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introduced the following basis property for metric spaces: Every basis of the topology contains a countable family of sets with vanishing diameters that covers the space. Soon thereafter, Witold
Hurewicz observed that Menger's basis property can be reformulated to the above form using sequences
355:
every Menger metric space is σ-compact. A. W. Miller and D. H. Fremlin proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. The
Fremlin-Miller proof was dichotomic, and the set witnessing the failure of the conjecture
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991:
692:. Hurewicz proved that a subset of the real line is Menger iff every continuous image of that space into the Baire space is not dominating. In particular, every subset of the real line of cardinality less than the
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269:{\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots }
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For subsets of the real line, the Menger property can be characterized using continuous functions into the
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The cardinality of
Bartoszyński and Tsaban's counter-example to Menger's conjecture is
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gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.
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Hurewicz, Witold (1926). "Über eine verallgemeinerung des
Borelschen Theorems".
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943:"Hereditary topological diagonalizations and the Menger–Hurewicz Conjectures"
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heavily depends on whether a certain (undecidable) axiom holds or not.
134:. A Menger space is a space in which for every sequence of open covers
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323:{\displaystyle {\mathcal {F}}_{1}\cup {\mathcal {F}}_{2}\cup \cdots }
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Chodounský, David; Repovš, Dušan; Zdomskyy, Lyubomyr (2015-12-01).
1008:
992:"Mathias Forcing and Combinatorial Covering Properties of Filters"
181:{\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }
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909:"On some properties of Hurewicz, Menger and Rothberger"
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Every compact, and even σ-compact, space is Menger.
46:. Unsourced material may be challenged and removed.
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441:{\displaystyle f,g\in \mathbb {N} ^{\mathbb {N} }}
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947:Proceedings of the American Mathematical Society
626:{\displaystyle f\in \mathbb {N} ^{\mathbb {N} }}
797:Menger's property characterizes filters whose
1059:
906:
764:Continuous image of a Menger space is Menger
767:The Menger property is closed under taking
589:{\displaystyle \mathbb {N} ^{\mathbb {N} }}
398:{\displaystyle \mathbb {N} ^{\mathbb {N} }}
1427:
1400:
1066:
1052:
941:Bartoszyński, Tomek; Tsaban, Boaz (2006).
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525:for all but finitely many natural numbers
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801:notion does not add dominating functions.
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106:Learn how and when to remove this message
871:
907:Fremlin, David; Miller, Arnold (1988).
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44:adding citations to reliable sources
15:
823:. Vol. 133. pp. 421–444.
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596:is dominating if for each function
188:of the space there are finite sets
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739:{\displaystyle {\mathfrak {d}}}
711:{\displaystyle {\mathfrak {d}}}
126:that satisfies a certain basic
31:needs additional citations for
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367:Combinatorial characterization
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996:The Journal of Symbolic Logic
969:10.1090/s0002-9939-05-07997-9
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518:{\displaystyle f(n)\leq g(n)}
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829:10.1007/978-3-7091-6110-4_14
7:
787:{\displaystyle F_{\sigma }}
685:{\displaystyle f\leq ^{*}g}
474:{\displaystyle f\leq ^{*}g}
351:Menger conjectured that in
10:
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1320:Banach fixed-point theorem
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874:Mathematische Zeitschrift
758:Every Menger space is a
916:Fundamenta Mathematicae
1375:Mathematics portal
1275:Metrics and properties
1261:Second-countable space
928:10.4064/fm-129-1-17-33
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652:{\displaystyle g\in A}
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276:such that the family
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1330:Invariance of domain
1282:Euler characteristic
1256:Bundle (mathematics)
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633:there is a function
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40:improve this article
1340:Tychonoff's theorem
1335:Poincaré conjecture
1089:General (point-set)
1018:10.1017/jsl.2014.73
821:Selecta Mathematica
347:Menger's conjecture
128:selection principle
1325:De Rham cohomology
1246:Polyhedral complex
1236:Simplicial complex
886:10.1007/bf01216792
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694:dominating number
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330:covers the space.
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118:In mathematics, a
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1229:fundamental group
838:978-3-7091-7282-7
558:{\displaystyle A}
538:{\displaystyle n}
359:Bartoszyński and
130:that generalizes
124:topological space
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1002:(4): 1398–1410.
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405:. For functions
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343:of open covers.
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111:
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100:
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89:
48:
24:
16:
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1467:
1466:
1464:
1463:
1462:
1443:
1442:
1441:
1436:
1367:
1349:
1345:Urysohn's lemma
1306:
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1156:
1147:
1119:low-dimensional
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853:
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843:
839:
817:
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799:Mathias forcing
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410:
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388:
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376:
375:
369:
349:
336:
308:
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37:
25:
12:
11:
5:
1471:
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1292:Winding number
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1284:
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1268:
1263:
1258:
1253:
1248:
1243:
1238:
1233:
1232:
1231:
1226:
1224:homotopy group
1216:
1215:
1214:
1209:
1204:
1199:
1194:
1184:
1179:
1174:
1164:
1162:
1158:
1157:
1150:
1148:
1146:
1145:
1140:
1135:
1134:
1133:
1123:
1122:
1121:
1111:
1106:
1101:
1096:
1091:
1085:
1083:
1079:
1078:
1071:
1070:
1063:
1056:
1048:
1040:
1039:
982:
953:(2): 605–615.
933:
899:
880:(1): 401–421.
864:
855:|journal=
837:
810:
809:
807:
804:
803:
802:
795:
781:
777:
765:
762:
760:Lindelöf space
756:
751:
748:
733:
705:
681:
676:
672:
668:
648:
645:
642:
619:
613:
608:
605:
582:
576:
554:
534:
514:
511:
508:
505:
502:
499:
496:
493:
490:
470:
465:
461:
457:
434:
428:
423:
420:
417:
414:
391:
385:
368:
365:
348:
345:
335:
332:
319:
316:
311:
305:
299:
294:
288:
265:
262:
257:
251:
245:
240:
234:
228:
223:
217:
211:
206:
200:
177:
174:
169:
163:
157:
152:
146:
114:
113:
55:"Menger space"
28:
26:
19:
9:
6:
4:
3:
2:
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1459:
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1454:
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1450:
1448:
1433:
1425:
1421:
1418:
1416:
1413:
1411:
1408:
1407:
1406:
1398:
1396:
1392:
1388:
1386:
1382:
1378:
1376:
1371:
1366:
1364:
1356:
1355:
1352:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1317:
1315:
1313:
1309:
1303:
1302:Orientability
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1298:
1295:
1293:
1290:
1288:
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1279:
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1234:
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1227:
1225:
1222:
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1217:
1213:
1210:
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1205:
1203:
1200:
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1190:
1189:
1188:
1185:
1183:
1180:
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1159:
1154:
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1139:
1138:Set-theoretic
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1132:
1129:
1128:
1127:
1124:
1120:
1117:
1116:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1099:Combinatorial
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1095:
1092:
1090:
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1086:
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1076:
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1046:
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1023:
1019:
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1001:
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826:
822:
815:
811:
800:
796:
779:
775:
766:
763:
761:
757:
754:
753:
747:
720:
718:
679:
674:
670:
666:
646:
643:
640:
606:
603:
552:
532:
509:
503:
500:
494:
488:
468:
463:
459:
455:
421:
418:
415:
412:
374:
364:
362:
357:
354:
344:
341:
331:
317:
314:
309:
297:
292:
263:
260:
255:
243:
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226:
221:
209:
204:
175:
172:
167:
155:
150:
133:
132:σ-compactness
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57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1432:Publications
1297:Chern number
1287:Betti number
1170: /
1161:Key concepts
1109:Differential
999:
995:
985:
960:math/0208224
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337:
120:Menger space
119:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1395:Wikiversity
1312:Key results
719:is Menger.
545:. A subset
373:Baire space
340:Karl Menger
96:August 2016
1447:Categories
1241:CW complex
1182:Continuity
1172:Closed set
1131:cohomology
806:References
750:Properties
659:such that
66:newspapers
1420:geometric
1415:algebraic
1266:Cobordism
1202:Hausdorff
1197:connected
1114:Geometric
1104:Continuum
1094:Algebraic
1026:0022-4812
1009:1401.2283
922:: 17–33.
894:119867793
857:ignored (
847:cite book
780:σ
675:∗
671:≤
644:∈
607:∈
501:≤
464:∗
460:≤
422:∈
338:In 1924,
318:⋯
315:∪
298:∪
264:…
244:⊂
210:⊂
176:…
1458:Topology
1385:Wikibook
1363:Category
1251:Manifold
1219:Homotopy
1177:Interior
1168:Open set
1126:Homology
1075:Topology
1034:15867466
448:, write
1410:general
1212:uniform
1192:compact
1143:Digital
977:9931601
794:subsets
334:History
80:scholar
1405:Topics
1207:metric
1082:Fields
1032:
1024:
975:
892:
835:
361:Tsaban
82:
75:
68:
61:
53:
1187:Space
1030:S2CID
1004:arXiv
973:S2CID
955:arXiv
912:(PDF)
890:S2CID
122:is a
87:JSTOR
73:books
1022:ISSN
859:help
833:ISBN
59:news
1014:doi
965:doi
951:134
924:doi
920:129
882:doi
825:doi
565:of
481:if
353:ZFC
42:by
1449::
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851::
849:}}
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884::
861:)
841:.
827::
776:F
732:d
704:d
680:g
667:f
647:A
641:g
618:N
612:N
604:f
581:N
575:N
553:A
533:n
513:)
510:n
507:(
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498:)
495:n
492:(
489:f
469:g
456:f
433:N
427:N
419:g
416:,
413:f
390:N
384:N
310:2
304:F
293:1
287:F
261:,
256:2
250:U
239:2
233:F
227:,
222:1
216:U
205:1
199:F
173:,
168:2
162:U
156:,
151:1
145:U
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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