192:
418:
440:
378:
537:
573:
341:
217:
472:
315:
241:
161:
141:
117:
97:
73:
41:
797:
589:
of two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself.
739:
761:
780:
294:
220:
351:. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.
285:
A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of
451:
673:
775:
821:
383:
603:
540:
166:
423:
361:
785:
348:
286:
504:
837:
793:
771:
757:
735:
727:
598:
501:
may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space
44:
546:
320:
647:
20:
807:
803:
789:
344:
380:. As an example that is dense-in-itself but not dense in its topological space, consider
197:
457:
300:
226:
146:
126:
102:
82:
76:
58:
26:
831:
478:
586:
290:
608:
579:
251:
120:
817:
749:
247:
494:
355:
258:
648:"On Two questions of Arhangel'skii and Collins regarding submaximal spaces"
254:. (In other words, a perfect set is a closed set without isolated point.)
474:
can never be dense-in-itself, because its unique point is isolated in it.
486:
482:
343:. On the other hand, the set of irrationals is not closed because every
354:
The above examples, the irrationals and the rationals, are also
816:
This article incorporates material from Dense in-itself on
672:
Dontchev, Julian; Ganster, Maximilian; Rose, David (1977).
477:
The dense-in-itself subsets of any space are closed under
549:
507:
460:
426:
386:
364:
323:
303:
229:
200:
169:
149:
129:
105:
85:
61:
29:
567:
531:
466:
434:
412:
372:
335:
309:
235:
211:
186:
155:
135:
111:
91:
67:
35:
822:Creative Commons Attribution/Share-Alike License
481:. In a dense-in-itself space, they include all
317:contains at least one other irrational number
788:reprint of 1978 ed.). Berlin, New York:
293:). This set is dense-in-itself because every
8:
578:The closure of any dense-in-itself set is a
562:
556:
526:
514:
548:
506:
459:
428:
427:
425:
388:
387:
385:
366:
365:
363:
322:
302:
277:is dense-in-itself" (no isolated point).
228:
199:
168:
148:
128:
104:
84:
60:
28:
16:Topological subset with no isolated point
265:. This can sometimes be confusing, as "
620:
575:is dense, but is not dense-in-itself.
99:is dense-in-itself if every point of
7:
273:" (always true) is not the same as "
646:Levy, Ronnie; Porter, Jack (1996).
358:in their topological space, namely
163:is dense-in-itself if and only if
14:
413:{\displaystyle \mathbb {Q} \cap }
497:. However, spaces that are not T
289:(considered as a subset of the
820:, which is licensed under the
407:
395:
1:
734:. Heldermann Verlag, Berlin.
187:{\displaystyle A\subseteq A'}
435:{\displaystyle \mathbb {R} }
373:{\displaystyle \mathbb {R} }
781:Counterexamples in Topology
420:. This set is not dense in
854:
627:Steen & Seebach, p. 6
532:{\displaystyle X=\{a,b\}}
686:Engelking, 1.7.10, p. 59
442:but is dense-in-itself.
297:of an irrational number
674:"α-Scattered spaces II"
568:{\displaystyle A=\{a\}}
485:. In a dense-in-itself
336:{\displaystyle y\neq x}
776:Seebach, J. Arthur Jr.
569:
533:
468:
436:
414:
374:
337:
311:
237:
213:
188:
157:
137:
113:
93:
69:
37:
570:
534:
469:
437:
415:
375:
338:
312:
238:
214:
189:
158:
138:
114:
94:
70:
38:
655:Topology Proceedings
604:Glossary of topology
547:
505:
458:
424:
384:
362:
321:
301:
227:
198:
167:
147:
127:
103:
83:
59:
27:
541:indiscrete topology
772:Steen, Lynn Arthur
756:. Academic Press.
728:Engelking, Ryszard
565:
529:
464:
454:subset of a space
432:
410:
370:
333:
307:
287:irrational numbers
246:A dense-in-itself
233:
212:{\displaystyle A'}
209:
184:
153:
133:
109:
89:
65:
33:
799:978-0-486-68735-3
713:Kuratowski, p. 77
704:Kuratowski, p. 78
695:Kuratowski, p. 78
599:Nowhere dense set
493:they include all
467:{\displaystyle X}
310:{\displaystyle x}
261:is distinct from
236:{\displaystyle A}
156:{\displaystyle A}
136:{\displaystyle A}
112:{\displaystyle A}
92:{\displaystyle A}
68:{\displaystyle A}
45:topological space
36:{\displaystyle A}
845:
811:
767:
745:
732:General Topology
714:
711:
705:
702:
696:
693:
687:
684:
678:
677:
669:
663:
662:
652:
643:
637:
636:Engelking, p. 25
634:
628:
625:
585:In general, the
574:
572:
571:
566:
538:
536:
535:
530:
473:
471:
470:
465:
441:
439:
438:
433:
431:
419:
417:
416:
411:
391:
379:
377:
376:
371:
369:
342:
340:
339:
334:
316:
314:
313:
308:
242:
240:
239:
234:
218:
216:
215:
210:
208:
193:
191:
190:
185:
183:
162:
160:
159:
154:
142:
140:
139:
134:
118:
116:
115:
110:
98:
96:
95:
90:
79:. Equivalently,
74:
72:
71:
66:
42:
40:
39:
34:
21:general topology
853:
852:
848:
847:
846:
844:
843:
842:
828:
827:
800:
790:Springer-Verlag
770:
764:
754:Topology Vol. I
748:
742:
726:
723:
718:
717:
712:
708:
703:
699:
694:
690:
685:
681:
671:
670:
666:
650:
645:
644:
640:
635:
631:
626:
622:
617:
595:
545:
544:
503:
502:
500:
490:
456:
455:
448:
422:
421:
382:
381:
360:
359:
345:rational number
319:
318:
299:
298:
283:
263:dense-in-itself
225:
224:
201:
196:
195:
176:
165:
164:
145:
144:
125:
124:
101:
100:
81:
80:
57:
56:
49:dense-in-itself
25:
24:
17:
12:
11:
5:
851:
849:
841:
840:
830:
829:
813:
812:
798:
768:
762:
750:Kuratowski, K.
746:
740:
722:
719:
716:
715:
706:
697:
688:
679:
664:
638:
629:
619:
618:
616:
613:
612:
611:
606:
601:
594:
591:
564:
561:
558:
555:
552:
528:
525:
522:
519:
516:
513:
510:
498:
488:
463:
447:
444:
430:
409:
406:
403:
400:
397:
394:
390:
368:
332:
329:
326:
306:
282:
279:
257:The notion of
232:
207:
204:
182:
179:
175:
172:
152:
132:
108:
88:
77:isolated point
64:
47:is said to be
32:
15:
13:
10:
9:
6:
4:
3:
2:
850:
839:
836:
835:
833:
826:
825:
823:
819:
809:
805:
801:
795:
791:
787:
783:
782:
777:
773:
769:
765:
759:
755:
751:
747:
743:
741:3-88538-006-4
737:
733:
729:
725:
724:
720:
710:
707:
701:
698:
692:
689:
683:
680:
675:
668:
665:
660:
656:
649:
642:
639:
633:
630:
624:
621:
614:
610:
607:
605:
602:
600:
597:
596:
592:
590:
588:
583:
581:
576:
559:
553:
550:
542:
523:
520:
517:
511:
508:
496:
492:
484:
480:
475:
461:
453:
445:
443:
404:
401:
398:
392:
357:
352:
350:
346:
330:
327:
324:
304:
296:
292:
288:
280:
278:
276:
272:
268:
264:
260:
255:
253:
249:
244:
230:
222:
205:
202:
180:
177:
173:
170:
150:
130:
122:
106:
86:
78:
62:
54:
50:
46:
30:
22:
815:
814:
779:
753:
731:
709:
700:
691:
682:
667:
658:
654:
641:
632:
623:
587:intersection
584:
577:
476:
449:
353:
347:lies in its
295:neighborhood
291:real numbers
284:
274:
270:
269:is dense in
266:
262:
256:
250:is called a
245:
52:
48:
18:
609:Dense order
580:perfect set
252:perfect set
221:derived set
121:limit point
23:, a subset
818:PlanetMath
763:012429202X
721:References
661:: 143–154.
543:, the set
495:dense sets
446:Properties
356:dense sets
248:closed set
539:with the
483:open sets
452:singleton
393:∩
328:≠
259:dense set
174:⊆
838:Topology
832:Category
778:(1978).
752:(1966).
730:(1989).
593:See also
281:Examples
206:′
194:, where
181:′
808:0507446
349:closure
219:is the
143:. Thus
75:has no
53:crowded
806:
796:
760:
738:
479:unions
786:Dover
651:(PDF)
615:Notes
491:space
119:is a
43:of a
794:ISBN
758:ISBN
736:ISBN
223:of
123:of
55:if
51:or
19:In
834::
804:MR
802:.
792:.
774:;
659:21
657:.
653:.
582:.
450:A
243:.
824:.
810:.
784:(
766:.
744:.
676:.
563:}
560:a
557:{
554:=
551:A
527:}
524:b
521:,
518:a
515:{
512:=
509:X
499:1
489:1
487:T
462:X
429:R
408:]
405:1
402:,
399:0
396:[
389:Q
367:R
331:x
325:y
305:x
275:X
271:X
267:X
231:A
203:A
178:A
171:A
151:A
131:A
107:A
87:A
63:A
31:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.