667:
1664:
1659:
1654:
1650:
1646:
1641:
1636:
1632:
1609:
1605:
1582:
1578:
1572:
1569:
1566:
1562:
1543:
1529:open cylinders
1515:
1510:
1506:
1502:
1496:
1493:
1488:
1484:
1463:
1459:
1454:
1450:
1446:
1440:
1437:
1432:
1428:
1399:
1375:
1372:
1345:
1339:
1335:
1324:
1320:
1316:
1311:
1307:
1298:
1295:
1292:
1287:
1281:
1276:
1272:
1268:
1262:
1259:
1254:
1250:
1245:
1222:
1217:
1213:
1190:
1186:
1165:
1162:
1159:
1139:
1135:
1130:
1126:
1122:
1116:
1113:
1108:
1104:
1077:
1073:
1067:
1064:
1061:
1057:
1034:
1029:
1025:
1019:
1016:
1013:
1009:
988:
983:
979:
956:
952:
931:
926:
922:
916:
913:
910:
906:
885:
880:
876:
870:
867:
864:
860:
837:
833:
812:
809:
787:
783:
779:
774:
770:
747:
743:
720:
716:
695:
690:
686:
680:
677:
674:
670:
658:
637:
633:
627:
624:
621:
617:
613:
610:
584:
580:
576:
571:
567:
563:
560:
555:
551:
524:
520:
514:
511:
508:
504:
492:
482:
457:
452:
448:
444:
441:
439:
435:
432:
429:
425:
419:
415:
411:
408:
407:
404:
399:
395:
391:
388:
386:
382:
378:
372:
369:
366:
362:
355:
350:
346:
342:
341:
328:
311:
291:
288:
285:
282:
258:
254:
248:
245:
242:
238:
234:
229:
225:
221:
218:
215:
191:
187:
156:
152:
131:
128:
125:
122:
98:
86:
83:
26:
9:
6:
4:
3:
2:
4418:
4407:
4404:
4402:
4399:
4398:
4396:
4378:
4372:
4368:
4367:
4361:
4357:
4353:
4349:
4343:
4339:
4335:
4334:
4329:
4328:
4323:
4319:
4318:
4306:
4300:
4296:
4292:
4291:
4283:
4275:
4268:
4261:
4260:Bourbaki 1989
4256:
4254:
4249:
4238:
4237:Weak topology
4235:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4210:Inverse limit
4208:
4205:
4202:
4199:
4196:
4190:
4187:
4186:
4180:
4177:
4172:
4169:
4156:
4152:
4151:metric spaces
4148:
4147:
4146:
4145:
4144:Metric spaces
4138:
4135:
4131:
4130:
4129:
4128:
4127:Connectedness
4119:
4115:
4114:
4111:A product of
4110:
4107:
4103:
4102:
4101:
4100:
4091:
4090:
4089:normal spaces
4087:A product of
4086:
4084:is Tychonoff.
4083:
4079:
4076:
4072:
4070:is Hausdorff.
4069:
4065:
4058:
4050:
4043:
4035:
4034:
4033:
4032:
4023:
4009:
4001:
3996:
3982:
3978:
3968:
3964:
3960:
3955:
3951:
3940:
3937:
3932:
3929:
3926:
3921:
3916:
3912:
3908:
3903:
3900:
3896:
3887:
3873:
3870:
3865:
3862:
3859:
3854:
3849:
3845:
3841:
3836:
3833:
3824:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3796:
3791:
3778:
3766:
3762:
3753:
3749:
3744:
3730:
3727:
3724:
3720:
3716:
3704:
3700:
3694:
3691:
3688:
3684:
3670:
3666:
3656:
3642:
3620:
3616:
3610:
3607:
3604:
3600:
3579:
3574:
3570:
3547:
3543:
3522:
3500:
3496:
3490:
3487:
3484:
3480:
3469:
3453:
3449:
3428:
3425:
3422:
3419:
3397:
3393:
3389:
3384:
3380:
3357:
3353:
3347:
3344:
3341:
3337:
3327:
3310:
3279:
3275:
3271:
3268:
3265:
3262:
3259:
3254:
3244:
3238:
3235:
3232:
3225:
3216:
3200:
3195:
3187:
3184:
3181:
3170:
3160:
3157:
3137:
3117:
3095:
3091:
3082:
3066:
3046:
3041:
3037:
3028:
3010:
3006:
2999:
2996:
2991:
2987:
2977:
2961:
2957:
2936:
2930:
2908:
2904:
2883:
2880:
2877:
2874:
2854:
2851:
2846:
2842:
2838:
2833:
2829:
2821:
2805:
2799:
2796:
2793:
2785:
2781:
2774:
2769:
2767:
2751:
2748:
2745:
2725:
2719:
2716:
2713:
2704:precisely one
2703:
2687:
2683:
2676:
2673:
2668:
2664:
2644:
2641:
2638:
2635:
2615:
2607:
2591:
2588:
2579:
2577:
2561:
2558:
2550:
2532:
2528:
2513:
2511:
2506:
2504:
2500:
2481:
2478:
2475:
2465:
2461:
2457:
2453:
2448:
2434:
2429:
2414:
2398:
2378:
2373:
2358:
2320:
2312:
2287:
2277:
2275:
2259:
2256:
2248:
2244:
2226:
2216:
2206:
2203:
2200:
2196:
2175:
2150:
2145:
2141:
2118:
2115:
2112:
2107:
2102:
2097:
2093:
2089:
2083:
2079:
2074:
2045:
2042:
2039:
2034:
2029:
2024:
2020:
2016:
2010:
2006:
2001:
1977:
1973:
1969:
1964:
1960:
1956:
1952:
1947:
1943:
1939:
1933:
1929:
1908:
1905:
1902:
1899:
1877:
1873:
1849:
1841:
1837:
1829:
1824:
1820:
1816:
1810:
1806:
1783:
1779:
1773:
1770:
1767:
1763:
1759:
1756:
1734:
1731:
1728:
1723:
1718:
1714:
1710:
1705:
1700:
1696:
1668:
1665:
1662:
1657:
1652:
1648:
1644:
1639:
1634:
1630:
1607:
1603:
1580:
1576:
1570:
1567:
1564:
1560:
1551:
1547:
1542:
1539:
1536:
1534:
1533:cylinder sets
1530:
1513:
1508:
1504:
1500:
1494:
1491:
1486:
1482:
1461:
1457:
1452:
1448:
1444:
1438:
1435:
1430:
1426:
1417:
1416:intersections
1413:
1397:
1389:
1373:
1370:
1362:
1357:
1343:
1337:
1333:
1322:
1318:
1314:
1309:
1305:
1296:
1293:
1290:
1279:
1274:
1270:
1266:
1260:
1257:
1252:
1248:
1243:
1234:
1220:
1215:
1211:
1188:
1184:
1163:
1160:
1157:
1137:
1133:
1128:
1124:
1120:
1114:
1111:
1106:
1102:
1093:
1075:
1071:
1065:
1062:
1059:
1055:
1045:
1032:
1027:
1023:
1017:
1014:
1011:
1007:
986:
981:
977:
954:
950:
929:
924:
920:
914:
911:
908:
904:
883:
878:
874:
868:
865:
862:
858:
835:
831:
810:
807:
785:
781:
777:
772:
768:
745:
741:
718:
714:
693:
688:
684:
678:
675:
672:
668:
659:
655:product space
652:
635:
631:
625:
622:
619:
615:
611:
608:
600:
582:
578:
569:
565:
561:
558:
553:
549:
540:
522:
518:
512:
509:
506:
502:
493:
486:
483:
476:
472:
455:
450:
446:
440:
433:
430:
427:
417:
413:
402:
397:
393:
387:
380:
376:
370:
367:
364:
360:
353:
348:
344:
331:
329:
327:
324:
309:
289:
286:
283:
280:
271:
256:
252:
246:
243:
240:
236:
232:
227:
223:
219:
216:
213:
205:
189:
185:
176:
173:. Denote the
172:
154:
150:
129:
126:
123:
120:
112:
96:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
47:product space
44:
40:
33:
19:
18:Product space
4380:. Retrieved
4365:
4331:
4326:
4289:
4282:
4273:
4267:
4173:
4165:
4143:
4142:
4126:
4125:
4098:
4097:
4030:
4029:
4000:dense subset
3997:
3888:
3825:
3799:
3792:
3657:
3328:
2978:
2777:
2580:
2549:box topology
2519:
2507:
2456:homeomorphic
2449:
2413:box topology
2283:
1537:
1358:
1235:
1046:
651:
485:
475:
473:
332:
323:
272:
206:
177:of the sets
89:Throughout,
88:
67:box topology
62:
46:
36:
4382:13 February
4099:Compactness
4077:is regular.
3215:closed maps
733:is open in
706:where each
302:denote the
61:called the
43:mathematics
4395:Categories
4376:0486434796
4315:References
4094:be normal.
4031:Separation
3412:for every
2516:Properties
2452:Cantor set
2333:copies of
1544:because a
599:continuous
85:Definition
4324:(1989) .
4134:connected
3938:∈
3930:∈
3871:∈
3863:∈
3821:Hausdorff
3728:∈
3721:∏
3692:∈
3685:∏
3608:∈
3601:∏
3488:∈
3481:∏
3470:non-empty
3423:∈
3390:⊆
3345:∈
3338:∏
3305:∖
3245:∈
3176:∖
3027:open maps
3003:→
2934:→
2878:∈
2852:∘
2803:→
2749:∈
2723:→
2680:→
2639:∈
2391:(Because
2286:real line
2247:functions
2204:∈
2197:∏
2116:∈
2051:∞
1978:∙
1970:∘
1948:∙
1903:∈
1834:→
1825:∙
1771:∈
1764:∏
1760:∈
1732:∈
1701:∙
1674:∞
1635:∙
1568:∈
1561:∏
1492:−
1436:−
1315:⊆
1294:∈
1258:−
1161:∈
1112:−
1092:generated
1063:∈
1056:∏
1015:∈
1008:∏
912:∈
905:∏
866:∈
859:∏
778:≠
676:∈
669:∏
623:∈
616:∏
575:→
570:∙
562:∏
510:∈
503:∏
443:↦
431:∈
390:→
368:∈
361:∏
284:∈
244:∈
237:∏
228:∙
220:∏
124:∈
111:index set
4356:18588129
4290:Topology
4183:See also
4116:need not
4092:need not
3329:Suppose
3081:subspace
2766:commutes
2280:Examples
2245:-valued
1992:denotes
1546:sequence
39:topology
4336:].
3441:If all
2782:in the
2780:product
2284:If the
2241:of all
1361:subbase
1359:form a
49:is the
4373:
4354:
4344:
4301:
4057:spaces
4042:spaces
1921:where
1388:subset
1150:where
942:where
357:
71:agrees
4330:[
4244:Notes
3998:is a
3472:then
3079:is a
2608:: if
2576:finer
1552:) in
1412:union
494:, on
169:be a
4384:2013
4371:ISBN
4352:OCLC
4342:ISBN
4299:ISBN
4153:are
4059:is T
4044:is T
3468:are
3025:are
2450:The
2243:real
1474:The
1176:and
760:and
597:are
474:The
322:-th
142:let
79:fine
45:, a
3826:If
3326:).
2551:on
2454:is
2415:on
2359:on
2249:on
1865:in
1550:net
1414:of
1390:of
330:by
204:by
37:In
4397::
4350:.
4297:,
4295:28
4252:^
4120:is
4108:).
4022:.
3745:Cl
3667:Cl
2768::
2512:.
2447:)
1957::=
1535:.
1386:A
612::=
233::=
217::=
4386:.
4358:.
4157:.
4063:.
4061:1
4055:1
4053:T
4048:.
4046:0
4040:0
4038:T
4010:X
3983:}
3979:i
3969:i
3965:z
3961:=
3956:i
3952:x
3946:|
3941:X
3933:I
3927:i
3922:)
3917:i
3913:x
3909:(
3904:=
3901:x
3897:{
3874:X
3866:I
3860:i
3855:)
3850:i
3846:z
3842:(
3837:=
3834:z
3779:.
3774:)
3767:i
3763:S
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3738:(
3731:I
3725:i
3717:=
3712:)
3705:i
3701:S
3695:I
3689:i
3679:(
3671:X
3643:X
3621:i
3617:S
3611:I
3605:i
3580:.
3575:i
3571:X
3548:i
3544:S
3523:X
3501:i
3497:S
3491:I
3485:i
3454:i
3450:S
3429:.
3426:I
3420:i
3398:i
3394:X
3385:i
3381:S
3358:i
3354:S
3348:I
3342:i
3314:}
3311:0
3308:{
3301:R
3280:,
3276:}
3272:1
3269:=
3266:y
3263:x
3260::
3255:2
3250:R
3242:)
3239:y
3236:,
3233:x
3230:(
3226:{
3201:.
3196:2
3192:)
3188:1
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3182:0
3179:(
3171:2
3166:R
3161:=
3158:W
3138:X
3118:W
3096:i
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3067:W
3047:.
3042:i
3038:X
3011:i
3007:X
3000:X
2997::
2992:i
2988:p
2962:i
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2937:Y
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2884:.
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2875:i
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2839:=
2834:i
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2806:X
2800:Y
2797::
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2752:I
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2726:X
2720:Y
2717::
2714:f
2688:i
2684:X
2677:Y
2674::
2669:i
2665:f
2645:,
2642:I
2636:i
2616:Y
2592:,
2589:X
2562:.
2559:X
2533:i
2529:X
2485:}
2482:1
2479:,
2476:0
2473:{
2435:.
2430:n
2425:R
2399:n
2379:.
2374:n
2369:R
2342:R
2321:n
2296:R
2260:,
2257:I
2227:I
2222:R
2217:=
2213:R
2207:I
2201:i
2176:i
2155:R
2151:=
2146:i
2142:X
2119:A
2113:a
2108:)
2103:)
2098:a
2094:s
2090:(
2084:i
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2075:(
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2040:n
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2030:)
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2017:(
2011:i
2007:p
2002:(
1974:s
1965:i
1961:p
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1711:(
1706:=
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1640:=
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