423:
288:
in the lexicographic product. Some authors take as definition the same space without the two isolated points. (In the point splitting description this corresponds to not splitting the endpoints
627:
575:
531:
203:
107:
722:
449:
332:
286:
254:
655:
326:
306:
163:
629:
is homeomorphic to the
Sorgenfrey line with half-open intervals to the right as a base, like two parallel arrows going in opposite directions, hence the name.
143:
967:
900:
668:
221:
932:
917:
827:
342:
994:
950:
725:
687:
945:
691:
777:
57:
588:
536:
672:
458:
205:, and giving the resulting linearly ordered set the order topology. The space is also known as the
168:
62:
955:
756:
661:
862:
Ostaszewski, A. J. (February 1974), "A Characterization of
Compact, Separable, Ordered Spaces",
701:
963:
941:
928:
913:
905:
896:
29:
871:
789:
733:
676:
658:
428:
41:
977:
259:
227:
973:
959:
664:
582:
336:
33:
585:
with half-open intervals to the left as a base for the topology, and the upper subspace
40:. It satisfies various interesting properties and serves as a useful counterexample in
640:
311:
291:
148:
110:
37:
116:
988:
533:, which are simultaneously closed intervals and open intervals.) The lower subspace
747:, separable ordered spaces are order-isomorphic to a subset of the split interval.
737:
578:
794:
744:
875:
680:
328:
of the interval.) The resulting space has essentially the same properties.
113:. Equivalently, the space can be constructed by taking the closed interval
17:
452:
339:
for the double arrow space topology consists of all sets of the form
36:
into two adjacent points and giving the resulting ordered set the
840:
Steen & Seebach, counterexample #95, under the name of
780:(6 July 1999), "Compact subsets of the first Baire class",
418:{\displaystyle ((a,b]\times \{0\})\cup ([a,b)\times \{1\})}
759: – List of concrete topologies and topological spaces
704:
643:
591:
539:
461:
451:. (In the point splitting description these are the
431:
345:
314:
294:
262:
230:
171:
151:
119:
65:
828:"The Lexicographic Order and The Double Arrow Space"
716:
649:
621:
569:
525:
443:
417:
320:
300:
280:
248:
197:
157:
137:
101:
683:; its metrizable subspaces are all countable.
958:reprint of 1978 ed.). Berlin, New York:
8:
782:Journal of the American Mathematical Society
616:
610:
564:
558:
409:
403:
373:
367:
331:The double arrow space is a subspace of the
96:
84:
32:that results from splitting each point in a
891:Arhangel'skii, A.V. and Sklyarenko, E.G..,
145:with its usual order, splitting each point
864:Journal of the London Mathematical Society
793:
703:
642:
590:
538:
514:
501:
482:
469:
460:
430:
344:
313:
293:
261:
229:
189:
176:
170:
150:
118:
64:
769:
335:. If we ignore the isolated points, a
724:of the space with itself is not even
333:lexicographically ordered unit square
7:
895:, Springer-Verlag, New York (1996)
912:, Heldermann Verlag Berlin, 1989.
669:linearly ordered topological space
222:linearly ordered topological space
14:
622:{\displaystyle [0,1)\times \{1\}}
570:{\displaystyle (0,1]\times \{0\}}
732:), as it contains a copy of the
690:, hereditarily separable, and
604:
592:
552:
540:
526:{\displaystyle =(a^{-},b^{+})}
520:
494:
488:
462:
412:
397:
385:
382:
376:
361:
349:
346:
275:
263:
243:
231:
198:{\displaystyle a^{-}<a^{+}}
132:
120:
102:{\displaystyle \times \{0,1\}}
78:
66:
1:
795:10.1090/S0894-0347-99-00312-4
211:Alexandrov double arrow space
951:Counterexamples in Topology
842:weak parallel line topology
1011:
224:with two isolated points,
852:Engelking, example 3.10.C
717:{\displaystyle X\times X}
165:into two adjacent points
925:Measure Theory, Volume 4
876:10.1112/jlms/s2-7.4.758
946:Seebach, J. Arthur Jr.
923:Fremlin, D.H. (2003),
718:
651:
623:
571:
527:
455:intervals of the form
445:
444:{\displaystyle a<b}
419:
322:
302:
282:
250:
199:
159:
139:
103:
56:can be defined as the
807:Fremlin, section 419L
719:
688:hereditarily Lindelöf
652:
624:
572:
528:
446:
420:
323:
303:
283:
281:{\displaystyle (1,1)}
251:
249:{\displaystyle (0,0)}
220:The space above is a
200:
160:
140:
104:
58:lexicographic product
816:Arhangel'skii, p. 39
702:
698:). But the product
641:
589:
537:
459:
429:
343:
312:
292:
260:
228:
169:
149:
117:
63:
893:General Topology II
726:hereditarily normal
637:The split interval
995:Topological spaces
942:Steen, Lynn Arthur
927:, Torres Fremlin,
906:Engelking, Ryszard
757:List of topologies
714:
647:
619:
567:
523:
441:
415:
318:
298:
278:
246:
207:double arrow space
195:
155:
135:
109:equipped with the
99:
26:double arrow space
969:978-0-486-68735-3
901:978-3-642-77032-6
778:Todorcevic, Stevo
650:{\displaystyle X}
321:{\displaystyle 1}
301:{\displaystyle 0}
158:{\displaystyle a}
30:topological space
1002:
981:
937:
910:General Topology
879:
878:
859:
853:
850:
844:
838:
832:
831:
823:
817:
814:
808:
805:
799:
798:
797:
774:
734:Sorgenfrey plane
723:
721:
720:
715:
692:perfectly normal
677:second countable
659:zero-dimensional
656:
654:
653:
648:
628:
626:
625:
620:
576:
574:
573:
568:
532:
530:
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327:
325:
324:
319:
307:
305:
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299:
287:
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284:
279:
255:
253:
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247:
215:two arrows space
204:
202:
201:
196:
194:
193:
181:
180:
164:
162:
161:
156:
144:
142:
141:
138:{\displaystyle }
136:
108:
106:
105:
100:
42:general topology
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1005:
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1001:
1000:
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985:
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970:
960:Springer-Verlag
940:
935:
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839:
835:
825:
824:
820:
815:
811:
806:
802:
776:
775:
771:
766:
753:
736:, which is not
731:
700:
699:
697:
665:Hausdorff space
639:
638:
635:
587:
586:
583:Sorgenfrey line
535:
534:
510:
497:
478:
465:
457:
456:
427:
426:
341:
340:
310:
309:
290:
289:
258:
257:
226:
225:
185:
172:
167:
166:
147:
146:
115:
114:
61:
60:
50:
34:closed interval
12:
11:
5:
1008:
1006:
998:
997:
987:
986:
983:
982:
968:
938:
933:
920:
903:
887:
884:
881:
880:
870:(4): 758–760,
854:
845:
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749:
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369:
366:
363:
360:
357:
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351:
348:
317:
297:
277:
274:
271:
268:
265:
245:
242:
239:
236:
233:
192:
188:
184:
179:
175:
154:
134:
131:
128:
125:
122:
111:order topology
98:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
54:split interval
49:
46:
38:order topology
22:split interval
13:
10:
9:
6:
4:
3:
2:
1007:
996:
993:
992:
990:
979:
975:
971:
965:
961:
957:
953:
952:
947:
943:
939:
936:
934:0-9538129-4-4
930:
926:
921:
919:
918:3-88538-006-4
915:
911:
907:
904:
902:
898:
894:
890:
889:
885:
877:
873:
869:
865:
858:
855:
849:
846:
843:
837:
834:
829:
822:
819:
813:
810:
804:
801:
796:
791:
788:: 1179–1212,
787:
783:
779:
773:
770:
763:
758:
755:
754:
750:
748:
746:
741:
739:
735:
727:
711:
708:
705:
693:
689:
684:
682:
678:
674:
670:
666:
663:
660:
644:
632:
630:
613:
607:
601:
598:
595:
584:
580:
561:
555:
549:
546:
543:
515:
511:
507:
502:
498:
491:
483:
479:
475:
470:
466:
454:
438:
435:
432:
406:
400:
394:
391:
388:
379:
370:
364:
358:
355:
352:
338:
334:
329:
315:
295:
272:
269:
266:
240:
237:
234:
223:
218:
216:
212:
208:
190:
186:
182:
177:
173:
152:
129:
126:
123:
112:
93:
90:
87:
81:
75:
72:
69:
59:
55:
47:
45:
43:
39:
35:
31:
27:
23:
19:
949:
924:
909:
892:
867:
863:
857:
848:
841:
836:
821:
812:
803:
785:
781:
772:
742:
685:
679:, hence not
636:
579:homeomorphic
330:
219:
214:
210:
206:
53:
51:
25:
21:
15:
667:. It is a
886:References
681:metrizable
633:Properties
48:Definition
948:(1995) .
826:Ma, Dan.
709:×
673:separable
608:×
556:×
503:−
484:−
401:×
380:∪
365:×
178:−
82:×
989:Category
751:See also
675:but not
671:that is
18:topology
978:0507446
745:compact
662:compact
581:to the
28:, is a
976:
966:
931:
916:
899:
738:normal
686:It is
453:clopen
20:, the
956:Dover
764:Notes
657:is a
425:with
24:, or
964:ISBN
929:ISBN
914:ISBN
897:ISBN
868:s2-7
743:All
436:<
337:base
308:and
256:and
183:<
52:The
872:doi
790:doi
577:is
213:or
16:In
991::
974:MR
972:.
962:.
944:;
908:,
866:,
786:12
784:,
740:.
728:(T
694:(T
217:.
209:,
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980:.
954:(
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830:.
792::
730:5
712:X
706:X
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614:1
611:{
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602:1
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596:0
593:[
565:}
562:0
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550:1
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544:0
541:(
521:)
516:+
512:b
508:,
499:a
495:(
492:=
489:]
480:b
476:,
471:+
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463:[
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410:}
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383:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.