4182:
4023:
1536:
709:
is nonmeager, no set is at the same time meagre and comeager, every comeagre set is nonmeagre, and there can be nonmeagre sets that are not comeagre, that is, with nonmeagre complement. See the
Examples section below.
4687:
4848:
from reals to reals where the image of a null set of reals is a meagre set, and vice versa. In fact, the image of a set of reals under the map is null if and only if the original set was meagre, and vice versa.
3579:
1352:
951:
2336:
1603:
1433:
4558:
1725:
4228:
3507:
1465:
991:
1569:
4464:
4781:
4625:
4417:
1287:
1234:
3998:
3665:
3441:
2137:
1777:
1697:
1629:
1396:
1374:
1262:
1209:
1187:
1165:
589:
4361:
4328:
4293:
4264:
2107:
1751:
3606:
1128:
2166:
1950:
1821:
4601:
4507:
3800:
3728:
3629:
3250:
3187:
3141:
3058:
3013:
2930:
2885:
2842:
2792:
2709:
2664:
2581:
2536:
2493:
2446:
2383:
2283:
2022:, on each separable Banach space, there exists a discontinuous linear functional whose kernel is meagre (this statement disproves the Wilansky–Klee conjecture).
1064:
660:
559:
528:
415:
384:
353:
271:
240:
160:
4825:
4801:
4747:
4727:
4707:
4578:
4484:
4437:
4393:
4018:
3909:
3874:
3822:
3748:
3527:
3461:
3412:
3392:
3372:
3352:
3313:
3293:
3270:
3227:
3207:
3164:
3118:
3098:
3078:
3035:
2990:
2970:
2950:
2907:
2862:
2819:
2769:
2749:
2729:
2686:
2641:
2621:
2601:
2558:
2513:
2470:
2423:
2403:
2356:
2249:
2229:
2209:
2189:
2009:
1970:
1877:
1675:
839:
819:
795:
775:
751:
731:
707:
687:
633:
609:
498:
478:
436:
324:
299:
211:
180:
126:
3973:
3941:
3854:
3780:
3701:
1909:
1853:
1096:
1023:
4177:{\displaystyle \bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-\left({\tfrac {1}{2}}\right)^{n+m},r_{n}+\left({\tfrac {1}{2}}\right)^{n+m}\right)}
777:, one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right). In this case
4238:
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, its closure), a meagre set need not be an
2011:
is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.
1473:
849:
some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space.
845:
will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space. Be aware however that in the context of
4630:
4913:
5072:
5083:
3324:
821:, meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space
5348:
5311:
5252:
3532:
5303:
5214:
5283:
3707:
1299:
5244:
5380:
2015:
1973:
2288:
All subsets and all countable unions of meagre sets are meagre. Thus the meagre subsets of a fixed space form a
1650:
is nonmeagre (because no set containing the isolated point can be nowhere dense). In particular, every nonempty
886:
2300:
and all countable intersections of comeagre sets are comeagre. Every superset of a nonmeagre set is nonmeagre.
2306:
1574:
1404:
4512:
1702:
17:
4928:
4187:
3466:
841:. (See the Properties and Examples sections below for the relationship between the two.) Similarly, a
1438:
956:
1545:
5375:
5302:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
4442:
2043:
880:
The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space.
846:
636:
4762:
4606:
4398:
3804:
Dually, there can be nonmeagre sets with measure zero. The complement of any meagre set of measure
1635:
is meagre, whereas any topological space that contains an isolated point is nonmeagre. Because the
689:
is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space
4845:
1699:
that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set
1639:
are countable, they are meagre as a subset of the reals and as a space—that is, they do not form a
1267:
1214:
5188:
5073:
https://mathoverflow.net/questions/3188/are-proper-linear-subspaces-of-banach-spaces-always-meager
3981:
3648:
3420:
3354:
is meagre. Consequently, any closed subset with empty interior is meagre. Thus a closed subset of
2116:
1756:
1680:
1612:
1379:
1357:
1245:
1192:
1170:
1148:
568:
4956:
4339:
4306:
4271:
4242:
563:
5084:
https://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11547-1/S0002-9904-1966-11547-1.pdf
2077:
Any subset of a meagre set is a meagre set, as is the union of countably many meagre sets. If
2071:
2035:
93:
2080:
5271:
4372:
2054:
2039:
1730:
1290:
1145:
is meagre. So it is also meagre in any space that contains it as a subspace. For example,
4841:
4840:, i.e. sets of Lebesgue measure 0. The Erdos–Sierpinski duality theorem states that if the
3710:, are closed nowhere dense and they can be constructed with a measure arbitrarily close to
3584:
1101:
2142:
8:
4439:
that have nonempty interiors such that every nonempty open set has a subset belonging to
4299:
1914:
1856:
1785:
187:
97:
4583:
4489:
4298:
Dually, just as the complement of a nowhere dense set need not be open, but has a dense
3785:
3713:
3611:
3232:
3169:
3123:
3040:
2995:
2912:
2867:
2824:
2774:
2691:
2646:
2563:
2518:
2475:
2428:
2365:
2265:
1031:
642:
541:
510:
397:
366:
335:
253:
222:
142:
5340:
5167:
4948:
4810:
4786:
4732:
4712:
4692:
4563:
4469:
4422:
4378:
4003:
3879:
3859:
3807:
3733:
3512:
3446:
3397:
3377:
3357:
3337:
3298:
3278:
3255:
3212:
3192:
3149:
3103:
3083:
3063:
3020:
2975:
2955:
2935:
2892:
2847:
2804:
2754:
2734:
2714:
2671:
2626:
2606:
2586:
2543:
2498:
2455:
2408:
2388:
2341:
2234:
2214:
2194:
2174:
2057:
1979:
1955:
1862:
1660:
824:
804:
780:
760:
736:
716:
692:
672:
618:
594:
483:
463:
421:
309:
284:
196:
183:
165:
111:
78:
3946:
3914:
3827:
3753:
3674:
1882:
1826:
1069:
996:
5354:
5344:
5317:
5307:
5297:
5279:
5258:
5248:
5220:
5210:
4876:
3638:
There exist nowhere dense subsets (which are thus meagre subsets) that have positive
2359:
2285:
the union of any family of open sets of the first category is of the first category.
2019:
754:
129:
55:
5232:
5038:
4965:
4864:
4804:
3668:
3639:
3272:
is equivalent to being meagre in itself, and similarly for the nonmeagre property.
1636:
869:
35:
1531:{\displaystyle S=(\mathbb {Q} \times \mathbb {Q} )\cup (\mathbb {R} \times \{0\})}
1398:. It is nonmeagre in itself (since as a subspace it contains an isolated point).
611:. (This use of the prefix "co" is consistent with its use in other terms such as "
5330:
4858:
2067:
2064:
4870:
3703:
3414:(because otherwise it would be nowhere dense and thus of the first category).
3328:
3295:
is nonmeagre if and only if every countable intersection of dense open sets in
2293:
2171:
Every nowhere dense subset is a meagre set. Consequently, any closed subset of
2053:
is nonmeagre but there exist nonmeagre spaces that are not Baire spaces. Since
1651:
1647:
1632:
1142:
59:
5369:
5336:
5262:
5224:
5209:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
2110:
303:
136:
82:
5321:
4969:
861:
669:
The notions of nonmeagre and comeagre should not be confused. If the space
438:" can be omitted if the ambient space is fixed and understood from context.
5293:
5148:
5043:
5030:
2060:
612:
5358:
5166:
Quintanilla, M. (2022). "The real numbers in inner models of set theory".
4974:"Following Bourbaki , a topological space is called a Baire space if ..."
4873: – Mathematical set regarded as insignificant, for analogs to meagre
2289:
2050:
2031:
1640:
1631:
is not a meagre topological space). A countable
Hausdorff space without
89:
74:
31:
5247:. Vol. 4. Berlin New York: Springer Science & Business Media.
4371:
Meagre sets have a useful alternative characterization in terms of the
1239:
88:
Meagre sets play an important role in the formulation of the notion of
62:
in a precise sense detailed below. A set that is not meagre is called
5007:
5005:
5003:
5001:
4999:
4997:
4995:
4867: – Property holding for typical examples, for analogs to residual
4295:
set made from nowhere dense sets (by taking the closure of each set).
3730:
The union of a countable number of such sets with measure approaching
4682:{\displaystyle W_{1}\supseteq W_{2}\supseteq W_{3}\supseteq \cdots .}
4303:
4239:
3671:
zero, and can even have full measure. For example, in the interval
1467:
But it is a nonmeagre subspace, that is, it is nonmeagre in itself.
5172:
4992:
4837:
4333:
2297:
1952:
is a complete metric space, it is nonmeagre. So the complement of
1189:(that is, meagre in itself with the subspace topology induced from
1134:
77:
of subsets; that is, any subset of a meagre set is meagre, and the
4895:
4893:
457:) if it is a meagre (respectively, nonmeagre) subset of itself.
4268:(countable union of closed sets), but is always contained in an
5117:
5025:
4890:
3229:
is nonmeagre in itself. And for an open set or a dense set in
96:, which is used in the proof of several fundamental results of
51:
5051:
4709:
wins if the intersection of this sequence contains a point in
2025:
3856:(for example the one in the previous paragraph) has measure
4302:(contains a dense open set), a comeagre set need not be a
5026:"Über die Baire'sche Kategorie gewisser Funktionenmengen"
4982:
4980:
1098:
is nonmeagre. But it is not comeagre, as its complement
5278:(Second ed.). New York: Springer. pp. 62–65.
1911:
that have a derivative at some point is meagre. Since
5129:
5095:
5093:
5091:
4977:
4142:
4093:
3633:
3574:{\displaystyle B\subseteq S_{1}\cup S_{2}\cup \cdots }
4813:
4789:
4765:
4735:
4715:
4695:
4633:
4609:
4586:
4566:
4515:
4492:
4472:
4445:
4425:
4401:
4381:
4342:
4309:
4274:
4245:
4190:
4026:
4006:
3984:
3949:
3917:
3882:
3862:
3830:
3810:
3788:
3756:
3736:
3716:
3677:
3651:
3614:
3587:
3535:
3515:
3469:
3449:
3423:
3400:
3380:
3360:
3340:
3301:
3281:
3258:
3235:
3215:
3195:
3172:
3152:
3126:
3106:
3086:
3066:
3043:
3023:
2998:
2978:
2958:
2938:
2915:
2895:
2870:
2850:
2827:
2807:
2777:
2757:
2737:
2717:
2694:
2674:
2649:
2629:
2609:
2589:
2566:
2546:
2521:
2501:
2478:
2458:
2431:
2411:
2391:
2368:
2344:
2309:
2268:
2237:
2217:
2197:
2177:
2145:
2119:
2083:
1982:
1958:
1917:
1885:
1865:
1829:
1788:
1759:
1733:
1705:
1683:
1663:
1615:
1577:
1548:
1476:
1441:
1407:
1382:
1360:
1302:
1270:
1248:
1217:
1195:
1173:
1151:
1104:
1072:
1034:
999:
959:
889:
827:
807:
783:
763:
739:
719:
695:
675:
645:
621:
597:
571:
544:
513:
486:
466:
424:
400:
369:
338:
312:
287:
256:
225:
199:
168:
145:
114:
70:. See below for definitions of other related terms.
5105:
5068:
5066:
4603:
alternately choose successively smaller elements of
4230:
is a sequence that enumerates the rational numbers.
5088:
713:As an additional point of terminology, if a subset
190:. See the corresponding article for more details.
4819:
4795:
4775:
4741:
4721:
4701:
4681:
4619:
4595:
4572:
4552:
4501:
4478:
4458:
4431:
4411:
4387:
4355:
4322:
4287:
4258:
4222:
4176:
4012:
3992:
3967:
3935:
3903:
3868:
3848:
3816:
3794:
3774:
3742:
3722:
3695:
3659:
3623:
3600:
3573:
3521:
3501:
3455:
3435:
3406:
3386:
3366:
3346:
3307:
3287:
3264:
3244:
3221:
3201:
3181:
3158:
3135:
3112:
3092:
3072:
3052:
3029:
3007:
2984:
2964:
2944:
2924:
2901:
2879:
2856:
2836:
2813:
2786:
2763:
2743:
2723:
2703:
2680:
2658:
2635:
2615:
2595:
2575:
2552:
2530:
2507:
2487:
2464:
2440:
2417:
2397:
2377:
2350:
2330:
2277:
2243:
2223:
2203:
2183:
2160:
2131:
2101:
2014:On an infinite-dimensional Banach, there exists a
2003:
1964:
1944:
1903:
1871:
1847:
1815:
1771:
1745:
1719:
1691:
1669:
1623:
1597:
1563:
1530:
1459:
1427:
1390:
1368:
1346:
1281:
1256:
1228:
1203:
1181:
1159:
1122:
1090:
1058:
1017:
985:
945:
833:
813:
789:
769:
745:
725:
701:
681:
654:
627:
603:
583:
553:
522:
492:
472:
430:
409:
378:
347:
318:
293:
265:
234:
205:
174:
154:
135:The definition of meagre set uses the notion of a
120:
5204:
5149:"Is there a measure zero set which isn't meagre?"
5063:
5011:
4899:
4879: – Difference of an open set by a meager set
5367:
4836:Many arguments about meagre sets also apply to
1972:, which consists of the continuous real-valued
5205:Narici, Lawrence; Beckenstein, Edward (2011).
4233:
3978:Here is another example of a nonmeagre set in
1347:{\displaystyle (\cap \mathbb {Q} )\cup \{2\}}
1289:But it is nonmeagre in itself, since it is a
1592:
1586:
1522:
1516:
1422:
1416:
1341:
1335:
639:of sets, each of whose interior is dense in
5165:
4831:
2026:Characterizations and sufficient conditions
73:The meagre subsets of a fixed space form a
5023:
946:{\displaystyle X=\cup (\cap \mathbb {Q} )}
635:if and only if it is equal to a countable
5171:
5042:
3986:
3653:
1713:
1685:
1617:
1579:
1551:
1509:
1495:
1487:
1444:
1409:
1384:
1362:
1325:
1272:
1250:
1219:
1197:
1175:
1153:
979:
936:
5231:
4986:
4929:"Sur les fonctions de variables réelles"
4911:
4861: – Type of topological vector space
2797:And correspondingly for nonmeagre sets:
2331:{\displaystyle A\subseteq Y\subseteq X,}
1598:{\displaystyle \mathbb {R} \times \{0\}}
1428:{\displaystyle \mathbb {R} \times \{0\}}
5328:
5123:
5057:
4560:In the Banach–Mazur game, two players,
4553:{\displaystyle MZ(X,Y,{\mathcal {W}}).}
3325:locally convex topological vector space
2018:whose kernel is nonmeagre. Also, under
1879:of continuous real-valued functions on
1823:of continuous real-valued functions on
1720:{\displaystyle U\subseteq \mathbb {R} }
1646:Any topological space that contains an
14:
5368:
5269:
5111:
4946:
3166:that is meagre in itself is meagre in
5292:
5135:
5099:
4926:
2211:is empty is of the first category of
2034:is nonmeagre. In particular, by the
27:"Small" subset of a topological space
5304:McGraw-Hill Science/Engineering/Math
5189:The Erdos-Sierpinski Duality Theorem
4949:"Cartesian products of Baire spaces"
4366:
2448:However the following results hold:
4783:meeting the above criteria, player
4223:{\displaystyle r_{1},r_{2},\ldots }
3634:Meagre subsets and Lebesgue measure
3502:{\displaystyle S_{1},S_{2},\ldots }
2231:(that is, it is a meager subset of
2074:, they are also nonmeagre spaces.
24:
5191:, notes. Accessed 18 January 2023.
4768:
4612:
4539:
4509:Then there is a Banach–Mazur game
4448:
4404:
4064:
4043:
3374:that is of the second category in
326:. Otherwise, the subset is called
25:
5392:
5237:General Topology 2: Chapters 5–10
4363:set formed from dense open sets.
2292:of subsets, a suitable notion of
1763:
1460:{\displaystyle \mathbb {R} ^{2}.}
986:{\displaystyle \cap \mathbb {Q} }
575:
3394:must have non-empty interior in
1974:nowhere differentiable functions
1564:{\displaystyle \mathbb {R} ^{2}}
5198:
5181:
5159:
5141:
5077:
4459:{\displaystyle {\mathcal {W}},}
3146:In particular, every subset of
2016:discontinuous linear functional
860:were the original ones used by
5017:
4940:
4920:
4905:
4776:{\displaystyle {\mathcal {W}}}
4620:{\displaystyle {\mathcal {W}}}
4544:
4522:
4412:{\displaystyle {\mathcal {W}}}
3962:
3950:
3930:
3918:
3895:
3883:
3843:
3831:
3769:
3757:
3690:
3678:
3334:Every nowhere dense subset of
2155:
2149:
2093:
1995:
1983:
1939:
1936:
1924:
1921:
1898:
1886:
1842:
1830:
1810:
1807:
1795:
1792:
1571:even though its meagre subset
1525:
1505:
1499:
1483:
1329:
1318:
1306:
1303:
1117:
1105:
1085:
1073:
1053:
1041:
1012:
1000:
972:
960:
940:
929:
917:
914:
908:
896:
868:terminology was introduced by
441:A topological space is called
302:if it is a countable union of
103:
13:
1:
5272:"The Banach Category Theorem"
5012:Narici & Beckenstein 2011
4900:Narici & Beckenstein 2011
3608:is of the second category in
3443:is of the second category in
3318:
1282:{\displaystyle \mathbb {R} .}
1229:{\displaystyle \mathbb {R} .}
1167:is both a meagre subspace of
4912:Schaefer, Helmut H. (1966).
4336:sets), but contains a dense
3993:{\displaystyle \mathbb {R} }
3660:{\displaystyle \mathbb {R} }
3436:{\displaystyle B\subseteq X}
2132:{\displaystyle S\subseteq X}
1772:{\displaystyle U\setminus H}
1692:{\displaystyle \mathbb {R} }
1624:{\displaystyle \mathbb {R} }
1391:{\displaystyle \mathbb {R} }
1369:{\displaystyle \mathbb {R} }
1257:{\displaystyle \mathbb {R} }
1204:{\displaystyle \mathbb {R} }
1182:{\displaystyle \mathbb {R} }
1160:{\displaystyle \mathbb {Q} }
864:in his thesis of 1899. The
615:".) A subset is comeagre in
584:{\displaystyle X\setminus A}
85:many meagre sets is meagre.
7:
4933:Annali di Mat. Pura ed Appl
4914:"Topological Vector Spaces"
4852:
4356:{\displaystyle G_{\delta }}
4332:(countable intersection of
4323:{\displaystyle G_{\delta }}
4288:{\displaystyle F_{\sigma }}
4259:{\displaystyle F_{\sigma }}
4234:Relation to Borel hierarchy
1025:is nonmeagre and comeagre.
875:
10:
5397:
5329:Willard, Stephen (2004) .
4419:be a family of subsets of
5207:Topological Vector Spaces
3750:gives a meagre subset of
3708:Smith–Volterra–Cantor set
2262:states that in any space
2139:is meagre if and only if
2044:locally compact Hausdorff
1211:) and a meagre subset of
847:topological vector spaces
5270:Oxtoby, John C. (1980).
5245:Éléments de mathématique
4883:
4832:Erdos–Sierpinski duality
4395:be a topological space,
2425:without being meagre in
2102:{\displaystyle h:X\to X}
1354:is not nowhere dense in
4970:10.4064/fm-49-2-157-166
4957:Fundamenta Mathematicae
3911:and hence nonmeagre in
2258:Banach category theorem
1746:{\displaystyle U\cap H}
1435:is meagre in the plane
1028:In the nonmeagre space
883:In the nonmeagre space
733:of a topological space
5381:Descriptive set theory
5044:10.4064/sm-3-1-174-179
4821:
4797:
4777:
4743:
4723:
4703:
4683:
4627:to produce a sequence
4621:
4597:
4574:
4554:
4503:
4480:
4460:
4433:
4413:
4389:
4357:
4324:
4289:
4260:
4224:
4178:
4068:
4047:
4014:
3994:
3969:
3937:
3905:
3870:
3850:
3818:
3796:
3776:
3744:
3724:
3697:
3661:
3625:
3602:
3575:
3523:
3503:
3457:
3437:
3408:
3388:
3368:
3348:
3309:
3289:
3266:
3246:
3223:
3203:
3183:
3160:
3137:
3114:
3094:
3074:
3054:
3031:
3009:
2986:
2966:
2946:
2926:
2903:
2881:
2858:
2838:
2815:
2788:
2765:
2745:
2725:
2705:
2682:
2660:
2637:
2617:
2597:
2577:
2554:
2532:
2509:
2489:
2466:
2442:
2419:
2399:
2379:
2352:
2332:
2279:
2245:
2225:
2205:
2185:
2162:
2133:
2103:
2036:Baire category theorem
2005:
1966:
1946:
1905:
1873:
1849:
1817:
1773:
1747:
1721:
1693:
1671:
1625:
1599:
1565:
1532:
1461:
1429:
1392:
1376:, but it is meagre in
1370:
1348:
1283:
1258:
1230:
1205:
1183:
1161:
1124:
1092:
1060:
1019:
987:
947:
835:
815:
791:
771:
747:
727:
703:
683:
665:Remarks on terminology
656:
629:
605:
585:
555:
524:
494:
474:
432:
411:
380:
349:
320:
295:
267:
236:
207:
176:
156:
122:
94:Baire category theorem
68:of the second category
4822:
4798:
4778:
4744:
4724:
4704:
4684:
4622:
4598:
4575:
4555:
4504:
4481:
4461:
4434:
4414:
4390:
4358:
4325:
4290:
4261:
4225:
4179:
4048:
4027:
4015:
3995:
3970:
3938:
3906:
3871:
3851:
3819:
3797:
3777:
3745:
3725:
3698:
3662:
3626:
3603:
3601:{\displaystyle S_{n}}
3576:
3524:
3504:
3458:
3438:
3409:
3389:
3369:
3349:
3310:
3290:
3267:
3247:
3224:
3209:that is nonmeagre in
3204:
3184:
3161:
3138:
3115:
3095:
3075:
3055:
3032:
3010:
2987:
2967:
2947:
2927:
2904:
2882:
2859:
2839:
2816:
2789:
2766:
2746:
2726:
2706:
2683:
2661:
2638:
2618:
2598:
2578:
2555:
2533:
2510:
2490:
2467:
2443:
2420:
2400:
2380:
2353:
2333:
2280:
2246:
2226:
2206:
2186:
2163:
2134:
2104:
2040:complete metric space
2006:
1967:
1947:
1906:
1874:
1855:with the topology of
1850:
1818:
1774:
1748:
1722:
1694:
1672:
1626:
1600:
1566:
1533:
1462:
1430:
1393:
1371:
1349:
1291:complete metric space
1284:
1259:
1231:
1206:
1184:
1162:
1125:
1123:{\displaystyle (1,2]}
1093:
1061:
1020:
988:
948:
836:
816:
797:can also be called a
792:
772:
748:
728:
704:
684:
657:
630:
606:
586:
556:
525:
495:
475:
433:
412:
381:
350:
321:
296:
268:
237:
208:
177:
162:that is, a subset of
157:
123:
48:set of first category
5276:Measure and Category
4927:Baire, René (1899).
4842:continuum hypothesis
4811:
4787:
4763:
4733:
4729:; otherwise, player
4713:
4693:
4631:
4607:
4584:
4564:
4513:
4490:
4470:
4443:
4423:
4399:
4379:
4340:
4307:
4272:
4243:
4188:
4024:
4004:
3982:
3947:
3915:
3880:
3860:
3828:
3808:
3786:
3754:
3734:
3714:
3675:
3649:
3612:
3585:
3533:
3513:
3467:
3447:
3421:
3398:
3378:
3358:
3338:
3299:
3279:
3275:A topological space
3256:
3233:
3213:
3193:
3170:
3150:
3124:
3104:
3084:
3064:
3041:
3021:
2996:
2976:
2956:
2936:
2913:
2893:
2868:
2848:
2825:
2805:
2775:
2755:
2735:
2715:
2692:
2672:
2647:
2627:
2607:
2587:
2564:
2544:
2519:
2499:
2476:
2456:
2429:
2409:
2389:
2366:
2342:
2307:
2266:
2235:
2215:
2195:
2175:
2161:{\displaystyle h(S)}
2143:
2117:
2081:
2046:space is nonmeagre.
1980:
1956:
1915:
1883:
1863:
1827:
1786:
1779:are both nonmeagre.
1757:
1731:
1703:
1681:
1677:of the real numbers
1661:
1613:
1575:
1546:
1474:
1439:
1405:
1380:
1358:
1300:
1268:
1264:and hence meagre in
1246:
1242:is nowhere dense in
1215:
1193:
1171:
1149:
1102:
1070:
1032:
997:
993:is meagre. The set
957:
887:
825:
805:
781:
761:
737:
717:
693:
673:
643:
619:
595:
569:
542:
511:
484:
464:
422:
398:
367:
336:
310:
285:
254:
223:
197:
166:
143:
112:
5299:Functional Analysis
5024:Banach, S. (1931).
5014:, pp. 371–423.
4947:Oxtoby, J. (1961).
4844:holds, there is an
4757: —
3975:is a Baire space.
3876:and is comeagre in
2042:and every nonempty
1945:{\displaystyle C()}
1857:uniform convergence
1816:{\displaystyle C()}
1130:is also nonmeagre.
98:functional analysis
5341:Dover Publications
5241:Topologie Générale
4817:
4793:
4773:
4755:
4739:
4719:
4699:
4679:
4617:
4596:{\displaystyle Q,}
4593:
4570:
4550:
4502:{\displaystyle Y.}
4499:
4476:
4456:
4429:
4409:
4385:
4353:
4320:
4285:
4256:
4220:
4174:
4151:
4102:
4010:
3990:
3965:
3933:
3901:
3866:
3846:
3814:
3795:{\displaystyle 1.}
3792:
3772:
3740:
3723:{\displaystyle 1.}
3720:
3693:
3657:
3624:{\displaystyle X.}
3621:
3598:
3581:then at least one
3571:
3519:
3499:
3453:
3433:
3404:
3384:
3364:
3344:
3305:
3285:
3262:
3245:{\displaystyle X,}
3242:
3219:
3199:
3182:{\displaystyle X.}
3179:
3156:
3136:{\displaystyle X.}
3133:
3110:
3090:
3070:
3053:{\displaystyle X,}
3050:
3027:
3008:{\displaystyle X.}
3005:
2982:
2962:
2942:
2925:{\displaystyle X,}
2922:
2899:
2880:{\displaystyle Y.}
2877:
2854:
2837:{\displaystyle X,}
2834:
2811:
2787:{\displaystyle X.}
2784:
2761:
2741:
2721:
2704:{\displaystyle X,}
2701:
2678:
2659:{\displaystyle X.}
2656:
2633:
2613:
2593:
2576:{\displaystyle X,}
2573:
2550:
2531:{\displaystyle X.}
2528:
2505:
2488:{\displaystyle Y,}
2485:
2462:
2441:{\displaystyle Y.}
2438:
2415:
2395:
2378:{\displaystyle X.}
2375:
2348:
2328:
2278:{\displaystyle X,}
2275:
2241:
2221:
2201:
2191:whose interior in
2181:
2158:
2129:
2099:
2001:
1962:
1942:
1901:
1869:
1845:
1813:
1769:
1743:
1717:
1689:
1667:
1657:There is a subset
1621:
1605:is a nonmeagre sub
1595:
1561:
1528:
1457:
1425:
1388:
1366:
1344:
1279:
1254:
1226:
1201:
1179:
1157:
1120:
1088:
1059:{\displaystyle X=}
1056:
1015:
983:
943:
843:nonmeagre subspace
831:
811:
787:
767:
743:
723:
699:
679:
655:{\displaystyle X.}
652:
625:
601:
581:
554:{\displaystyle X,}
551:
523:{\displaystyle X,}
520:
490:
470:
428:
418:The qualifier "in
410:{\displaystyle X.}
407:
379:{\displaystyle X,}
376:
348:{\displaystyle X,}
345:
316:
291:
266:{\displaystyle X,}
263:
235:{\displaystyle X,}
232:
203:
172:
155:{\displaystyle X,}
152:
118:
5350:978-0-486-43479-7
5313:978-0-07-054236-5
5254:978-3-540-64563-4
5233:Bourbaki, Nicolas
5138:, pp. 42–43.
4877:Property of Baire
4820:{\displaystyle X}
4796:{\displaystyle Q}
4753:
4742:{\displaystyle Q}
4722:{\displaystyle X}
4702:{\displaystyle P}
4573:{\displaystyle P}
4486:be any subset of
4479:{\displaystyle X}
4432:{\displaystyle Y}
4388:{\displaystyle Y}
4373:Banach–Mazur game
4367:Banach–Mazur game
4150:
4101:
4013:{\displaystyle 0}
3904:{\displaystyle ,}
3869:{\displaystyle 0}
3817:{\displaystyle 1}
3743:{\displaystyle 1}
3522:{\displaystyle X}
3456:{\displaystyle X}
3407:{\displaystyle X}
3387:{\displaystyle X}
3367:{\displaystyle X}
3347:{\displaystyle X}
3308:{\displaystyle X}
3288:{\displaystyle X}
3265:{\displaystyle X}
3222:{\displaystyle X}
3202:{\displaystyle X}
3159:{\displaystyle X}
3113:{\displaystyle A}
3093:{\displaystyle Y}
3073:{\displaystyle A}
3030:{\displaystyle Y}
2985:{\displaystyle A}
2965:{\displaystyle Y}
2945:{\displaystyle A}
2902:{\displaystyle Y}
2857:{\displaystyle A}
2814:{\displaystyle A}
2764:{\displaystyle A}
2744:{\displaystyle Y}
2724:{\displaystyle A}
2681:{\displaystyle Y}
2636:{\displaystyle A}
2616:{\displaystyle Y}
2596:{\displaystyle A}
2553:{\displaystyle Y}
2508:{\displaystyle A}
2465:{\displaystyle A}
2418:{\displaystyle X}
2405:may be meagre in
2398:{\displaystyle A}
2360:subspace topology
2351:{\displaystyle Y}
2244:{\displaystyle X}
2224:{\displaystyle X}
2204:{\displaystyle X}
2184:{\displaystyle X}
2004:{\displaystyle ,}
1965:{\displaystyle A}
1872:{\displaystyle A}
1670:{\displaystyle H}
834:{\displaystyle X}
814:{\displaystyle X}
790:{\displaystyle A}
770:{\displaystyle X}
755:subspace topology
746:{\displaystyle X}
726:{\displaystyle A}
702:{\displaystyle X}
682:{\displaystyle X}
628:{\displaystyle X}
604:{\displaystyle X}
493:{\displaystyle X}
473:{\displaystyle A}
431:{\displaystyle X}
319:{\displaystyle X}
294:{\displaystyle X}
206:{\displaystyle X}
175:{\displaystyle X}
130:topological space
121:{\displaystyle X}
58:that is small or
56:topological space
16:(Redirected from
5388:
5376:General topology
5362:
5332:General Topology
5325:
5289:
5266:
5228:
5192:
5185:
5179:
5177:
5175:
5163:
5157:
5156:
5145:
5139:
5133:
5127:
5121:
5115:
5109:
5103:
5097:
5086:
5081:
5075:
5070:
5061:
5055:
5049:
5048:
5046:
5021:
5015:
5009:
4990:
4984:
4975:
4973:
4953:
4944:
4938:
4936:
4924:
4918:
4917:
4909:
4903:
4897:
4865:Generic property
4826:
4824:
4823:
4818:
4805:winning strategy
4802:
4800:
4799:
4794:
4782:
4780:
4779:
4774:
4772:
4771:
4758:
4748:
4746:
4745:
4740:
4728:
4726:
4725:
4720:
4708:
4706:
4705:
4700:
4688:
4686:
4685:
4680:
4669:
4668:
4656:
4655:
4643:
4642:
4626:
4624:
4623:
4618:
4616:
4615:
4602:
4600:
4599:
4594:
4579:
4577:
4576:
4571:
4559:
4557:
4556:
4551:
4543:
4542:
4508:
4506:
4505:
4500:
4485:
4483:
4482:
4477:
4465:
4463:
4462:
4457:
4452:
4451:
4438:
4436:
4435:
4430:
4418:
4416:
4415:
4410:
4408:
4407:
4394:
4392:
4391:
4386:
4362:
4360:
4359:
4354:
4352:
4351:
4329:
4327:
4326:
4321:
4319:
4318:
4294:
4292:
4291:
4286:
4284:
4283:
4265:
4263:
4262:
4257:
4255:
4254:
4229:
4227:
4226:
4221:
4213:
4212:
4200:
4199:
4183:
4181:
4180:
4175:
4173:
4169:
4168:
4167:
4156:
4152:
4143:
4132:
4131:
4119:
4118:
4107:
4103:
4094:
4083:
4082:
4067:
4062:
4046:
4041:
4019:
4017:
4016:
4011:
3999:
3997:
3996:
3991:
3989:
3974:
3972:
3971:
3968:{\displaystyle }
3966:
3942:
3940:
3939:
3936:{\displaystyle }
3934:
3910:
3908:
3907:
3902:
3875:
3873:
3872:
3867:
3855:
3853:
3852:
3849:{\displaystyle }
3847:
3823:
3821:
3820:
3815:
3801:
3799:
3798:
3793:
3781:
3779:
3778:
3775:{\displaystyle }
3773:
3749:
3747:
3746:
3741:
3729:
3727:
3726:
3721:
3702:
3700:
3699:
3696:{\displaystyle }
3694:
3669:Lebesgue measure
3666:
3664:
3663:
3658:
3656:
3645:A meagre set in
3640:Lebesgue measure
3630:
3628:
3627:
3622:
3607:
3605:
3604:
3599:
3597:
3596:
3580:
3578:
3577:
3572:
3564:
3563:
3551:
3550:
3528:
3526:
3525:
3520:
3508:
3506:
3505:
3500:
3492:
3491:
3479:
3478:
3462:
3460:
3459:
3454:
3442:
3440:
3439:
3434:
3413:
3411:
3410:
3405:
3393:
3391:
3390:
3385:
3373:
3371:
3370:
3365:
3353:
3351:
3350:
3345:
3314:
3312:
3311:
3306:
3294:
3292:
3291:
3286:
3271:
3269:
3268:
3263:
3252:being meagre in
3251:
3249:
3248:
3243:
3228:
3226:
3225:
3220:
3208:
3206:
3205:
3200:
3189:Every subset of
3188:
3186:
3185:
3180:
3165:
3163:
3162:
3157:
3142:
3140:
3139:
3134:
3120:is nonmeagre in
3119:
3117:
3116:
3111:
3099:
3097:
3096:
3091:
3080:is nonmeagre in
3079:
3077:
3076:
3071:
3059:
3057:
3056:
3051:
3036:
3034:
3033:
3028:
3014:
3012:
3011:
3006:
2992:is nonmeagre in
2991:
2989:
2988:
2983:
2971:
2969:
2968:
2963:
2952:is nonmeagre in
2951:
2949:
2948:
2943:
2931:
2929:
2928:
2923:
2908:
2906:
2905:
2900:
2886:
2884:
2883:
2878:
2864:is nonmeagre in
2863:
2861:
2860:
2855:
2843:
2841:
2840:
2835:
2821:is nonmeagre in
2820:
2818:
2817:
2812:
2793:
2791:
2790:
2785:
2770:
2768:
2767:
2762:
2750:
2748:
2747:
2742:
2730:
2728:
2727:
2722:
2710:
2708:
2707:
2702:
2687:
2685:
2684:
2679:
2665:
2663:
2662:
2657:
2642:
2640:
2639:
2634:
2622:
2620:
2619:
2614:
2602:
2600:
2599:
2594:
2582:
2580:
2579:
2574:
2559:
2557:
2556:
2551:
2537:
2535:
2534:
2529:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2471:
2469:
2468:
2463:
2447:
2445:
2444:
2439:
2424:
2422:
2421:
2416:
2404:
2402:
2401:
2396:
2384:
2382:
2381:
2376:
2357:
2355:
2354:
2349:
2337:
2335:
2334:
2329:
2296:. Dually, all
2284:
2282:
2281:
2276:
2261:
2260:
2250:
2248:
2247:
2242:
2230:
2228:
2227:
2222:
2210:
2208:
2207:
2202:
2190:
2188:
2187:
2182:
2167:
2165:
2164:
2159:
2138:
2136:
2135:
2130:
2108:
2106:
2105:
2100:
2072:are Baire spaces
2010:
2008:
2007:
2002:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1910:
1908:
1907:
1904:{\displaystyle }
1902:
1878:
1876:
1875:
1870:
1854:
1852:
1851:
1848:{\displaystyle }
1846:
1822:
1820:
1819:
1814:
1778:
1776:
1775:
1770:
1752:
1750:
1749:
1744:
1726:
1724:
1723:
1718:
1716:
1698:
1696:
1695:
1690:
1688:
1676:
1674:
1673:
1668:
1637:rational numbers
1630:
1628:
1627:
1622:
1620:
1604:
1602:
1601:
1596:
1582:
1570:
1568:
1567:
1562:
1560:
1559:
1554:
1537:
1535:
1534:
1529:
1512:
1498:
1490:
1466:
1464:
1463:
1458:
1453:
1452:
1447:
1434:
1432:
1431:
1426:
1412:
1397:
1395:
1394:
1389:
1387:
1375:
1373:
1372:
1367:
1365:
1353:
1351:
1350:
1345:
1328:
1288:
1286:
1285:
1280:
1275:
1263:
1261:
1260:
1255:
1253:
1235:
1233:
1232:
1227:
1222:
1210:
1208:
1207:
1202:
1200:
1188:
1186:
1185:
1180:
1178:
1166:
1164:
1163:
1158:
1156:
1129:
1127:
1126:
1121:
1097:
1095:
1094:
1091:{\displaystyle }
1089:
1065:
1063:
1062:
1057:
1024:
1022:
1021:
1018:{\displaystyle }
1016:
992:
990:
989:
984:
982:
952:
950:
949:
944:
939:
840:
838:
837:
832:
820:
818:
817:
812:
796:
794:
793:
788:
776:
774:
773:
768:
752:
750:
749:
744:
732:
730:
729:
724:
708:
706:
705:
700:
688:
686:
685:
680:
661:
659:
658:
653:
634:
632:
631:
626:
610:
608:
607:
602:
590:
588:
587:
582:
560:
558:
557:
552:
537:
536:
529:
527:
526:
521:
506:
505:
499:
497:
496:
491:
479:
477:
476:
471:
455:
454:
447:
446:
437:
435:
434:
429:
416:
414:
413:
408:
393:
392:
385:
383:
382:
377:
362:
361:
360:nonmeagre subset
354:
352:
351:
346:
332:
331:
325:
323:
322:
317:
300:
298:
297:
292:
280:
279:
272:
270:
269:
264:
249:
248:
241:
239:
238:
233:
219:
218:
212:
210:
209:
204:
181:
179:
178:
173:
161:
159:
158:
153:
127:
125:
124:
119:
36:general topology
21:
5396:
5395:
5391:
5390:
5389:
5387:
5386:
5385:
5366:
5365:
5351:
5314:
5286:
5255:
5217:
5201:
5196:
5195:
5186:
5182:
5164:
5160:
5147:
5146:
5142:
5134:
5130:
5126:, Theorem 25.2.
5122:
5118:
5110:
5106:
5098:
5089:
5082:
5078:
5071:
5064:
5060:, Theorem 25.5.
5056:
5052:
5022:
5018:
5010:
4993:
4985:
4978:
4951:
4945:
4941:
4925:
4921:
4910:
4906:
4898:
4891:
4886:
4859:Barrelled space
4855:
4834:
4829:
4812:
4809:
4808:
4807:if and only if
4788:
4785:
4784:
4767:
4766:
4764:
4761:
4760:
4756:
4734:
4731:
4730:
4714:
4711:
4710:
4694:
4691:
4690:
4664:
4660:
4651:
4647:
4638:
4634:
4632:
4629:
4628:
4611:
4610:
4608:
4605:
4604:
4585:
4582:
4581:
4565:
4562:
4561:
4538:
4537:
4514:
4511:
4510:
4491:
4488:
4487:
4471:
4468:
4467:
4447:
4446:
4444:
4441:
4440:
4424:
4421:
4420:
4403:
4402:
4400:
4397:
4396:
4380:
4377:
4376:
4369:
4347:
4343:
4341:
4338:
4337:
4314:
4310:
4308:
4305:
4304:
4279:
4275:
4273:
4270:
4269:
4250:
4246:
4244:
4241:
4240:
4236:
4208:
4204:
4195:
4191:
4189:
4186:
4185:
4157:
4141:
4137:
4136:
4127:
4123:
4108:
4092:
4088:
4087:
4078:
4074:
4073:
4069:
4063:
4052:
4042:
4031:
4025:
4022:
4021:
4005:
4002:
4001:
3985:
3983:
3980:
3979:
3948:
3945:
3944:
3916:
3913:
3912:
3881:
3878:
3877:
3861:
3858:
3857:
3829:
3826:
3825:
3809:
3806:
3805:
3787:
3784:
3783:
3755:
3752:
3751:
3735:
3732:
3731:
3715:
3712:
3711:
3704:fat Cantor sets
3676:
3673:
3672:
3652:
3650:
3647:
3646:
3636:
3613:
3610:
3609:
3592:
3588:
3586:
3583:
3582:
3559:
3555:
3546:
3542:
3534:
3531:
3530:
3514:
3511:
3510:
3509:are subsets of
3487:
3483:
3474:
3470:
3468:
3465:
3464:
3448:
3445:
3444:
3422:
3419:
3418:
3399:
3396:
3395:
3379:
3376:
3375:
3359:
3356:
3355:
3339:
3336:
3335:
3321:
3300:
3297:
3296:
3280:
3277:
3276:
3257:
3254:
3253:
3234:
3231:
3230:
3214:
3211:
3210:
3194:
3191:
3190:
3171:
3168:
3167:
3151:
3148:
3147:
3125:
3122:
3121:
3105:
3102:
3101:
3100:if and only if
3085:
3082:
3081:
3065:
3062:
3061:
3042:
3039:
3038:
3022:
3019:
3018:
2997:
2994:
2993:
2977:
2974:
2973:
2972:if and only if
2957:
2954:
2953:
2937:
2934:
2933:
2914:
2911:
2910:
2894:
2891:
2890:
2869:
2866:
2865:
2849:
2846:
2845:
2826:
2823:
2822:
2806:
2803:
2802:
2776:
2773:
2772:
2756:
2753:
2752:
2751:if and only if
2736:
2733:
2732:
2716:
2713:
2712:
2693:
2690:
2689:
2673:
2670:
2669:
2648:
2645:
2644:
2628:
2625:
2624:
2623:if and only if
2608:
2605:
2604:
2588:
2585:
2584:
2565:
2562:
2561:
2545:
2542:
2541:
2520:
2517:
2516:
2500:
2497:
2496:
2477:
2474:
2473:
2457:
2454:
2453:
2430:
2427:
2426:
2410:
2407:
2406:
2390:
2387:
2386:
2367:
2364:
2363:
2343:
2340:
2339:
2308:
2305:
2304:
2267:
2264:
2263:
2256:
2255:
2236:
2233:
2232:
2216:
2213:
2212:
2196:
2193:
2192:
2176:
2173:
2172:
2144:
2141:
2140:
2118:
2115:
2114:
2082:
2079:
2078:
2068:locally compact
2049:Every nonempty
2038:every nonempty
2030:Every nonempty
2028:
1981:
1978:
1977:
1957:
1954:
1953:
1916:
1913:
1912:
1884:
1881:
1880:
1864:
1861:
1860:
1828:
1825:
1824:
1787:
1784:
1783:
1758:
1755:
1754:
1732:
1729:
1728:
1712:
1704:
1701:
1700:
1684:
1682:
1679:
1678:
1662:
1659:
1658:
1633:isolated points
1616:
1614:
1611:
1610:
1578:
1576:
1573:
1572:
1555:
1550:
1549:
1547:
1544:
1543:
1538:is a meagre sub
1508:
1494:
1486:
1475:
1472:
1471:
1448:
1443:
1442:
1440:
1437:
1436:
1408:
1406:
1403:
1402:
1383:
1381:
1378:
1377:
1361:
1359:
1356:
1355:
1324:
1301:
1298:
1297:
1271:
1269:
1266:
1265:
1249:
1247:
1244:
1243:
1218:
1216:
1213:
1212:
1196:
1194:
1191:
1190:
1174:
1172:
1169:
1168:
1152:
1150:
1147:
1146:
1138:
1103:
1100:
1099:
1071:
1068:
1067:
1033:
1030:
1029:
998:
995:
994:
978:
958:
955:
954:
935:
888:
885:
884:
878:
858:second category
826:
823:
822:
806:
803:
802:
799:meagre subspace
782:
779:
778:
762:
759:
758:
738:
735:
734:
718:
715:
714:
694:
691:
690:
674:
671:
670:
644:
641:
640:
620:
617:
616:
596:
593:
592:
570:
567:
566:
543:
540:
539:
534:
533:
512:
509:
508:
503:
502:
485:
482:
481:
465:
462:
461:
452:
451:
449:(respectively,
444:
443:
423:
420:
419:
399:
396:
395:
391:second category
390:
389:
368:
365:
364:
359:
358:
337:
334:
333:
329:
328:
311:
308:
307:
286:
283:
282:
277:
276:
255:
252:
251:
246:
245:
224:
221:
220:
216:
215:
198:
195:
194:
167:
164:
163:
144:
141:
140:
113:
110:
109:
106:
42:(also called a
28:
23:
22:
15:
12:
11:
5:
5394:
5384:
5383:
5378:
5364:
5363:
5349:
5326:
5312:
5290:
5284:
5267:
5253:
5229:
5216:978-1584888666
5215:
5200:
5197:
5194:
5193:
5180:
5158:
5140:
5128:
5116:
5104:
5087:
5076:
5062:
5050:
5037:(1): 174–179.
5016:
4991:
4989:, p. 192.
4976:
4964:(2): 157–166.
4939:
4919:
4904:
4902:, p. 389.
4888:
4887:
4885:
4882:
4881:
4880:
4874:
4871:Negligible set
4868:
4862:
4854:
4851:
4833:
4830:
4816:
4792:
4770:
4751:
4738:
4718:
4698:
4678:
4675:
4672:
4667:
4663:
4659:
4654:
4650:
4646:
4641:
4637:
4614:
4592:
4589:
4569:
4549:
4546:
4541:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4498:
4495:
4475:
4455:
4450:
4428:
4406:
4384:
4368:
4365:
4350:
4346:
4317:
4313:
4282:
4278:
4253:
4249:
4235:
4232:
4219:
4216:
4211:
4207:
4203:
4198:
4194:
4172:
4166:
4163:
4160:
4155:
4149:
4146:
4140:
4135:
4130:
4126:
4122:
4117:
4114:
4111:
4106:
4100:
4097:
4091:
4086:
4081:
4077:
4072:
4066:
4061:
4058:
4055:
4051:
4045:
4040:
4037:
4034:
4030:
4009:
3988:
3964:
3961:
3958:
3955:
3952:
3932:
3929:
3926:
3923:
3920:
3900:
3897:
3894:
3891:
3888:
3885:
3865:
3845:
3842:
3839:
3836:
3833:
3813:
3791:
3771:
3768:
3765:
3762:
3759:
3739:
3719:
3692:
3689:
3686:
3683:
3680:
3667:need not have
3655:
3635:
3632:
3620:
3617:
3595:
3591:
3570:
3567:
3562:
3558:
3554:
3549:
3545:
3541:
3538:
3518:
3498:
3495:
3490:
3486:
3482:
3477:
3473:
3452:
3432:
3429:
3426:
3403:
3383:
3363:
3343:
3329:barreled space
3320:
3317:
3304:
3284:
3261:
3241:
3238:
3218:
3198:
3178:
3175:
3155:
3144:
3143:
3132:
3129:
3109:
3089:
3069:
3049:
3046:
3026:
3015:
3004:
3001:
2981:
2961:
2941:
2921:
2918:
2898:
2887:
2876:
2873:
2853:
2833:
2830:
2810:
2795:
2794:
2783:
2780:
2760:
2740:
2720:
2700:
2697:
2677:
2666:
2655:
2652:
2632:
2612:
2592:
2572:
2569:
2549:
2538:
2527:
2524:
2504:
2484:
2481:
2461:
2437:
2434:
2414:
2394:
2374:
2371:
2347:
2327:
2324:
2321:
2318:
2315:
2312:
2294:negligible set
2274:
2271:
2240:
2220:
2200:
2180:
2157:
2154:
2151:
2148:
2128:
2125:
2122:
2113:then a subset
2098:
2095:
2092:
2089:
2086:
2027:
2024:
2020:Martin's axiom
2000:
1997:
1994:
1991:
1988:
1985:
1961:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1900:
1897:
1894:
1891:
1888:
1868:
1844:
1841:
1838:
1835:
1832:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1768:
1765:
1762:
1742:
1739:
1736:
1715:
1711:
1708:
1687:
1666:
1654:is nonmeagre.
1652:discrete space
1648:isolated point
1619:
1608:
1594:
1591:
1588:
1585:
1581:
1558:
1553:
1541:
1527:
1524:
1521:
1518:
1515:
1511:
1507:
1504:
1501:
1497:
1493:
1489:
1485:
1482:
1479:
1456:
1451:
1446:
1424:
1421:
1418:
1415:
1411:
1386:
1364:
1343:
1340:
1337:
1334:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1278:
1274:
1252:
1225:
1221:
1199:
1177:
1155:
1143:isolated point
1136:
1119:
1116:
1113:
1110:
1107:
1087:
1084:
1081:
1078:
1075:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1014:
1011:
1008:
1005:
1002:
981:
977:
974:
971:
968:
965:
962:
942:
938:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
877:
874:
854:first category
830:
810:
786:
766:
742:
722:
698:
678:
651:
648:
624:
600:
580:
577:
574:
550:
547:
519:
516:
489:
469:
427:
406:
403:
375:
372:
344:
341:
315:
290:
278:first category
262:
259:
231:
228:
202:
171:
151:
148:
117:
105:
102:
26:
9:
6:
4:
3:
2:
5393:
5382:
5379:
5377:
5374:
5373:
5371:
5360:
5356:
5352:
5346:
5342:
5338:
5337:Mineola, N.Y.
5334:
5333:
5327:
5323:
5319:
5315:
5309:
5305:
5301:
5300:
5295:
5294:Rudin, Walter
5291:
5287:
5285:0-387-90508-1
5281:
5277:
5273:
5268:
5264:
5260:
5256:
5250:
5246:
5242:
5238:
5234:
5230:
5226:
5222:
5218:
5212:
5208:
5203:
5202:
5190:
5184:
5174:
5169:
5162:
5154:
5150:
5144:
5137:
5132:
5125:
5120:
5114:, p. 62.
5113:
5108:
5102:, p. 43.
5101:
5096:
5094:
5092:
5085:
5080:
5074:
5069:
5067:
5059:
5054:
5045:
5040:
5036:
5033:
5032:
5027:
5020:
5013:
5008:
5006:
5004:
5002:
5000:
4998:
4996:
4988:
4987:Bourbaki 1989
4983:
4981:
4971:
4967:
4963:
4959:
4958:
4950:
4943:
4934:
4930:
4923:
4915:
4908:
4901:
4896:
4894:
4889:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4856:
4850:
4847:
4843:
4839:
4828:
4814:
4806:
4790:
4750:
4736:
4716:
4696:
4676:
4673:
4670:
4665:
4661:
4657:
4652:
4648:
4644:
4639:
4635:
4590:
4587:
4567:
4547:
4534:
4531:
4528:
4525:
4519:
4516:
4496:
4493:
4473:
4453:
4426:
4382:
4374:
4364:
4348:
4344:
4335:
4331:
4315:
4311:
4301:
4296:
4280:
4276:
4267:
4251:
4247:
4231:
4217:
4214:
4209:
4205:
4201:
4196:
4192:
4170:
4164:
4161:
4158:
4153:
4147:
4144:
4138:
4133:
4128:
4124:
4120:
4115:
4112:
4109:
4104:
4098:
4095:
4089:
4084:
4079:
4075:
4070:
4059:
4056:
4053:
4049:
4038:
4035:
4032:
4028:
4007:
4000:with measure
3976:
3959:
3956:
3953:
3927:
3924:
3921:
3898:
3892:
3889:
3886:
3863:
3840:
3837:
3834:
3811:
3802:
3789:
3782:with measure
3766:
3763:
3760:
3737:
3717:
3709:
3705:
3687:
3684:
3681:
3670:
3643:
3641:
3631:
3618:
3615:
3593:
3589:
3568:
3565:
3560:
3556:
3552:
3547:
3543:
3539:
3536:
3516:
3496:
3493:
3488:
3484:
3480:
3475:
3471:
3450:
3430:
3427:
3424:
3415:
3401:
3381:
3361:
3341:
3332:
3330:
3326:
3316:
3315:is nonempty.
3302:
3282:
3273:
3259:
3239:
3236:
3216:
3196:
3176:
3173:
3153:
3130:
3127:
3107:
3087:
3067:
3047:
3044:
3024:
3016:
3002:
2999:
2979:
2959:
2939:
2919:
2916:
2896:
2888:
2874:
2871:
2851:
2831:
2828:
2808:
2800:
2799:
2798:
2781:
2778:
2771:is meagre in
2758:
2738:
2731:is meagre in
2718:
2698:
2695:
2675:
2667:
2653:
2650:
2643:is meagre in
2630:
2610:
2603:is meagre in
2590:
2570:
2567:
2547:
2539:
2525:
2522:
2515:is meagre in
2502:
2482:
2479:
2472:is meagre in
2459:
2451:
2450:
2449:
2435:
2432:
2412:
2392:
2372:
2369:
2362:induced from
2361:
2345:
2325:
2322:
2319:
2316:
2313:
2310:
2301:
2299:
2295:
2291:
2286:
2272:
2269:
2259:
2252:
2238:
2218:
2198:
2178:
2169:
2152:
2146:
2126:
2123:
2120:
2112:
2111:homeomorphism
2096:
2090:
2087:
2084:
2075:
2073:
2069:
2066:
2062:
2061:metric spaces
2059:
2056:
2052:
2047:
2045:
2041:
2037:
2033:
2023:
2021:
2017:
2012:
1998:
1992:
1989:
1986:
1975:
1959:
1933:
1930:
1927:
1918:
1895:
1892:
1889:
1866:
1858:
1839:
1836:
1833:
1804:
1801:
1798:
1789:
1782:In the space
1780:
1766:
1760:
1740:
1737:
1734:
1709:
1706:
1664:
1655:
1653:
1649:
1644:
1642:
1638:
1634:
1606:
1589:
1583:
1556:
1539:
1519:
1513:
1502:
1491:
1480:
1477:
1468:
1454:
1449:
1419:
1413:
1399:
1338:
1332:
1321:
1315:
1312:
1309:
1294:
1292:
1276:
1241:
1236:
1223:
1144:
1140:
1131:
1114:
1111:
1108:
1082:
1079:
1076:
1050:
1047:
1044:
1038:
1035:
1026:
1009:
1006:
1003:
975:
969:
966:
963:
932:
926:
923:
920:
911:
905:
902:
899:
893:
890:
881:
873:
871:
867:
863:
859:
855:
850:
848:
844:
828:
808:
800:
784:
764:
757:induced from
756:
753:is given the
740:
720:
711:
696:
676:
667:
666:
662:
649:
646:
638:
622:
614:
598:
591:is meagre in
578:
572:
565:
561:
548:
545:
530:
517:
514:
487:
467:
458:
456:
448:
439:
425:
417:
404:
401:
386:
373:
370:
355:
342:
339:
313:
305:
304:nowhere dense
301:
288:
273:
260:
257:
247:meagre subset
242:
229:
226:
200:
191:
189:
185:
169:
149:
146:
138:
137:nowhere dense
133:
131:
115:
101:
99:
95:
91:
86:
84:
80:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5331:
5298:
5275:
5240:
5236:
5206:
5199:Bibliography
5183:
5161:
5153:MathOverflow
5152:
5143:
5131:
5124:Willard 2004
5119:
5107:
5079:
5058:Willard 2004
5053:
5034:
5031:Studia Math.
5029:
5019:
4961:
4955:
4942:
4932:
4922:
4916:. Macmillan.
4907:
4835:
4752:
4370:
4297:
4237:
3977:
3803:
3644:
3637:
3416:
3333:
3323:A nonmeagre
3322:
3274:
3145:
3037:is dense in
2796:
2688:is dense in
2302:
2287:
2257:
2253:
2170:
2168:is meagre.
2076:
2048:
2029:
2013:
1781:
1656:
1645:
1469:
1400:
1295:
1237:
1133:A countable
1132:
1027:
882:
879:
865:
857:
853:
851:
842:
798:
712:
668:
664:
663:
637:intersection
532:
501:
459:
450:
442:
440:
388:
357:
330:nonmeagre in
327:
275:
244:
214:
193:A subset of
192:
134:
108:Throughout,
107:
87:
72:
67:
63:
47:
43:
39:
32:mathematical
29:
18:Comeagre set
5112:Oxtoby 1980
4935:. 3: 1–123.
4827:is meagre.
3706:, like the
2909:is open in
2560:is open in
2063:as well as
2051:Baire space
2032:Baire space
1727:, the sets
1641:Baire space
306:subsets of
104:Definitions
92:and of the
90:Baire space
5370:Categories
5187:S. Saito,
5173:2206.10754
5136:Rudin 1991
5100:Rudin 1991
4846:involution
3529:such that
3319:Properties
1859:, the set
1609:(that is,
1240:Cantor set
862:René Baire
852:The terms
564:complement
500:is called
387:or of the
274:or of the
213:is called
186:has empty
139:subset of
128:will be a
60:negligible
44:meager set
40:meagre set
5263:246032063
5235:(1989) .
5225:144216834
4937:, page 65
4838:null sets
4674:⋯
4671:⊇
4658:⊇
4645:⊇
4349:δ
4316:δ
4281:σ
4252:σ
4218:…
4085:−
4065:∞
4050:⋃
4044:∞
4029:⋂
3569:⋯
3566:∪
3553:∪
3540:⊆
3497:…
3428:⊆
2320:⊆
2314:⊆
2298:supersets
2124:⊆
2094:→
2065:Hausdorff
1764:∖
1738:∩
1710:⊆
1584:×
1514:×
1503:∪
1492:×
1414:×
1401:The line
1333:∪
1322:∩
976:∩
933:∩
912:∪
872:in 1948.
576:∖
460:A subset
453:nonmeagre
217:meagre in
83:countably
64:nonmeagre
34:field of
5322:21163277
5296:(1991).
4853:See also
4759:For any
4300:interior
2385:The set
2358:has the
2303:Suppose
2058:(pseudo)
2055:complete
1470:The set
1296:The set
1141:without
1066:the set
953:the set
876:Examples
870:Bourbaki
613:cofinite
535:residual
504:comeagre
188:interior
5243:].
4754:Theorem
4749:wins.
4689:Player
4375:. Let
3463:and if
2290:σ-ideal
2070:spaces
562:if its
184:closure
75:σ-ideal
50:) is a
30:In the
5359:115240
5357:
5347:
5320:
5310:
5282:
5261:
5251:
5223:
5213:
5178:(p.25)
4803:has a
4184:where
3943:since
2338:where
866:meagre
445:meagre
182:whose
52:subset
5239:[
5168:arXiv
4952:(PDF)
4884:Notes
3327:is a
3060:then
2932:then
2844:then
2711:then
2583:then
2495:then
2109:is a
1607:space
1139:space
79:union
66:, or
54:of a
46:or a
5355:OCLC
5345:ISBN
5318:OCLC
5308:ISBN
5280:ISBN
5259:OCLC
5249:ISBN
5221:OCLC
5211:ISBN
4580:and
4466:and
4334:open
2254:The
2251:).
1753:and
1238:The
856:and
38:, a
5039:doi
4966:doi
4330:set
4266:set
3824:in
3642:.
3417:If
3331:.
3017:If
2889:If
2801:If
2668:If
2540:If
2452:If
1976:on
1643:.
1542:of
1540:set
801:of
538:in
531:or
507:in
480:of
394:in
363:of
281:in
250:of
132:.
81:of
5372::
5353:.
5343:.
5339::
5335:.
5316:.
5306:.
5274:.
5257:.
5219:.
5151:.
5090:^
5065:^
5028:.
4994:^
4979:^
4962:49
4960:.
4954:.
4931:.
4892:^
4020::
3790:1.
3718:1.
1293:.
356:a
243:a
100:.
5361:.
5324:.
5288:.
5265:.
5227:.
5176:.
5170::
5155:.
5047:.
5041::
5035:3
4972:.
4968::
4815:X
4791:Q
4769:W
4737:Q
4717:X
4697:P
4677:.
4666:3
4662:W
4653:2
4649:W
4640:1
4636:W
4613:W
4591:,
4588:Q
4568:P
4548:.
4545:)
4540:W
4535:,
4532:Y
4529:,
4526:X
4523:(
4520:Z
4517:M
4497:.
4494:Y
4474:X
4454:,
4449:W
4427:Y
4405:W
4383:Y
4345:G
4312:G
4277:F
4248:F
4215:,
4210:2
4206:r
4202:,
4197:1
4193:r
4171:)
4165:m
4162:+
4159:n
4154:)
4148:2
4145:1
4139:(
4134:+
4129:n
4125:r
4121:,
4116:m
4113:+
4110:n
4105:)
4099:2
4096:1
4090:(
4080:n
4076:r
4071:(
4060:1
4057:=
4054:n
4039:1
4036:=
4033:m
4008:0
3987:R
3963:]
3960:1
3957:,
3954:0
3951:[
3931:]
3928:1
3925:,
3922:0
3919:[
3899:,
3896:]
3893:1
3890:,
3887:0
3884:[
3864:0
3844:]
3841:1
3838:,
3835:0
3832:[
3812:1
3770:]
3767:1
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3761:0
3758:[
3738:1
3691:]
3688:1
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3682:0
3679:[
3654:R
3619:.
3616:X
3594:n
3590:S
3561:2
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3494:,
3489:2
3485:S
3481:,
3476:1
3472:S
3451:X
3431:X
3425:B
3402:X
3382:X
3362:X
3342:X
3303:X
3283:X
3260:X
3240:,
3237:X
3217:X
3197:X
3177:.
3174:X
3154:X
3131:.
3128:X
3108:A
3088:Y
3068:A
3048:,
3045:X
3025:Y
3003:.
3000:X
2980:A
2960:Y
2940:A
2920:,
2917:X
2897:Y
2875:.
2872:Y
2852:A
2832:,
2829:X
2809:A
2782:.
2779:X
2759:A
2739:Y
2719:A
2699:,
2696:X
2676:Y
2654:.
2651:X
2631:A
2611:Y
2591:A
2571:,
2568:X
2548:Y
2526:.
2523:X
2503:A
2483:,
2480:Y
2460:A
2436:.
2433:Y
2413:X
2393:A
2373:.
2370:X
2346:Y
2326:,
2323:X
2317:Y
2311:A
2273:,
2270:X
2239:X
2219:X
2199:X
2179:X
2156:)
2153:S
2150:(
2147:h
2127:X
2121:S
2097:X
2091:X
2088::
2085:h
1999:,
1996:]
1993:1
1990:,
1987:0
1984:[
1960:A
1940:)
1937:]
1934:1
1931:,
1928:0
1925:[
1922:(
1919:C
1899:]
1896:1
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1890:0
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1867:A
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1840:1
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1811:)
1808:]
1805:1
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1796:[
1793:(
1790:C
1767:H
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1593:}
1590:0
1587:{
1580:R
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1526:)
1523:}
1520:0
1517:{
1510:R
1506:(
1500:)
1496:Q
1488:Q
1484:(
1481:=
1478:S
1455:.
1450:2
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1423:}
1420:0
1417:{
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1342:}
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1277:.
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1109:1
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1086:]
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1054:]
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1004:0
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915:(
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906:1
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894:=
891:X
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765:X
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697:X
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650:.
647:X
623:X
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546:X
518:,
515:X
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468:A
426:X
405:.
402:X
374:,
371:X
343:,
340:X
314:X
289:X
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258:X
230:,
227:X
201:X
170:X
150:,
147:X
116:X
20:)
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