4171:
4012:
1525:
698:
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.
4676:
4837:
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.
3568:
1341:
940:
2325:
1592:
1422:
4547:
1714:
4217:
3496:
1454:
980:
1558:
4453:
4770:
4614:
4406:
1276:
1223:
3987:
3654:
3430:
2126:
1766:
1686:
1618:
1385:
1363:
1251:
1198:
1176:
1154:
578:
4350:
4317:
4282:
4253:
2096:
1740:
3595:
1117:
2155:
1939:
1810:
4590:
4496:
3789:
3717:
3618:
3239:
3176:
3130:
3047:
3002:
2919:
2874:
2831:
2781:
2698:
2653:
2570:
2525:
2482:
2435:
2372:
2272:
2011:, on each separable Banach space, there exists a discontinuous linear functional whose kernel is meagre (this statement disproves the Wilansky–Klee conjecture).
1053:
649:
548:
517:
404:
373:
342:
260:
229:
149:
4814:
4790:
4736:
4716:
4696:
4567:
4473:
4426:
4382:
4007:
3898:
3863:
3811:
3737:
3516:
3450:
3401:
3381:
3361:
3341:
3302:
3282:
3259:
3216:
3196:
3153:
3107:
3087:
3067:
3024:
2979:
2959:
2939:
2896:
2851:
2808:
2758:
2738:
2718:
2675:
2630:
2610:
2590:
2547:
2502:
2459:
2412:
2392:
2345:
2238:
2218:
2198:
2178:
1998:
1959:
1866:
1664:
828:
808:
784:
764:
740:
720:
696:
676:
622:
598:
487:
467:
425:
313:
288:
200:
169:
115:
3962:
3930:
3843:
3769:
3690:
1898:
1842:
1085:
1012:
4166:{\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)}
766:, 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
4227:
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
2000:
is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.
1462:
838:
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.
834:
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
4619:
4902:
5061:
5072:
3313:
810:, meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space
5337:
5300:
5241:
3521:
5292:
5203:
5272:
3696:
1288:
5233:
5369:
2004:
1962:
2277:
All subsets and all countable unions of meagre sets are meagre. Thus the meagre subsets of a fixed space form a
1639:
is nonmeagre (because no set containing the isolated point can be nowhere dense). In particular, every nonempty
875:
2289:
and all countable intersections of comeagre sets are comeagre. Every superset of a nonmeagre set is nonmeagre.
2295:
1563:
1393:
4501:
1691:
4917:
4176:
3455:
830:. (See the Properties and Examples sections below for the relationship between the two.) Similarly, a
1427:
945:
1534:
5364:
5291:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
4431:
2032:
869:
The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space.
835:
625:
4751:
4595:
4387:
3793:
Dually, there can be nonmeagre sets with measure zero. The complement of any meagre set of measure
1624:
is meagre, whereas any topological space that contains an isolated point is nonmeagre. Because the
678:
is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space
4834:
1688:
that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set
1628:
are countable, they are meagre as a subset of the reals and as a space—that is, they do not form a
1256:
1203:
5177:
5062:
https://mathoverflow.net/questions/3188/are-proper-linear-subspaces-of-banach-spaces-always-meager
3970:
3637:
3409:
3343:
is meagre. Consequently, any closed subset with empty interior is meagre. Thus a closed subset of
2105:
1745:
1669:
1601:
1368:
1346:
1234:
1181:
1159:
1137:
557:
4945:
4328:
4295:
4260:
4231:
552:
5073:
https://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11547-1/S0002-9904-1966-11547-1.pdf
2066:
Any subset of a meagre set is a meagre set, as is the union of countably many meagre sets. If
2060:
2024:
82:
2069:
5260:
4361:
2043:
2028:
1719:
1279:
1134:
is meagre. So it is also meagre in any space that contains it as a subspace. For example,
4830:
4829:, i.e. sets of Lebesgue measure 0. The Erdos–Sierpinski duality theorem states that if the
3699:, are closed nowhere dense and they can be constructed with a measure arbitrarily close to
3573:
1090:
2131:
8:
4428:
that have nonempty interiors such that every nonempty open set has a subset belonging to
4288:
1903:
1845:
1774:
176:
86:
4572:
4478:
4287:
Dually, just as the complement of a nowhere dense set need not be open, but has a dense
3774:
3702:
3600:
3221:
3158:
3112:
3029:
2984:
2901:
2856:
2813:
2763:
2680:
2635:
2552:
2507:
2464:
2417:
2354:
2254:
1020:
631:
530:
499:
386:
355:
324:
242:
211:
131:
5329:
5156:
4937:
4799:
4775:
4721:
4701:
4681:
4552:
4458:
4411:
4367:
3992:
3868:
3848:
3796:
3722:
3501:
3435:
3386:
3366:
3346:
3326:
3287:
3267:
3244:
3201:
3181:
3138:
3092:
3072:
3052:
3009:
2964:
2944:
2924:
2881:
2836:
2793:
2743:
2723:
2703:
2660:
2615:
2595:
2575:
2532:
2487:
2444:
2397:
2377:
2330:
2223:
2203:
2183:
2163:
2046:
1968:
1944:
1851:
1649:
813:
793:
769:
749:
725:
705:
681:
661:
607:
583:
472:
452:
410:
298:
273:
185:
172:
154:
100:
67:
3935:
3903:
3816:
3742:
3663:
1871:
1815:
1058:
985:
5343:
5333:
5306:
5296:
5286:
5268:
5247:
5237:
5209:
5199:
4865:
3627:
There exist nowhere dense subsets (which are thus meagre subsets) that have positive
2348:
2274:
the union of any family of open sets of the first category is of the first category.
2008:
743:
118:
44:
5221:
5027:
4954:
4853:
4793:
3657:
3628:
3261:
is equivalent to being meagre in itself, and similarly for the nonmeagre property.
1625:
858:
24:
1520:{\displaystyle S=(\mathbb {Q} \times \mathbb {Q} )\cup (\mathbb {R} \times \{0\})}
1387:. It is nonmeagre in itself (since as a subspace it contains an isolated point).
600:. (This use of the prefix "co" is consistent with its use in other terms such as "
5319:
4847:
2056:
2053:
4859:
3692:
3403:(because otherwise it would be nowhere dense and thus of the first category).
3317:
3284:
is nonmeagre if and only if every countable intersection of dense open sets in
2282:
2160:
Every nowhere dense subset is a meagre set. Consequently, any closed subset of
2042:
is nonmeagre but there exist nonmeagre spaces that are not Baire spaces. Since
1640:
1636:
1621:
1131:
48:
5358:
5325:
5251:
5213:
5198:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
2099:
292:
125:
71:
5310:
4958:
850:
658:
The notions of nonmeagre and comeagre should not be confused. If the space
427:" can be omitted if the ambient space is fixed and understood from context.
5282:
5137:
5032:
5019:
2049:
601:
5347:
5155:
Quintanilla, M. (2022). "The real numbers in inner models of set theory".
4963:"Following Bourbaki , a topological space is called a Baire space if ..."
4862: – Mathematical set regarded as insignificant, for analogs to meagre
2278:
2039:
2020:
1629:
1620:
is not a meagre topological space). A countable
Hausdorff space without
78:
63:
20:
5236:. Vol. 4. Berlin New York: Springer Science & Business Media.
4360:
Meagre sets have a useful alternative characterization in terms of the
1228:
77:
Meagre sets play an important role in the formulation of the notion of
51:
in a precise sense detailed below. A set that is not meagre is called
4996:
4994:
4992:
4990:
4988:
4986:
4984:
4856: – Property holding for typical examples, for analogs to residual
4284:
set made from nowhere dense sets (by taking the closure of each set).
3719:
The union of a countable number of such sets with measure approaching
4671:{\displaystyle W_{1}\supseteq W_{2}\supseteq W_{3}\supseteq \cdots .}
4292:
4228:
3660:
zero, and can even have full measure. For example, in the interval
1456:
But it is a nonmeagre subspace, that is, it is nonmeagre in itself.
5161:
4981:
4826:
4322:
2286:
1941:
is a complete metric space, it is nonmeagre. So the complement of
1178:(that is, meagre in itself with the subspace topology induced from
1123:
66:
of subsets; that is, any subset of a meagre set is meagre, and the
4884:
4882:
446:) if it is a meagre (respectively, nonmeagre) subset of itself.
4257:(countable union of closed sets), but is always contained in an
5106:
5014:
4879:
3218:
is nonmeagre in itself. And for an open set or a dense set in
85:, which is used in the proof of several fundamental results of
40:
5040:
4698:
wins if the intersection of this sequence contains a point in
2014:
3845:(for example the one in the previous paragraph) has measure
4291:(contains a dense open set), a comeagre set need not be a
5015:"Über die Baire'sche Kategorie gewisser Funktionenmengen"
4971:
4969:
1087:
is nonmeagre. But it is not comeagre, as its complement
5267:(Second ed.). New York: Springer. pp. 62–65.
1900:
that have a derivative at some point is meagre. Since
5118:
5084:
5082:
5080:
4966:
4131:
4082:
3622:
3563:{\displaystyle B\subseteq S_{1}\cup S_{2}\cup \cdots }
4802:
4778:
4754:
4724:
4704:
4684:
4622:
4598:
4575:
4555:
4504:
4481:
4461:
4434:
4414:
4390:
4370:
4331:
4298:
4263:
4234:
4179:
4015:
3995:
3973:
3938:
3906:
3871:
3851:
3819:
3799:
3777:
3745:
3725:
3705:
3666:
3640:
3603:
3576:
3524:
3504:
3458:
3438:
3412:
3389:
3369:
3349:
3329:
3290:
3270:
3247:
3224:
3204:
3184:
3161:
3141:
3115:
3095:
3075:
3055:
3032:
3012:
2987:
2967:
2947:
2927:
2904:
2884:
2859:
2839:
2816:
2796:
2766:
2746:
2726:
2706:
2683:
2663:
2638:
2618:
2598:
2578:
2555:
2535:
2510:
2490:
2467:
2447:
2420:
2400:
2380:
2357:
2333:
2298:
2257:
2226:
2206:
2186:
2166:
2134:
2108:
2072:
1971:
1947:
1906:
1874:
1854:
1818:
1777:
1748:
1722:
1694:
1672:
1652:
1604:
1566:
1537:
1465:
1430:
1396:
1371:
1349:
1291:
1259:
1237:
1206:
1184:
1162:
1140:
1093:
1061:
1023:
988:
948:
878:
816:
796:
772:
752:
728:
708:
684:
664:
634:
610:
586:
560:
533:
502:
475:
455:
413:
389:
358:
327:
301:
276:
245:
214:
188:
157:
134:
103:
59:. See below for definitions of other related terms.
5094:
5057:
5055:
4592:
alternately choose successively smaller elements of
4219:
is a sequence that enumerates the rational numbers.
5077:
702:As an additional point of terminology, if a subset
179:. See the corresponding article for more details.
4808:
4784:
4764:
4730:
4710:
4690:
4670:
4608:
4584:
4561:
4541:
4490:
4467:
4447:
4420:
4400:
4376:
4344:
4311:
4276:
4247:
4211:
4165:
4001:
3981:
3956:
3924:
3892:
3857:
3837:
3805:
3783:
3763:
3731:
3711:
3684:
3648:
3612:
3589:
3562:
3510:
3490:
3444:
3424:
3395:
3375:
3355:
3335:
3296:
3276:
3253:
3233:
3210:
3190:
3170:
3147:
3124:
3101:
3081:
3061:
3041:
3018:
2996:
2973:
2953:
2933:
2913:
2890:
2868:
2845:
2825:
2802:
2775:
2752:
2732:
2712:
2692:
2669:
2647:
2624:
2604:
2584:
2564:
2541:
2519:
2496:
2476:
2453:
2429:
2406:
2386:
2366:
2339:
2319:
2266:
2232:
2212:
2192:
2172:
2149:
2120:
2090:
2003:On an infinite-dimensional Banach, there exists a
1992:
1953:
1933:
1892:
1860:
1836:
1804:
1760:
1734:
1708:
1680:
1658:
1612:
1586:
1552:
1519:
1448:
1416:
1379:
1357:
1335:
1270:
1245:
1217:
1192:
1170:
1148:
1111:
1079:
1047:
1006:
974:
934:
822:
802:
778:
758:
734:
714:
690:
670:
643:
616:
592:
572:
542:
511:
481:
461:
419:
398:
367:
336:
307:
282:
254:
223:
194:
163:
143:
124:The definition of meagre set uses the notion of a
109:
5193:
5138:"Is there a measure zero set which isn't meagre?"
5052:
5000:
4888:
4868: – Difference of an open set by a meager set
5356:
4825:Many arguments about meagre sets also apply to
1961:, which consists of the continuous real-valued
5194:Narici, Lawrence; Beckenstein, Edward (2011).
4222:
3967:Here is another example of a nonmeagre set in
1336:{\displaystyle (\cap \mathbb {Q} )\cup \{2\}}
1278:But it is nonmeagre in itself, since it is a
1581:
1575:
1511:
1505:
1411:
1405:
1330:
1324:
628:of sets, each of whose interior is dense in
5154:
4820:
2015:Characterizations and sufficient conditions
62:The meagre subsets of a fixed space form a
5012:
935:{\displaystyle X=\cup (\cap \mathbb {Q} )}
624:if and only if it is equal to a countable
5160:
5031:
3975:
3642:
1702:
1674:
1606:
1568:
1540:
1498:
1484:
1476:
1433:
1398:
1373:
1351:
1314:
1261:
1239:
1208:
1186:
1164:
1142:
968:
925:
5220:
4975:
4918:"Sur les fonctions de variables réelles"
4900:
4850: – Type of topological vector space
2786:And correspondingly for nonmeagre sets:
2320:{\displaystyle A\subseteq Y\subseteq X,}
1587:{\displaystyle \mathbb {R} \times \{0\}}
1417:{\displaystyle \mathbb {R} \times \{0\}}
5317:
5112:
5046:
4549:In the Banach–Mazur game, two players,
4542:{\displaystyle MZ(X,Y,{\mathcal {W}}).}
3314:locally convex topological vector space
2007:whose kernel is nonmeagre. Also, under
1868:of continuous real-valued functions on
1812:of continuous real-valued functions on
1709:{\displaystyle U\subseteq \mathbb {R} }
1635:Any topological space that contains an
5357:
5258:
5100:
4935:
3155:that is meagre in itself is meagre in
5281:
5124:
5088:
4915:
2200:is empty is of the first category of
2023:is nonmeagre. In particular, by the
16:"Small" subset of a topological space
5293:McGraw-Hill Science/Engineering/Math
5178:The Erdos-Sierpinski Duality Theorem
4938:"Cartesian products of Baire spaces"
4355:
2437:However the following results hold:
4772:meeting the above criteria, player
4212:{\displaystyle r_{1},r_{2},\ldots }
3623:Meagre subsets and Lebesgue measure
3491:{\displaystyle S_{1},S_{2},\ldots }
2220:(that is, it is a meager subset of
2063:, they are also nonmeagre spaces.
13:
5180:, notes. Accessed 18 January 2023.
4757:
4601:
4528:
4498:Then there is a Banach–Mazur game
4437:
4393:
4053:
4032:
3363:that is of the second category in
315:. Otherwise, the subset is called
14:
5381:
5226:General Topology 2: Chapters 5–10
4352:set formed from dense open sets.
2281:of subsets, a suitable notion of
1752:
1449:{\displaystyle \mathbb {R} ^{2}.}
975:{\displaystyle \cap \mathbb {Q} }
564:
3383:must have non-empty interior in
1963:nowhere differentiable functions
1553:{\displaystyle \mathbb {R} ^{2}}
5187:
5170:
5148:
5130:
5066:
4448:{\displaystyle {\mathcal {W}},}
3135:In particular, every subset of
2005:discontinuous linear functional
849:were the original ones used by
5006:
4929:
4909:
4894:
4765:{\displaystyle {\mathcal {W}}}
4609:{\displaystyle {\mathcal {W}}}
4533:
4511:
4401:{\displaystyle {\mathcal {W}}}
3951:
3939:
3919:
3907:
3884:
3872:
3832:
3820:
3758:
3746:
3679:
3667:
3323:Every nowhere dense subset of
2144:
2138:
2082:
1984:
1972:
1928:
1925:
1913:
1910:
1887:
1875:
1831:
1819:
1799:
1796:
1784:
1781:
1560:even though its meagre subset
1514:
1494:
1488:
1472:
1318:
1307:
1295:
1292:
1106:
1094:
1074:
1062:
1042:
1030:
1001:
989:
961:
949:
929:
918:
906:
903:
897:
885:
857:terminology was introduced by
430:A topological space is called
291:if it is a countable union of
92:
1:
5261:"The Banach Category Theorem"
5001:Narici & Beckenstein 2011
4889:Narici & Beckenstein 2011
3597:is of the second category in
3432:is of the second category in
3307:
1271:{\displaystyle \mathbb {R} .}
1218:{\displaystyle \mathbb {R} .}
1156:is both a meagre subspace of
4901:Schaefer, Helmut H. (1966).
4325:sets), but contains a dense
3982:{\displaystyle \mathbb {R} }
3649:{\displaystyle \mathbb {R} }
3425:{\displaystyle B\subseteq X}
2121:{\displaystyle S\subseteq X}
1761:{\displaystyle U\setminus H}
1681:{\displaystyle \mathbb {R} }
1613:{\displaystyle \mathbb {R} }
1380:{\displaystyle \mathbb {R} }
1358:{\displaystyle \mathbb {R} }
1246:{\displaystyle \mathbb {R} }
1193:{\displaystyle \mathbb {R} }
1171:{\displaystyle \mathbb {R} }
1149:{\displaystyle \mathbb {Q} }
853:in his thesis of 1899. The
604:".) A subset is comeagre in
573:{\displaystyle X\setminus A}
74:many meagre sets is meagre.
7:
4922:Annali di Mat. Pura ed Appl
4903:"Topological Vector Spaces"
4841:
4345:{\displaystyle G_{\delta }}
4321:(countable intersection of
4312:{\displaystyle G_{\delta }}
4277:{\displaystyle F_{\sigma }}
4248:{\displaystyle F_{\sigma }}
4223:Relation to Borel hierarchy
1014:is nonmeagre and comeagre.
864:
10:
5386:
5318:Willard, Stephen (2004) .
4408:be a family of subsets of
5196:Topological Vector Spaces
3739:gives a meagre subset of
3697:Smith–Volterra–Cantor set
2251:states that in any space
2128:is meagre if and only if
2033:locally compact Hausdorff
1200:) and a meagre subset of
836:topological vector spaces
5259:Oxtoby, John C. (1980).
5234:Éléments de mathématique
4872:
4821:Erdos–Sierpinski duality
4384:be a topological space,
2414:without being meagre in
2091:{\displaystyle h:X\to X}
1343:is not nowhere dense in
4959:10.4064/fm-49-2-157-166
4946:Fundamenta Mathematicae
3900:and hence nonmeagre in
2247:Banach category theorem
1735:{\displaystyle U\cap H}
1424:is meagre in the plane
1017:In the nonmeagre space
872:In the nonmeagre space
722:of a topological space
5370:Descriptive set theory
5033:10.4064/sm-3-1-174-179
4810:
4786:
4766:
4732:
4712:
4692:
4672:
4616:to produce a sequence
4610:
4586:
4563:
4543:
4492:
4469:
4449:
4422:
4402:
4378:
4346:
4313:
4278:
4249:
4213:
4167:
4057:
4036:
4003:
3983:
3958:
3926:
3894:
3859:
3839:
3807:
3785:
3765:
3733:
3713:
3686:
3650:
3614:
3591:
3564:
3512:
3492:
3446:
3426:
3397:
3377:
3357:
3337:
3298:
3278:
3255:
3235:
3212:
3192:
3172:
3149:
3126:
3103:
3083:
3063:
3043:
3020:
2998:
2975:
2955:
2935:
2915:
2892:
2870:
2847:
2827:
2804:
2777:
2754:
2734:
2714:
2694:
2671:
2649:
2626:
2606:
2586:
2566:
2543:
2521:
2498:
2478:
2455:
2431:
2408:
2388:
2368:
2341:
2321:
2268:
2234:
2214:
2194:
2174:
2151:
2122:
2092:
2025:Baire category theorem
1994:
1955:
1935:
1894:
1862:
1838:
1806:
1762:
1736:
1710:
1682:
1660:
1614:
1588:
1554:
1521:
1450:
1418:
1381:
1365:, but it is meagre in
1359:
1337:
1272:
1247:
1219:
1194:
1172:
1150:
1113:
1081:
1049:
1008:
976:
936:
824:
804:
780:
760:
736:
716:
692:
672:
654:Remarks on terminology
645:
618:
594:
574:
544:
513:
483:
463:
421:
400:
369:
338:
309:
284:
256:
225:
196:
165:
145:
111:
83:Baire category theorem
57:of the second category
4811:
4787:
4767:
4733:
4713:
4693:
4673:
4611:
4587:
4564:
4544:
4493:
4470:
4450:
4423:
4403:
4379:
4347:
4314:
4279:
4250:
4214:
4168:
4037:
4016:
4004:
3984:
3959:
3927:
3895:
3860:
3840:
3808:
3786:
3766:
3734:
3714:
3687:
3651:
3615:
3592:
3590:{\displaystyle S_{n}}
3565:
3513:
3493:
3447:
3427:
3398:
3378:
3358:
3338:
3299:
3279:
3256:
3236:
3213:
3198:that is nonmeagre in
3193:
3173:
3150:
3127:
3104:
3084:
3064:
3044:
3021:
2999:
2976:
2956:
2936:
2916:
2893:
2871:
2848:
2828:
2805:
2778:
2755:
2735:
2715:
2695:
2672:
2650:
2627:
2607:
2587:
2567:
2544:
2522:
2499:
2479:
2456:
2432:
2409:
2389:
2369:
2342:
2322:
2269:
2235:
2215:
2195:
2175:
2152:
2123:
2093:
2029:complete metric space
1995:
1956:
1936:
1895:
1863:
1844:with the topology of
1839:
1807:
1763:
1737:
1711:
1683:
1661:
1615:
1589:
1555:
1522:
1451:
1419:
1382:
1360:
1338:
1280:complete metric space
1273:
1248:
1220:
1195:
1173:
1151:
1114:
1112:{\displaystyle (1,2]}
1082:
1050:
1009:
977:
937:
825:
805:
786:can also be called a
781:
761:
737:
717:
693:
673:
646:
619:
595:
575:
545:
514:
484:
464:
422:
401:
370:
339:
310:
285:
257:
226:
197:
166:
151:that is, a subset of
146:
112:
37:set of first category
5265:Measure and Category
4916:Baire, René (1899).
4831:continuum hypothesis
4800:
4776:
4752:
4722:
4718:; otherwise, player
4702:
4682:
4620:
4596:
4573:
4553:
4502:
4479:
4459:
4432:
4412:
4388:
4368:
4329:
4296:
4261:
4232:
4177:
4013:
3993:
3971:
3936:
3904:
3869:
3849:
3817:
3797:
3775:
3743:
3723:
3703:
3664:
3638:
3601:
3574:
3522:
3502:
3456:
3436:
3410:
3387:
3367:
3347:
3327:
3288:
3268:
3264:A topological space
3245:
3222:
3202:
3182:
3159:
3139:
3113:
3093:
3073:
3053:
3030:
3010:
2985:
2965:
2945:
2925:
2902:
2882:
2857:
2837:
2814:
2794:
2764:
2744:
2724:
2704:
2681:
2661:
2636:
2616:
2596:
2576:
2553:
2533:
2508:
2488:
2465:
2445:
2418:
2398:
2378:
2355:
2331:
2296:
2255:
2224:
2204:
2184:
2164:
2150:{\displaystyle h(S)}
2132:
2106:
2070:
2035:space is nonmeagre.
1969:
1945:
1904:
1872:
1852:
1816:
1775:
1768:are both nonmeagre.
1746:
1720:
1692:
1670:
1666:of the real numbers
1650:
1602:
1564:
1535:
1463:
1428:
1394:
1369:
1347:
1289:
1257:
1253:and hence meagre in
1235:
1231:is nowhere dense in
1204:
1182:
1160:
1138:
1091:
1059:
1021:
986:
982:is meagre. The set
946:
876:
814:
794:
770:
750:
726:
706:
682:
662:
632:
608:
584:
558:
531:
500:
473:
453:
411:
387:
356:
325:
299:
274:
243:
212:
186:
155:
132:
101:
5288:Functional Analysis
5013:Banach, S. (1931).
5003:, pp. 371–423.
4936:Oxtoby, J. (1961).
4833:holds, there is an
4746: —
3964:is a Baire space.
3865:and is comeagre in
2031:and every nonempty
1934:{\displaystyle C()}
1846:uniform convergence
1805:{\displaystyle C()}
1119:is also nonmeagre.
87:functional analysis
5330:Dover Publications
5230:Topologie Générale
4806:
4782:
4762:
4744:
4728:
4708:
4688:
4668:
4606:
4585:{\displaystyle Q,}
4582:
4559:
4539:
4491:{\displaystyle Y.}
4488:
4465:
4445:
4418:
4398:
4374:
4342:
4309:
4274:
4245:
4209:
4163:
4140:
4091:
3999:
3979:
3954:
3922:
3890:
3855:
3835:
3803:
3784:{\displaystyle 1.}
3781:
3761:
3729:
3712:{\displaystyle 1.}
3709:
3682:
3646:
3613:{\displaystyle X.}
3610:
3587:
3570:then at least one
3560:
3508:
3488:
3442:
3422:
3393:
3373:
3353:
3333:
3294:
3274:
3251:
3234:{\displaystyle X,}
3231:
3208:
3188:
3171:{\displaystyle X.}
3168:
3145:
3125:{\displaystyle X.}
3122:
3099:
3079:
3059:
3042:{\displaystyle X,}
3039:
3016:
2997:{\displaystyle X.}
2994:
2971:
2951:
2931:
2914:{\displaystyle X,}
2911:
2888:
2869:{\displaystyle Y.}
2866:
2843:
2826:{\displaystyle X,}
2823:
2800:
2776:{\displaystyle X.}
2773:
2750:
2730:
2710:
2693:{\displaystyle X,}
2690:
2667:
2648:{\displaystyle X.}
2645:
2622:
2602:
2582:
2565:{\displaystyle X,}
2562:
2539:
2520:{\displaystyle X.}
2517:
2494:
2477:{\displaystyle Y,}
2474:
2451:
2430:{\displaystyle Y.}
2427:
2404:
2384:
2367:{\displaystyle X.}
2364:
2337:
2317:
2267:{\displaystyle X,}
2264:
2230:
2210:
2190:
2180:whose interior in
2170:
2147:
2118:
2088:
1990:
1951:
1931:
1890:
1858:
1834:
1802:
1758:
1732:
1706:
1678:
1656:
1646:There is a subset
1610:
1594:is a nonmeagre sub
1584:
1550:
1517:
1446:
1414:
1377:
1355:
1333:
1268:
1243:
1215:
1190:
1168:
1146:
1109:
1077:
1048:{\displaystyle X=}
1045:
1004:
972:
932:
832:nonmeagre subspace
820:
800:
776:
756:
732:
712:
688:
668:
644:{\displaystyle X.}
641:
614:
590:
570:
543:{\displaystyle X,}
540:
512:{\displaystyle X,}
509:
479:
459:
417:
407:The qualifier "in
399:{\displaystyle X.}
396:
368:{\displaystyle X,}
365:
337:{\displaystyle X,}
334:
305:
280:
255:{\displaystyle X,}
252:
224:{\displaystyle X,}
221:
192:
161:
144:{\displaystyle X,}
141:
107:
5339:978-0-486-43479-7
5302:978-0-07-054236-5
5243:978-3-540-64563-4
5222:Bourbaki, Nicolas
5127:, pp. 42–43.
4866:Property of Baire
4809:{\displaystyle X}
4785:{\displaystyle Q}
4742:
4731:{\displaystyle Q}
4711:{\displaystyle X}
4691:{\displaystyle P}
4562:{\displaystyle P}
4475:be any subset of
4468:{\displaystyle X}
4421:{\displaystyle Y}
4377:{\displaystyle Y}
4362:Banach–Mazur game
4356:Banach–Mazur game
4139:
4090:
4002:{\displaystyle 0}
3893:{\displaystyle ,}
3858:{\displaystyle 0}
3806:{\displaystyle 1}
3732:{\displaystyle 1}
3511:{\displaystyle X}
3445:{\displaystyle X}
3396:{\displaystyle X}
3376:{\displaystyle X}
3356:{\displaystyle X}
3336:{\displaystyle X}
3297:{\displaystyle X}
3277:{\displaystyle X}
3254:{\displaystyle X}
3211:{\displaystyle X}
3191:{\displaystyle X}
3148:{\displaystyle X}
3102:{\displaystyle A}
3082:{\displaystyle Y}
3062:{\displaystyle A}
3019:{\displaystyle Y}
2974:{\displaystyle A}
2954:{\displaystyle Y}
2934:{\displaystyle A}
2891:{\displaystyle Y}
2846:{\displaystyle A}
2803:{\displaystyle A}
2753:{\displaystyle A}
2733:{\displaystyle Y}
2713:{\displaystyle A}
2670:{\displaystyle Y}
2625:{\displaystyle A}
2605:{\displaystyle Y}
2585:{\displaystyle A}
2542:{\displaystyle Y}
2497:{\displaystyle A}
2454:{\displaystyle A}
2407:{\displaystyle X}
2394:may be meagre in
2387:{\displaystyle A}
2349:subspace topology
2340:{\displaystyle Y}
2233:{\displaystyle X}
2213:{\displaystyle X}
2193:{\displaystyle X}
2173:{\displaystyle X}
1993:{\displaystyle ,}
1954:{\displaystyle A}
1861:{\displaystyle A}
1659:{\displaystyle H}
823:{\displaystyle X}
803:{\displaystyle X}
779:{\displaystyle A}
759:{\displaystyle X}
744:subspace topology
735:{\displaystyle X}
715:{\displaystyle A}
691:{\displaystyle X}
671:{\displaystyle X}
617:{\displaystyle X}
593:{\displaystyle X}
482:{\displaystyle X}
462:{\displaystyle A}
420:{\displaystyle X}
308:{\displaystyle X}
283:{\displaystyle X}
195:{\displaystyle X}
164:{\displaystyle X}
119:topological space
110:{\displaystyle X}
47:that is small or
45:topological space
5377:
5365:General topology
5351:
5321:General Topology
5314:
5278:
5255:
5217:
5181:
5174:
5168:
5166:
5164:
5152:
5146:
5145:
5134:
5128:
5122:
5116:
5110:
5104:
5098:
5092:
5086:
5075:
5070:
5064:
5059:
5050:
5044:
5038:
5037:
5035:
5010:
5004:
4998:
4979:
4973:
4964:
4962:
4942:
4933:
4927:
4925:
4913:
4907:
4906:
4898:
4892:
4886:
4854:Generic property
4815:
4813:
4812:
4807:
4794:winning strategy
4791:
4789:
4788:
4783:
4771:
4769:
4768:
4763:
4761:
4760:
4747:
4737:
4735:
4734:
4729:
4717:
4715:
4714:
4709:
4697:
4695:
4694:
4689:
4677:
4675:
4674:
4669:
4658:
4657:
4645:
4644:
4632:
4631:
4615:
4613:
4612:
4607:
4605:
4604:
4591:
4589:
4588:
4583:
4568:
4566:
4565:
4560:
4548:
4546:
4545:
4540:
4532:
4531:
4497:
4495:
4494:
4489:
4474:
4472:
4471:
4466:
4454:
4452:
4451:
4446:
4441:
4440:
4427:
4425:
4424:
4419:
4407:
4405:
4404:
4399:
4397:
4396:
4383:
4381:
4380:
4375:
4351:
4349:
4348:
4343:
4341:
4340:
4318:
4316:
4315:
4310:
4308:
4307:
4283:
4281:
4280:
4275:
4273:
4272:
4254:
4252:
4251:
4246:
4244:
4243:
4218:
4216:
4215:
4210:
4202:
4201:
4189:
4188:
4172:
4170:
4169:
4164:
4162:
4158:
4157:
4156:
4145:
4141:
4132:
4121:
4120:
4108:
4107:
4096:
4092:
4083:
4072:
4071:
4056:
4051:
4035:
4030:
4008:
4006:
4005:
4000:
3988:
3986:
3985:
3980:
3978:
3963:
3961:
3960:
3957:{\displaystyle }
3955:
3931:
3929:
3928:
3925:{\displaystyle }
3923:
3899:
3897:
3896:
3891:
3864:
3862:
3861:
3856:
3844:
3842:
3841:
3838:{\displaystyle }
3836:
3812:
3810:
3809:
3804:
3790:
3788:
3787:
3782:
3770:
3768:
3767:
3764:{\displaystyle }
3762:
3738:
3736:
3735:
3730:
3718:
3716:
3715:
3710:
3691:
3689:
3688:
3685:{\displaystyle }
3683:
3658:Lebesgue measure
3655:
3653:
3652:
3647:
3645:
3634:A meagre set in
3629:Lebesgue measure
3619:
3617:
3616:
3611:
3596:
3594:
3593:
3588:
3586:
3585:
3569:
3567:
3566:
3561:
3553:
3552:
3540:
3539:
3517:
3515:
3514:
3509:
3497:
3495:
3494:
3489:
3481:
3480:
3468:
3467:
3451:
3449:
3448:
3443:
3431:
3429:
3428:
3423:
3402:
3400:
3399:
3394:
3382:
3380:
3379:
3374:
3362:
3360:
3359:
3354:
3342:
3340:
3339:
3334:
3303:
3301:
3300:
3295:
3283:
3281:
3280:
3275:
3260:
3258:
3257:
3252:
3241:being meagre in
3240:
3238:
3237:
3232:
3217:
3215:
3214:
3209:
3197:
3195:
3194:
3189:
3178:Every subset of
3177:
3175:
3174:
3169:
3154:
3152:
3151:
3146:
3131:
3129:
3128:
3123:
3109:is nonmeagre in
3108:
3106:
3105:
3100:
3088:
3086:
3085:
3080:
3069:is nonmeagre in
3068:
3066:
3065:
3060:
3048:
3046:
3045:
3040:
3025:
3023:
3022:
3017:
3003:
3001:
3000:
2995:
2981:is nonmeagre in
2980:
2978:
2977:
2972:
2960:
2958:
2957:
2952:
2941:is nonmeagre in
2940:
2938:
2937:
2932:
2920:
2918:
2917:
2912:
2897:
2895:
2894:
2889:
2875:
2873:
2872:
2867:
2853:is nonmeagre in
2852:
2850:
2849:
2844:
2832:
2830:
2829:
2824:
2810:is nonmeagre in
2809:
2807:
2806:
2801:
2782:
2780:
2779:
2774:
2759:
2757:
2756:
2751:
2739:
2737:
2736:
2731:
2719:
2717:
2716:
2711:
2699:
2697:
2696:
2691:
2676:
2674:
2673:
2668:
2654:
2652:
2651:
2646:
2631:
2629:
2628:
2623:
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2571:
2569:
2568:
2563:
2548:
2546:
2545:
2540:
2526:
2524:
2523:
2518:
2503:
2501:
2500:
2495:
2483:
2481:
2480:
2475:
2460:
2458:
2457:
2452:
2436:
2434:
2433:
2428:
2413:
2411:
2410:
2405:
2393:
2391:
2390:
2385:
2373:
2371:
2370:
2365:
2346:
2344:
2343:
2338:
2326:
2324:
2323:
2318:
2285:. Dually, all
2273:
2271:
2270:
2265:
2250:
2249:
2239:
2237:
2236:
2231:
2219:
2217:
2216:
2211:
2199:
2197:
2196:
2191:
2179:
2177:
2176:
2171:
2156:
2154:
2153:
2148:
2127:
2125:
2124:
2119:
2097:
2095:
2094:
2089:
2061:are Baire spaces
1999:
1997:
1996:
1991:
1960:
1958:
1957:
1952:
1940:
1938:
1937:
1932:
1899:
1897:
1896:
1893:{\displaystyle }
1891:
1867:
1865:
1864:
1859:
1843:
1841:
1840:
1837:{\displaystyle }
1835:
1811:
1809:
1808:
1803:
1767:
1765:
1764:
1759:
1741:
1739:
1738:
1733:
1715:
1713:
1712:
1707:
1705:
1687:
1685:
1684:
1679:
1677:
1665:
1663:
1662:
1657:
1626:rational numbers
1619:
1617:
1616:
1611:
1609:
1593:
1591:
1590:
1585:
1571:
1559:
1557:
1556:
1551:
1549:
1548:
1543:
1526:
1524:
1523:
1518:
1501:
1487:
1479:
1455:
1453:
1452:
1447:
1442:
1441:
1436:
1423:
1421:
1420:
1415:
1401:
1386:
1384:
1383:
1378:
1376:
1364:
1362:
1361:
1356:
1354:
1342:
1340:
1339:
1334:
1317:
1277:
1275:
1274:
1269:
1264:
1252:
1250:
1249:
1244:
1242:
1224:
1222:
1221:
1216:
1211:
1199:
1197:
1196:
1191:
1189:
1177:
1175:
1174:
1169:
1167:
1155:
1153:
1152:
1147:
1145:
1118:
1116:
1115:
1110:
1086:
1084:
1083:
1080:{\displaystyle }
1078:
1054:
1052:
1051:
1046:
1013:
1011:
1010:
1007:{\displaystyle }
1005:
981:
979:
978:
973:
971:
941:
939:
938:
933:
928:
829:
827:
826:
821:
809:
807:
806:
801:
785:
783:
782:
777:
765:
763:
762:
757:
741:
739:
738:
733:
721:
719:
718:
713:
697:
695:
694:
689:
677:
675:
674:
669:
650:
648:
647:
642:
623:
621:
620:
615:
599:
597:
596:
591:
579:
577:
576:
571:
549:
547:
546:
541:
526:
525:
518:
516:
515:
510:
495:
494:
488:
486:
485:
480:
468:
466:
465:
460:
444:
443:
436:
435:
426:
424:
423:
418:
405:
403:
402:
397:
382:
381:
374:
372:
371:
366:
351:
350:
349:nonmeagre subset
343:
341:
340:
335:
321:
320:
314:
312:
311:
306:
289:
287:
286:
281:
269:
268:
261:
259:
258:
253:
238:
237:
230:
228:
227:
222:
208:
207:
201:
199:
198:
193:
170:
168:
167:
162:
150:
148:
147:
142:
116:
114:
113:
108:
25:general topology
5385:
5384:
5380:
5379:
5378:
5376:
5375:
5374:
5355:
5354:
5340:
5303:
5275:
5244:
5206:
5190:
5185:
5184:
5175:
5171:
5153:
5149:
5136:
5135:
5131:
5123:
5119:
5115:, Theorem 25.2.
5111:
5107:
5099:
5095:
5087:
5078:
5071:
5067:
5060:
5053:
5049:, Theorem 25.5.
5045:
5041:
5011:
5007:
4999:
4982:
4974:
4967:
4940:
4934:
4930:
4914:
4910:
4899:
4895:
4887:
4880:
4875:
4848:Barrelled space
4844:
4823:
4818:
4801:
4798:
4797:
4796:if and only if
4777:
4774:
4773:
4756:
4755:
4753:
4750:
4749:
4745:
4723:
4720:
4719:
4703:
4700:
4699:
4683:
4680:
4679:
4653:
4649:
4640:
4636:
4627:
4623:
4621:
4618:
4617:
4600:
4599:
4597:
4594:
4593:
4574:
4571:
4570:
4554:
4551:
4550:
4527:
4526:
4503:
4500:
4499:
4480:
4477:
4476:
4460:
4457:
4456:
4436:
4435:
4433:
4430:
4429:
4413:
4410:
4409:
4392:
4391:
4389:
4386:
4385:
4369:
4366:
4365:
4358:
4336:
4332:
4330:
4327:
4326:
4303:
4299:
4297:
4294:
4293:
4268:
4264:
4262:
4259:
4258:
4239:
4235:
4233:
4230:
4229:
4225:
4197:
4193:
4184:
4180:
4178:
4175:
4174:
4146:
4130:
4126:
4125:
4116:
4112:
4097:
4081:
4077:
4076:
4067:
4063:
4062:
4058:
4052:
4041:
4031:
4020:
4014:
4011:
4010:
3994:
3991:
3990:
3974:
3972:
3969:
3968:
3937:
3934:
3933:
3905:
3902:
3901:
3870:
3867:
3866:
3850:
3847:
3846:
3818:
3815:
3814:
3798:
3795:
3794:
3776:
3773:
3772:
3744:
3741:
3740:
3724:
3721:
3720:
3704:
3701:
3700:
3693:fat Cantor sets
3665:
3662:
3661:
3641:
3639:
3636:
3635:
3625:
3602:
3599:
3598:
3581:
3577:
3575:
3572:
3571:
3548:
3544:
3535:
3531:
3523:
3520:
3519:
3503:
3500:
3499:
3498:are subsets of
3476:
3472:
3463:
3459:
3457:
3454:
3453:
3437:
3434:
3433:
3411:
3408:
3407:
3388:
3385:
3384:
3368:
3365:
3364:
3348:
3345:
3344:
3328:
3325:
3324:
3310:
3289:
3286:
3285:
3269:
3266:
3265:
3246:
3243:
3242:
3223:
3220:
3219:
3203:
3200:
3199:
3183:
3180:
3179:
3160:
3157:
3156:
3140:
3137:
3136:
3114:
3111:
3110:
3094:
3091:
3090:
3089:if and only if
3074:
3071:
3070:
3054:
3051:
3050:
3031:
3028:
3027:
3011:
3008:
3007:
2986:
2983:
2982:
2966:
2963:
2962:
2961:if and only if
2946:
2943:
2942:
2926:
2923:
2922:
2903:
2900:
2899:
2883:
2880:
2879:
2858:
2855:
2854:
2838:
2835:
2834:
2815:
2812:
2811:
2795:
2792:
2791:
2765:
2762:
2761:
2745:
2742:
2741:
2740:if and only if
2725:
2722:
2721:
2705:
2702:
2701:
2682:
2679:
2678:
2662:
2659:
2658:
2637:
2634:
2633:
2617:
2614:
2613:
2612:if and only if
2597:
2594:
2593:
2577:
2574:
2573:
2554:
2551:
2550:
2534:
2531:
2530:
2509:
2506:
2505:
2489:
2486:
2485:
2466:
2463:
2462:
2446:
2443:
2442:
2419:
2416:
2415:
2399:
2396:
2395:
2379:
2376:
2375:
2356:
2353:
2352:
2332:
2329:
2328:
2297:
2294:
2293:
2256:
2253:
2252:
2245:
2244:
2225:
2222:
2221:
2205:
2202:
2201:
2185:
2182:
2181:
2165:
2162:
2161:
2133:
2130:
2129:
2107:
2104:
2103:
2071:
2068:
2067:
2057:locally compact
2038:Every nonempty
2027:every nonempty
2019:Every nonempty
2017:
1970:
1967:
1966:
1946:
1943:
1942:
1905:
1902:
1901:
1873:
1870:
1869:
1853:
1850:
1849:
1817:
1814:
1813:
1776:
1773:
1772:
1747:
1744:
1743:
1721:
1718:
1717:
1701:
1693:
1690:
1689:
1673:
1671:
1668:
1667:
1651:
1648:
1647:
1622:isolated points
1605:
1603:
1600:
1599:
1567:
1565:
1562:
1561:
1544:
1539:
1538:
1536:
1533:
1532:
1527:is a meagre sub
1497:
1483:
1475:
1464:
1461:
1460:
1437:
1432:
1431:
1429:
1426:
1425:
1397:
1395:
1392:
1391:
1372:
1370:
1367:
1366:
1350:
1348:
1345:
1344:
1313:
1290:
1287:
1286:
1260:
1258:
1255:
1254:
1238:
1236:
1233:
1232:
1207:
1205:
1202:
1201:
1185:
1183:
1180:
1179:
1163:
1161:
1158:
1157:
1141:
1139:
1136:
1135:
1127:
1092:
1089:
1088:
1060:
1057:
1056:
1022:
1019:
1018:
987:
984:
983:
967:
947:
944:
943:
924:
877:
874:
873:
867:
847:second category
815:
812:
811:
795:
792:
791:
788:meagre subspace
771:
768:
767:
751:
748:
747:
727:
724:
723:
707:
704:
703:
683:
680:
679:
663:
660:
659:
633:
630:
629:
609:
606:
605:
585:
582:
581:
559:
556:
555:
532:
529:
528:
523:
522:
501:
498:
497:
492:
491:
474:
471:
470:
454:
451:
450:
441:
440:
438:(respectively,
433:
432:
412:
409:
408:
388:
385:
384:
380:second category
379:
378:
357:
354:
353:
348:
347:
326:
323:
322:
318:
317:
300:
297:
296:
275:
272:
271:
266:
265:
244:
241:
240:
235:
234:
213:
210:
209:
205:
204:
187:
184:
183:
156:
153:
152:
133:
130:
129:
102:
99:
98:
95:
31:(also called a
17:
12:
11:
5:
5383:
5373:
5372:
5367:
5353:
5352:
5338:
5315:
5301:
5279:
5273:
5256:
5242:
5218:
5205:978-1584888666
5204:
5189:
5186:
5183:
5182:
5169:
5147:
5129:
5117:
5105:
5093:
5076:
5065:
5051:
5039:
5026:(1): 174–179.
5005:
4980:
4978:, p. 192.
4965:
4953:(2): 157–166.
4928:
4908:
4893:
4891:, p. 389.
4877:
4876:
4874:
4871:
4870:
4869:
4863:
4860:Negligible set
4857:
4851:
4843:
4840:
4822:
4819:
4805:
4781:
4759:
4740:
4727:
4707:
4687:
4667:
4664:
4661:
4656:
4652:
4648:
4643:
4639:
4635:
4630:
4626:
4603:
4581:
4578:
4558:
4538:
4535:
4530:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4487:
4484:
4464:
4444:
4439:
4417:
4395:
4373:
4357:
4354:
4339:
4335:
4306:
4302:
4271:
4267:
4242:
4238:
4224:
4221:
4208:
4205:
4200:
4196:
4192:
4187:
4183:
4161:
4155:
4152:
4149:
4144:
4138:
4135:
4129:
4124:
4119:
4115:
4111:
4106:
4103:
4100:
4095:
4089:
4086:
4080:
4075:
4070:
4066:
4061:
4055:
4050:
4047:
4044:
4040:
4034:
4029:
4026:
4023:
4019:
3998:
3977:
3953:
3950:
3947:
3944:
3941:
3921:
3918:
3915:
3912:
3909:
3889:
3886:
3883:
3880:
3877:
3874:
3854:
3834:
3831:
3828:
3825:
3822:
3802:
3780:
3760:
3757:
3754:
3751:
3748:
3728:
3708:
3681:
3678:
3675:
3672:
3669:
3656:need not have
3644:
3624:
3621:
3609:
3606:
3584:
3580:
3559:
3556:
3551:
3547:
3543:
3538:
3534:
3530:
3527:
3507:
3487:
3484:
3479:
3475:
3471:
3466:
3462:
3441:
3421:
3418:
3415:
3392:
3372:
3352:
3332:
3318:barreled space
3309:
3306:
3293:
3273:
3250:
3230:
3227:
3207:
3187:
3167:
3164:
3144:
3133:
3132:
3121:
3118:
3098:
3078:
3058:
3038:
3035:
3015:
3004:
2993:
2990:
2970:
2950:
2930:
2910:
2907:
2887:
2876:
2865:
2862:
2842:
2822:
2819:
2799:
2784:
2783:
2772:
2769:
2749:
2729:
2709:
2689:
2686:
2666:
2655:
2644:
2641:
2621:
2601:
2581:
2561:
2558:
2538:
2527:
2516:
2513:
2493:
2473:
2470:
2450:
2426:
2423:
2403:
2383:
2363:
2360:
2336:
2316:
2313:
2310:
2307:
2304:
2301:
2283:negligible set
2263:
2260:
2229:
2209:
2189:
2169:
2146:
2143:
2140:
2137:
2117:
2114:
2111:
2102:then a subset
2087:
2084:
2081:
2078:
2075:
2016:
2013:
2009:Martin's axiom
1989:
1986:
1983:
1980:
1977:
1974:
1950:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1889:
1886:
1883:
1880:
1877:
1857:
1833:
1830:
1827:
1824:
1821:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1757:
1754:
1751:
1731:
1728:
1725:
1704:
1700:
1697:
1676:
1655:
1643:is nonmeagre.
1641:discrete space
1637:isolated point
1608:
1597:
1583:
1580:
1577:
1574:
1570:
1547:
1542:
1530:
1516:
1513:
1510:
1507:
1504:
1500:
1496:
1493:
1490:
1486:
1482:
1478:
1474:
1471:
1468:
1445:
1440:
1435:
1413:
1410:
1407:
1404:
1400:
1375:
1353:
1332:
1329:
1326:
1323:
1320:
1316:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1267:
1263:
1241:
1214:
1210:
1188:
1166:
1144:
1132:isolated point
1125:
1108:
1105:
1102:
1099:
1096:
1076:
1073:
1070:
1067:
1064:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1003:
1000:
997:
994:
991:
970:
966:
963:
960:
957:
954:
951:
931:
927:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
866:
863:
843:first category
819:
799:
775:
755:
731:
711:
687:
667:
640:
637:
613:
589:
569:
566:
563:
539:
536:
508:
505:
478:
458:
416:
395:
392:
364:
361:
333:
330:
304:
279:
267:first category
251:
248:
220:
217:
191:
160:
140:
137:
106:
94:
91:
15:
9:
6:
4:
3:
2:
5382:
5371:
5368:
5366:
5363:
5362:
5360:
5349:
5345:
5341:
5335:
5331:
5327:
5326:Mineola, N.Y.
5323:
5322:
5316:
5312:
5308:
5304:
5298:
5294:
5290:
5289:
5284:
5283:Rudin, Walter
5280:
5276:
5274:0-387-90508-1
5270:
5266:
5262:
5257:
5253:
5249:
5245:
5239:
5235:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5201:
5197:
5192:
5191:
5179:
5173:
5163:
5158:
5151:
5143:
5139:
5133:
5126:
5121:
5114:
5109:
5103:, p. 62.
5102:
5097:
5091:, p. 43.
5090:
5085:
5083:
5081:
5074:
5069:
5063:
5058:
5056:
5048:
5043:
5034:
5029:
5025:
5022:
5021:
5016:
5009:
5002:
4997:
4995:
4993:
4991:
4989:
4987:
4985:
4977:
4976:Bourbaki 1989
4972:
4970:
4960:
4956:
4952:
4948:
4947:
4939:
4932:
4923:
4919:
4912:
4904:
4897:
4890:
4885:
4883:
4878:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4845:
4839:
4836:
4832:
4828:
4817:
4803:
4795:
4779:
4739:
4725:
4705:
4685:
4665:
4662:
4659:
4654:
4650:
4646:
4641:
4637:
4633:
4628:
4624:
4579:
4576:
4556:
4536:
4523:
4520:
4517:
4514:
4508:
4505:
4485:
4482:
4462:
4442:
4415:
4371:
4363:
4353:
4337:
4333:
4324:
4320:
4304:
4300:
4290:
4285:
4269:
4265:
4256:
4240:
4236:
4220:
4206:
4203:
4198:
4194:
4190:
4185:
4181:
4159:
4153:
4150:
4147:
4142:
4136:
4133:
4127:
4122:
4117:
4113:
4109:
4104:
4101:
4098:
4093:
4087:
4084:
4078:
4073:
4068:
4064:
4059:
4048:
4045:
4042:
4038:
4027:
4024:
4021:
4017:
3996:
3989:with measure
3965:
3948:
3945:
3942:
3916:
3913:
3910:
3887:
3881:
3878:
3875:
3852:
3829:
3826:
3823:
3800:
3791:
3778:
3771:with measure
3755:
3752:
3749:
3726:
3706:
3698:
3694:
3676:
3673:
3670:
3659:
3632:
3630:
3620:
3607:
3604:
3582:
3578:
3557:
3554:
3549:
3545:
3541:
3536:
3532:
3528:
3525:
3505:
3485:
3482:
3477:
3473:
3469:
3464:
3460:
3439:
3419:
3416:
3413:
3404:
3390:
3370:
3350:
3330:
3321:
3319:
3315:
3305:
3304:is nonempty.
3291:
3271:
3262:
3248:
3228:
3225:
3205:
3185:
3165:
3162:
3142:
3119:
3116:
3096:
3076:
3056:
3036:
3033:
3013:
3005:
2991:
2988:
2968:
2948:
2928:
2908:
2905:
2885:
2877:
2863:
2860:
2840:
2820:
2817:
2797:
2789:
2788:
2787:
2770:
2767:
2760:is meagre in
2747:
2727:
2720:is meagre in
2707:
2687:
2684:
2664:
2656:
2642:
2639:
2632:is meagre in
2619:
2599:
2592:is meagre in
2579:
2559:
2556:
2536:
2528:
2514:
2511:
2504:is meagre in
2491:
2471:
2468:
2461:is meagre in
2448:
2440:
2439:
2438:
2424:
2421:
2401:
2381:
2361:
2358:
2351:induced from
2350:
2334:
2314:
2311:
2308:
2305:
2302:
2299:
2290:
2288:
2284:
2280:
2275:
2261:
2258:
2248:
2241:
2227:
2207:
2187:
2167:
2158:
2141:
2135:
2115:
2112:
2109:
2101:
2100:homeomorphism
2085:
2079:
2076:
2073:
2064:
2062:
2058:
2055:
2051:
2050:metric spaces
2048:
2045:
2041:
2036:
2034:
2030:
2026:
2022:
2012:
2010:
2006:
2001:
1987:
1981:
1978:
1975:
1964:
1948:
1922:
1919:
1916:
1907:
1884:
1881:
1878:
1855:
1847:
1828:
1825:
1822:
1793:
1790:
1787:
1778:
1771:In the space
1769:
1755:
1749:
1729:
1726:
1723:
1698:
1695:
1653:
1644:
1642:
1638:
1633:
1631:
1627:
1623:
1595:
1578:
1572:
1545:
1528:
1508:
1502:
1491:
1480:
1469:
1466:
1457:
1443:
1438:
1408:
1402:
1388:
1327:
1321:
1310:
1304:
1301:
1298:
1283:
1281:
1265:
1230:
1225:
1212:
1133:
1129:
1120:
1103:
1100:
1097:
1071:
1068:
1065:
1039:
1036:
1033:
1027:
1024:
1015:
998:
995:
992:
964:
958:
955:
952:
921:
915:
912:
909:
900:
894:
891:
888:
882:
879:
870:
862:
860:
856:
852:
848:
844:
839:
837:
833:
817:
797:
789:
773:
753:
746:induced from
745:
742:is given the
729:
709:
700:
685:
665:
656:
655:
651:
638:
635:
627:
611:
603:
587:
580:is meagre in
567:
561:
554:
550:
537:
534:
519:
506:
503:
476:
456:
447:
445:
437:
428:
414:
406:
393:
390:
375:
362:
359:
344:
331:
328:
302:
294:
293:nowhere dense
290:
277:
262:
249:
246:
236:meagre subset
231:
218:
215:
189:
180:
178:
174:
158:
138:
135:
127:
126:nowhere dense
122:
120:
104:
90:
88:
84:
80:
75:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
5320:
5287:
5264:
5229:
5225:
5195:
5188:Bibliography
5172:
5150:
5142:MathOverflow
5141:
5132:
5120:
5113:Willard 2004
5108:
5096:
5068:
5047:Willard 2004
5042:
5023:
5020:Studia Math.
5018:
5008:
4950:
4944:
4931:
4921:
4911:
4905:. Macmillan.
4896:
4824:
4741:
4359:
4286:
4226:
3966:
3792:
3633:
3626:
3405:
3322:
3312:A nonmeagre
3311:
3263:
3134:
3026:is dense in
2785:
2677:is dense in
2291:
2276:
2246:
2242:
2159:
2157:is meagre.
2065:
2037:
2018:
2002:
1770:
1645:
1634:
1458:
1389:
1284:
1226:
1122:A countable
1121:
1016:
871:
868:
854:
846:
842:
840:
831:
787:
701:
657:
653:
652:
626:intersection
521:
490:
448:
439:
431:
429:
377:
346:
319:nonmeagre in
316:
264:
233:
203:
182:A subset of
181:
123:
97:Throughout,
96:
76:
61:
56:
52:
36:
32:
28:
21:mathematical
18:
5101:Oxtoby 1980
4924:. 3: 1–123.
4816:is meagre.
3695:, like the
2898:is open in
2549:is open in
2052:as well as
2040:Baire space
2021:Baire space
1716:, the sets
1630:Baire space
295:subsets of
93:Definitions
81:and of the
79:Baire space
5359:Categories
5176:S. Saito,
5162:2206.10754
5125:Rudin 1991
5089:Rudin 1991
4835:involution
3518:such that
3308:Properties
1848:, the set
1598:(that is,
1229:Cantor set
851:René Baire
841:The terms
553:complement
489:is called
376:or of the
263:or of the
202:is called
175:has empty
128:subset of
117:will be a
49:negligible
33:meager set
29:meagre set
5252:246032063
5224:(1989) .
5214:144216834
4926:, page 65
4827:null sets
4663:⋯
4660:⊇
4647:⊇
4634:⊇
4338:δ
4305:δ
4270:σ
4241:σ
4207:…
4074:−
4054:∞
4039:⋃
4033:∞
4018:⋂
3558:⋯
3555:∪
3542:∪
3529:⊆
3486:…
3417:⊆
2309:⊆
2303:⊆
2287:supersets
2113:⊆
2083:→
2054:Hausdorff
1753:∖
1727:∩
1699:⊆
1573:×
1503:×
1492:∪
1481:×
1403:×
1390:The line
1322:∪
1311:∩
965:∩
922:∩
901:∪
861:in 1948.
565:∖
449:A subset
442:nonmeagre
206:meagre in
72:countably
53:nonmeagre
23:field of
5311:21163277
5285:(1991).
4842:See also
4748:For any
4289:interior
2374:The set
2347:has the
2292:Suppose
2047:(pseudo)
2044:complete
1459:The set
1285:The set
1130:without
1055:the set
942:the set
865:Examples
859:Bourbaki
602:cofinite
524:residual
493:comeagre
177:interior
5232:].
4743:Theorem
4738:wins.
4678:Player
4364:. Let
3452:and if
2279:σ-ideal
2059:spaces
551:if its
173:closure
64:σ-ideal
39:) is a
19:In the
5348:115240
5346:
5336:
5309:
5299:
5271:
5250:
5240:
5212:
5202:
5167:(p.25)
4792:has a
4173:where
3932:since
2327:where
855:meagre
434:meagre
171:whose
41:subset
5228:[
5157:arXiv
4941:(PDF)
4873:Notes
3316:is a
3049:then
2921:then
2833:then
2700:then
2572:then
2484:then
2098:is a
1596:space
1128:space
68:union
55:, or
43:of a
35:or a
5344:OCLC
5334:ISBN
5307:OCLC
5297:ISBN
5269:ISBN
5248:OCLC
5238:ISBN
5210:OCLC
5200:ISBN
4569:and
4455:and
4323:open
2243:The
2240:).
1742:and
1227:The
845:and
27:, a
5028:doi
4955:doi
4319:set
4255:set
3813:in
3631:.
3406:If
3320:.
3006:If
2878:If
2790:If
2657:If
2529:If
2441:If
1965:on
1632:.
1531:of
1529:set
790:of
527:in
520:or
496:in
469:of
383:in
352:of
270:in
239:of
121:.
70:of
5361::
5342:.
5332:.
5328::
5324:.
5305:.
5295:.
5263:.
5246:.
5208:.
5140:.
5079:^
5054:^
5017:.
4983:^
4968:^
4951:49
4949:.
4943:.
4920:.
4881:^
4009::
3779:1.
3707:1.
1282:.
345:a
232:a
89:.
5350:.
5313:.
5277:.
5254:.
5216:.
5165:.
5159::
5144:.
5036:.
5030::
5024:3
4961:.
4957::
4804:X
4780:Q
4758:W
4726:Q
4706:X
4686:P
4666:.
4655:3
4651:W
4642:2
4638:W
4629:1
4625:W
4602:W
4580:,
4577:Q
4557:P
4537:.
4534:)
4529:W
4524:,
4521:Y
4518:,
4515:X
4512:(
4509:Z
4506:M
4486:.
4483:Y
4463:X
4443:,
4438:W
4416:Y
4394:W
4372:Y
4334:G
4301:G
4266:F
4237:F
4204:,
4199:2
4195:r
4191:,
4186:1
4182:r
4160:)
4154:m
4151:+
4148:n
4143:)
4137:2
4134:1
4128:(
4123:+
4118:n
4114:r
4110:,
4105:m
4102:+
4099:n
4094:)
4088:2
4085:1
4079:(
4069:n
4065:r
4060:(
4049:1
4046:=
4043:n
4028:1
4025:=
4022:m
3997:0
3976:R
3952:]
3949:1
3946:,
3943:0
3940:[
3920:]
3917:1
3914:,
3911:0
3908:[
3888:,
3885:]
3882:1
3879:,
3876:0
3873:[
3853:0
3833:]
3830:1
3827:,
3824:0
3821:[
3801:1
3759:]
3756:1
3753:,
3750:0
3747:[
3727:1
3680:]
3677:1
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3671:0
3668:[
3643:R
3608:.
3605:X
3583:n
3579:S
3550:2
3546:S
3537:1
3533:S
3526:B
3506:X
3483:,
3478:2
3474:S
3470:,
3465:1
3461:S
3440:X
3420:X
3414:B
3391:X
3371:X
3351:X
3331:X
3292:X
3272:X
3249:X
3229:,
3226:X
3206:X
3186:X
3166:.
3163:X
3143:X
3120:.
3117:X
3097:A
3077:Y
3057:A
3037:,
3034:X
3014:Y
2992:.
2989:X
2969:A
2949:Y
2929:A
2909:,
2906:X
2886:Y
2864:.
2861:Y
2841:A
2821:,
2818:X
2798:A
2771:.
2768:X
2748:A
2728:Y
2708:A
2688:,
2685:X
2665:Y
2643:.
2640:X
2620:A
2600:Y
2580:A
2560:,
2557:X
2537:Y
2515:.
2512:X
2492:A
2472:,
2469:Y
2449:A
2425:.
2422:Y
2402:X
2382:A
2362:.
2359:X
2335:Y
2315:,
2312:X
2306:Y
2300:A
2262:,
2259:X
2228:X
2208:X
2188:X
2168:X
2145:)
2142:S
2139:(
2136:h
2116:X
2110:S
2086:X
2080:X
2077::
2074:h
1988:,
1985:]
1982:1
1979:,
1976:0
1973:[
1949:A
1929:)
1926:]
1923:1
1920:,
1917:0
1914:[
1911:(
1908:C
1888:]
1885:1
1882:,
1879:0
1876:[
1856:A
1832:]
1829:1
1826:,
1823:0
1820:[
1800:)
1797:]
1794:1
1791:,
1788:0
1785:[
1782:(
1779:C
1756:H
1750:U
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1607:R
1582:}
1579:0
1576:{
1569:R
1546:2
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1515:)
1512:}
1509:0
1506:{
1499:R
1495:(
1489:)
1485:Q
1477:Q
1473:(
1470:=
1467:S
1444:.
1439:2
1434:R
1412:}
1409:0
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1352:R
1331:}
1328:2
1325:{
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1299:0
1296:[
1293:(
1266:.
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1213:.
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1107:]
1104:2
1101:,
1098:1
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1075:]
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1066:0
1063:[
1043:]
1040:2
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1031:[
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1025:X
1002:]
999:1
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990:[
969:Q
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959:3
956:,
953:2
950:[
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910:2
907:[
904:(
898:]
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886:[
883:=
880:X
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754:X
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710:A
686:X
666:X
639:.
636:X
612:X
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562:X
538:,
535:X
507:,
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477:X
457:A
415:X
394:.
391:X
363:,
360:X
332:,
329:X
303:X
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250:,
247:X
219:,
216:X
190:X
159:X
139:,
136:X
105:X
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