284:
483:
357:
612:
90:
655:
746:
521:
200:
137:
sets in the collection are almost disjoint. Often the prefix 'pairwise' is dropped, and a pairwise almost disjoint collection is simply called "almost disjoint".
372:
532:
551:. Here are some alternative definitions of "almost disjoint" that are sometimes used (similar definitions apply to infinite collections):
904:
859:
896:
847:
837:
Vaughan, Jerry E. (1990). "Chapter 11: Small uncountable cardinals and topology". In van Mill, Jan; Reed, George M. (eds.).
310:
572:
50:
304: } is an almost disjoint collection consisting of more than one set, then clearly its intersection is finite:
32:
is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint".
926:
503:
The possible cardinalities of a maximal almost disjoint family (commonly referred to as a MAD family) on the set
29:
627:
544:
878:(January 12, 2010). "Chapter 6 : Combinatorial Cardinal Characteristics of the Continuum". In
811:
Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118
710:
851:
838:
900:
855:
759:
21:
799:
Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North
Holland, p. 47
506:
883:
363:
279:{\displaystyle A_{i}\neq A_{j}\quad \implies \quad \left|A_{i}\cap A_{j}\right|<\infty .}
879:
820:
528:
290:
116:
108:
111:
and are almost disjoint, because their intersection is the finite set {1}. However, the
679:
524:
44:. In this case, two sets are almost disjoint if their intersection is finite, i.e. if
920:
875:
683:
619:
112:
563:
are almost disjoint if the cardinality of their intersection is less than κ, i.e. if
661:
is simply the definition of almost disjoint given above, where the intersection of
478:{\displaystyle \{\{1,2,3,\ldots \},\{2,3,4,\ldots \},\{3,4,5,\ldots \},\ldots \}}
887:
775:
548:
100:
17:
41:
543:
Sometimes "almost disjoint" is used in some other sense, or in the sense of
489:
145:
125:
This definition extends to any collection of sets. A collection of sets is
295:
is almost disjoint, because any two of them only meet at the origin. If {
122:
are not almost disjoint, because their intersection is infinite.
823:. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds)
701:
are almost disjoint if their intersection is a null-set, i.e. if
527:
has been the object of intense study. The minimum infinite such
289:
For example, the collection of all lines through the origin in
713:
630:
575:
509:
375:
313:
203:
53:
618:The case of κ = 1 is simply the definition of
740:
649:
606:
515:
500:two distinct sets in this collection is infinite.
477:
352:{\displaystyle \bigcap _{i\in I}A_{i}<\infty .}
351:
278:
84:
40:The most common choice is to take "small" to mean
107:, and '< ∞' means 'finite'.) For example, the
607:{\displaystyle \left|A\cap B\right|<\kappa .}
366:is not true—the intersection of the collection
85:{\displaystyle \left|A\cap B\right|<\infty .}
496:almost disjoint; in fact, the intersection of
774:are almost disjoint if their intersection is
8:
555:Let κ be any cardinal number. Then two sets
472:
463:
439:
433:
409:
403:
379:
376:
232:
228:
712:
641:
629:
574:
533:cardinal characteristics of the continuum
508:
374:
334:
318:
312:
256:
243:
221:
208:
202:
52:
792:
182: } is almost disjoint if for any
165:be a set. Then the collection of sets {
807:
805:
7:
825:Handbook of Set-Theoretic Topology.
650:{\displaystyle \kappa =\aleph _{0}}
638:
343:
270:
76:
14:
848:North-Holland Publishing Company
827:North-Holland, Amsterdam, 1984.
233:
227:
729:
717:
229:
1:
741:{\displaystyle m(A\cap B)=0.}
943:
840:Open Problems in Topology
531:is one of the classical
492:, but the collection is
131:mutually almost disjoint
127:pairwise almost disjoint
516:{\displaystyle \omega }
889:Handbook of Set Theory
742:
651:
608:
517:
479:
353:
280:
99: |' denotes the
86:
743:
652:
609:
518:
480:
354:
281:
87:
899:. pp. 395–490.
711:
628:
573:
549:topological category
507:
373:
311:
201:
51:
762:. Then two subsets
689:. Then two subsets
738:
647:
604:
513:
475:
349:
329:
276:
82:
884:Kanamori, Akihiro
760:topological space
314:
934:
927:Families of sets
911:
910:
894:
880:Foreman, Matthew
872:
866:
865:
845:
834:
828:
818:
812:
809:
800:
797:
747:
745:
744:
739:
680:complete measure
656:
654:
653:
648:
646:
645:
613:
611:
610:
605:
594:
590:
522:
520:
519:
514:
484:
482:
481:
476:
358:
356:
355:
350:
339:
338:
328:
285:
283:
282:
277:
266:
262:
261:
260:
248:
247:
226:
225:
213:
212:
117:rational numbers
109:closed intervals
91:
89:
88:
83:
72:
68:
942:
941:
937:
936:
935:
933:
932:
931:
917:
916:
915:
914:
907:
895:. Vol. 1.
892:
874:
873:
869:
862:
843:
836:
835:
831:
821:Eric van Douwen
819:
815:
810:
803:
798:
794:
789:
709:
708:
637:
626:
625:
580:
576:
571:
570:
541:
525:natural numbers
505:
504:
371:
370:
330:
309:
308:
303:
252:
239:
238:
234:
217:
204:
199:
198:
173:
164:
148:, and for each
115:and the set of
58:
54:
49:
48:
38:
26:almost disjoint
12:
11:
5:
940:
938:
930:
929:
919:
918:
913:
912:
905:
876:Blass, Andreas
867:
860:
829:
813:
801:
791:
790:
788:
785:
784:
783:
751:
750:
749:
748:
737:
734:
731:
728:
725:
722:
719:
716:
703:
702:
671:
670:
659:
658:
657:
644:
640:
636:
633:
622:; the case of
616:
615:
614:
603:
600:
597:
593:
589:
586:
583:
579:
565:
564:
545:measure theory
540:
539:Other meanings
537:
512:
486:
485:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
360:
359:
348:
345:
342:
337:
333:
327:
324:
321:
317:
299:
287:
286:
275:
272:
269:
265:
259:
255:
251:
246:
242:
237:
231:
224:
220:
216:
211:
207:
169:
160:
140:Formally, let
93:
92:
81:
78:
75:
71:
67:
64:
61:
57:
37:
34:
13:
10:
9:
6:
4:
3:
2:
939:
928:
925:
924:
922:
908:
906:1-4020-4843-2
902:
898:
891:
890:
885:
881:
877:
871:
868:
863:
861:0-444-88768-7
857:
853:
849:
846:. Amsterdam:
842:
841:
833:
830:
826:
822:
817:
814:
808:
806:
802:
796:
793:
786:
781:
777:
773:
769:
765:
761:
757:
753:
752:
735:
732:
726:
723:
720:
714:
707:
706:
705:
704:
700:
696:
692:
688:
685:
684:measure space
681:
677:
673:
672:
668:
664:
660:
642:
634:
631:
624:
623:
621:
620:disjoint sets
617:
601:
598:
595:
591:
587:
584:
581:
577:
569:
568:
567:
566:
562:
558:
554:
553:
552:
550:
546:
538:
536:
534:
530:
526:
510:
501:
499:
495:
491:
469:
466:
460:
457:
454:
451:
448:
445:
442:
436:
430:
427:
424:
421:
418:
415:
412:
406:
400:
397:
394:
391:
388:
385:
382:
369:
368:
367:
365:
362:However, the
346:
340:
335:
331:
325:
322:
319:
315:
307:
306:
305:
302:
298:
294:
293:
273:
267:
263:
257:
253:
249:
244:
240:
235:
222:
218:
214:
209:
205:
197:
196:
195:
193:
189:
185:
181:
177:
172:
168:
163:
159:
155:
151:
147:
143:
138:
136:
132:
128:
123:
121:
118:
114:
113:unit interval
110:
106:
102:
98:
79:
73:
69:
65:
62:
59:
55:
47:
46:
45:
43:
35:
33:
31:
27:
23:
19:
888:
870:
839:
832:
824:
816:
795:
779:
771:
767:
763:
755:
698:
694:
690:
686:
675:
666:
662:
560:
556:
542:
502:
497:
493:
487:
361:
300:
296:
291:
288:
191:
187:
183:
179:
175:
170:
166:
161:
157:
153:
149:
141:
139:
134:
130:
126:
124:
119:
104:
96:
94:
39:
30:intersection
25:
15:
850:. pp.
133:if any two
101:cardinality
18:mathematics
787:References
669:is finite.
36:Definition
724:∩
639:ℵ
632:κ
599:κ
585:∩
511:ω
470:…
461:…
431:…
401:…
344:∞
323:∈
316:⋂
271:∞
250:∩
230:⟹
215:≠
146:index set
95:(Here, '|
77:∞
63:∩
28:if their
921:Category
897:Springer
886:(eds.).
529:cardinal
364:converse
174: :
135:distinct
852:196–218
523:of the
903:
858:
776:meagre
156:, let
144:be an
42:finite
20:, two
893:(PDF)
844:(PDF)
758:be a
682:on a
678:be a
490:empty
901:ISBN
856:ISBN
766:and
754:Let
693:and
674:Let
665:and
596:<
559:and
341:<
268:<
186:and
74:<
24:are
22:sets
778:in
770:of
697:of
547:or
498:any
494:not
488:is
190:in
178:in
152:in
129:or
103:of
16:In
923::
882:;
854:.
804:^
736:0.
535:.
194:,
909:.
864:.
782:.
780:X
772:X
768:B
764:A
756:X
733:=
730:)
727:B
721:A
718:(
715:m
699:X
695:B
691:A
687:X
676:m
667:B
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643:0
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602:.
592:|
588:B
582:A
578:|
561:B
557:A
473:}
467:,
464:}
458:,
455:5
452:,
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446:,
443:3
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428:,
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413:2
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