247:
1287:
421:
if each is disjoint from the other's closure, though the closures themselves do not have to be disjoint. Equivalently, each subset is included in an open set disjoint from the other subset. All of the remaining conditions for separation of sets may also be applied to points (or to a point and a set)
525:
These conditions are given in order of increasing strength: Any two topologically distinguishable points must be distinct, and any two separated points must be topologically distinguishable. Any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so
1283:, and one goes from the left side to the right side by removing that requirement, using the Kolmogorov quotient operation. (The names in parentheses given on the left side of this table are generally ambiguous or at least less well known; but they are used in the diagram below.)
567:
Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.
1521:
There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they aren't discussed here; see their individual articles.
1471:
as follows: "P" = "perfectly", "C" = "completely", "N" = "normal", and "R" (without a subscript) = "regular". A bullet indicates that there is no special name for a space at that spot. The dash at the bottom indicates no condition.
1105:
The following table summarizes the separation axioms as well as the implications between them: cells which are merged represent equivalent properties, each axiom implies the ones in the cells to its left, and if we assume the
351:(in any of various ways). The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.
250:
An illustration of some of the separation axioms. Grey amorphous broken-outline regions indicate open sets surrounding disjoint closed sets or points: red solid-outline circles denote closed sets while black dots represent
1502:
together imply a host of other properties, since combining the two properties leads through the many nodes on the right-side branch. Since regularity is the most well known of these, spaces that are both normal and
1278:
for more information. When applied to the separation axioms, this leads to the relationships in the table to the left below. In this table, one goes from the right side to the left side by adding the requirement of
1475:
Two properties may be combined using this diagram by following the diagram upwards until both branches meet. For example, if a space is both completely normal ("CN") and completely
Hausdorff ("CT
1511:
are often called "normal
Hausdorff spaces" by people that wish to avoid the ambiguous "T" notation. These conventions can be generalised to other regular spaces and Hausdorff spaces.
290:, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the
378:(or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an
1071:
if any two disjoint closed sets are precisely separated by a continuous function. Every perfectly normal space is also both completely normal and completely regular.
430:
will be considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by a function, if and only if their singleton sets {
974:
are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is
1173:
775:
121:
103:
1551:. More briefly, every irreducible closed set has a unique generic point. Any Hausdorff space must be sober, and any sober space must be T
1872:
1037:
1079:
553:
301:
235:
1286:
287:
1844:
604:
534:
382:
that one point belongs to but the other point does not). That is, at least one of the points does not belong to the other's
1727:(Hausdorff) space satisfies the axiom that all compact subsets are closed, but the reverse is not necessarily true; for the
564:", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.
685:
is Fréchet"; one should avoid saying "Fréchet space" in this context, since there is another entirely different notion of
1715:
because every single-point set is necessarily compact and thus closed, but the reverse is not necessarily true; for the
1459:, the relationships between the separation axioms are indicated in the diagram to the right. In this diagram, the non-T
401:
if each of them has a neighbourhood that is not a neighbourhood of the other; that is, neither belongs to the other's
375:
1916:
1696:. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.
1178:
802:
20:
1203:
1183:
900:
1507:
are typically called "normal regular spaces". In a somewhat similar fashion, spaces that are both normal and T
1025:
873:
246:
1495:
space (less ambiguously known as a completely normal
Hausdorff space, as can be seen in the table above).
1067:
1002:
336:
332:
1619:
1029:
if any two separated sets are separated by neighbourhoods. Every completely normal space is also normal.
607:. (It will be a common theme among the separation axioms to have one version of an axiom that requires T
304:. Especially in older literature, different authors might have different definitions of each condition.
1900:
1728:
1563:
986:
1275:
690:
462:
297:("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties.
1262:
axiom is special in that it can not only be added to a property (so that completely regular plus T
1647:
1539:
that is not the (possibly nondisjoint) union of two smaller closed sets, there is a unique point
958:
and completely regular. Every
Tychonoff space is both regular Hausdorff and completely Hausdorff.
402:
383:
34:
975:
1921:
1868:
1840:
1716:
1595:
1238:
924:, they are separated by a continuous function. Every completely regular space is also regular.
576:
317:
312:
Before we define the separation axioms themselves, we give concrete meaning to the concept of
264:
41:
1110:
axiom, then each axiom also implies the ones in the cells above it (for example, all normal T
1749:
1600:
1147:
686:
276:
49:
1656:
1604:
1168:
932:
732:
291:
157:
85:
1735:
many points, the compact sets are all finite and hence all closed but the space is not T
1665:
1491:
side of the slash, so a completely normal completely
Hausdorff space is the same as a T
1244:
530:
313:
529:
For more on these conditions (including their use outside the separation axioms), see
1910:
1862:
1198:
829:
335:
points. It's not enough for elements of a topological space to be distinct (that is,
139:
1583:. Any Hausdorff space must be weak Hausdorff, and any weak Hausdorff space must be T
1895:
1468:
1223:
1208:
1017:
and normal. Every normal
Hausdorff space is also both Tychonoff and normal regular.
966:
328:
212:
192:
174:
817:
are separated by a continuous function. Every completely
Hausdorff space is also T
1834:
1732:
1693:
1531:
260:
1723:, every subset is compact but not every subset is closed. Furthermore, every T
1266:
is
Tychonoff) but also be subtracted from a property (so that Hausdorff minus T
861:
will also be separated by closed neighbourhoods.) Every regular space is also R
853:, they are separated by neighbourhoods. (In fact, in a regular space, any such
1652:
1101:. Every perfectly normal Hausdorff space is also completely normal Hausdorff.
661:
are separated. Equivalently, every single-point set is a closed set. Thus,
473:
327:
The separation axioms are about the use of topological means to distinguish
1193:
1163:
1142:
701:
642:
619:
588:
480:
379:
267:
that one wishes to consider. Some of these restrictions are given by the
256:
67:
1711:(Hausdorff) in strength. A space satisfying this axiom is necessarily T
1680:
and fully normal. Every fully normal space is normal and every fully T
263:, there are several restrictions that one often makes on the kinds of
344:
681:
space", "Fréchet topology", and "suppose that the topological space
1059:. Every completely normal Hausdorff space is also normal Hausdorff.
994:
and normal. Every normal regular space is also completely regular.
1487:(even though completely normal spaces may not be), one takes the T
1463:
version of a condition is on the left side of the slash, and the T
283:
245:
1885:
axioms mentioned in the Main
Definitions, with these definitions)
347:
of a topological space to be disjoint; we may want them to be
1479:"), then following both branches up, one finds the spot "•/T
1285:
438:} are separated according to the corresponding criterion.
560:" are sometimes interchanged, similarly "regular" and "T
554:
alternative meanings in some of mathematical literature
457:
if they have disjoint closed neighbourhoods. They are
712:, if any two topologically distinguishable points in
630:, if any two topologically distinguishable points in
888:
and regular. Every regular
Hausdorff space is also T
1467:version is on the right side. Letters are used for
211:
191:
173:
156:
138:
120:
102:
84:
66:
48:
40:
28:
16:
Axioms in topology defining notions of "separation"
790:are separated by closed neighbourhoods. Every T
453:if they have disjoint neighbourhoods. They are
556:; for example, the meanings of "normal" and "T
1611:. Any regular space must also be semiregular.
1575:from a compact Hausdorff space, the image of
1498:As can be seen from the diagram, normal and R
8:
1901:Table of separation and metrisability axioms
546:
492:precisely separated by a continuous function
1688:. Moreover, one can show that every fully T
1867:. Reading, Mass.: Addison-Wesley Pub. Co.
1483:". Since completely Hausdorff spaces are T
1455:Other than the inclusion or exclusion of T
545:These definitions all use essentially the
282:The separation axioms are not fundamental
1816:
1804:
1792:
1780:
1768:
677:. (Although one may say such things as "T
1836:Handbook of Analysis and its Foundations
1293:
1116:
755:is Hausdorff if and only if it is both T
716:are separated by neighbourhoods. Every R
486:({0}) and B is a subset of the preimage
358:be a topological space. Then two points
1761:
1290:Hasse diagram of the separation axioms.
1055:, if it is both completely normal and T
751:are separated by neighbourhoods. Thus,
302:separation axioms have varied over time
1097:, if it is both perfectly normal and T
970:if any two disjoint closed subsets of
494:if there exists a continuous function
320:. (Separated sets are not the same as
25:
1719:on infinitely many points, which is T
1114:spaces are also completely regular).
571:In all of the following definitions,
374:if they do not have exactly the same
7:
1274:), in a fairly precise sense; see
1128:Separated by closed neighborhoods
459:separated by a continuous function
455:separated by closed neighbourhoods
14:
763:. Every Hausdorff space is also T
343:. Similarly, it's not enough for
19:For the axiom of set theory, see
1254:Relationships between the axioms
1134:Precisely separated by function
813:, if any two distinct points in
786:, if any two distinct points in
747:, if any two distinct points in
657:, if any two distinct points in
599:, if any two distinct points in
422:by using singleton sets. Points
324:, defined in the next section.)
1896:Separation Axioms at ProvenMath
1627:, there is a nonempty open set
479:such that A is a subset of the
300:The precise definitions of the
1701:all compact subsets are closed
1430:Completely normal Hausdorff (T
611:and one version that doesn't.)
535:Topological distinguishability
405:. More generally, two subsets
1:
1839:. San Diego: Academic Press.
1623:if for any nonempty open set
1567:if, for every continuous map
1442:Perfectly normal Hausdorff (T
1190:Closed set and point outside
605:topologically distinguishable
372:topologically distinguishable
341:topologically distinguishable
271:. These are sometimes called
952:completely regular Hausdorff
1125:Separated by neighborhoods
1038:completely normal Hausdorff
669:if and only if it is both T
451:separated by neighbourhoods
273:Tychonoff separation axioms
1938:
1543:such that the closure of {
1437:Completely normal regular
1080:perfectly normal Hausdorff
528:
339:); we may want them to be
21:Axiom schema of separation
18:
1861:Willard, Stephen (1970).
1631:such that the closure of
1535:if, for every closed set
1237:
1222:
1207:
1197:
552:Many of these names have
490:({1}). Finally, they are
231:
1833:Schechter, Eric (1997).
794:space is also Hausdorff.
1517:Other separation axioms
1139:Distinguishable points
547:preliminary definitions
308:Preliminary definitions
1674:fully normal Hausdorff
1291:
1131:Separated by function
259:and related fields of
252:
134:(completely Hausdorff)
1879:(has all of the non-R
1857:axioms, among others)
1703:is strictly between T
1607:for the open sets of
1289:
1215:Disjoint closed sets
249:
1729:cocountable topology
1372:Regular Hausdorff (T
1364:Completely Hausdorff
1179:Completely Hausdorff
904:if, given any point
833:if, given any point
803:completely Hausdorff
510:equals the preimage
1419:Completely normal T
1407:Normal Hausdorff (T
1391:Completely regular
1276:Kolmogorov quotient
1184:Perfectly Hausdorff
920:does not belong to
849:does not belong to
691:functional analysis
463:continuous function
152:(regular Hausdorff)
1819:, 16.6(C), p. 438.
1807:, 16.6(D), p. 438.
1425:Completely normal
1367:(No special name)
1359:(No special name)
1292:
1204:Completely regular
901:completely regular
461:if there exists a
318:topological spaces
265:topological spaces
253:
205:(completely normal
187:(normal Hausdorff)
35:topological spaces
1917:Separation axioms
1717:cofinite topology
1676:, if it is both T
1601:regular open sets
1453:
1452:
1449:Perfectly normal
1319:(No requirement)
1251:
1250:
1239:Completely normal
1026:completely normal
1013:, if it is both T
954:, if it is both T
884:, if it is both T
874:regular Hausdorff
577:topological space
269:separation axioms
244:
243:
225:(perfectly normal
30:Separation axioms
1929:
1878:
1864:General topology
1850:
1820:
1814:
1808:
1802:
1796:
1795:, 16.17, p. 443.
1790:
1784:
1783:, 16.16, p. 442.
1778:
1772:
1766:
1750:General topology
1635:is contained in
1294:
1209:Perfectly normal
1160:Distinct points
1117:
1068:perfectly normal
1003:normal Hausdorff
541:Main definitions
322:separated spaces
316:(and points) in
277:Andrey Tychonoff
227: Hausdorff)
222:
217:
207: Hausdorff)
202:
197:
184:
179:
164:
163:
149:
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131:
126:
111:
110:
95:
90:
77:
72:
59:
54:
26:
1937:
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1907:
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1892:
1884:
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1779:
1775:
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1746:
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1726:
1722:
1714:
1710:
1706:
1699:The axiom that
1691:
1687:
1683:
1679:
1669:
1657:star refinement
1586:
1554:
1519:
1510:
1506:
1501:
1494:
1490:
1486:
1482:
1478:
1466:
1462:
1458:
1445:
1433:
1422:
1414:Normal regular
1410:
1399:
1387:
1375:
1356:
1348:
1341:
1333:
1327:
1316:
1307:
1300:
1282:
1273:
1269:
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1256:
1230:Separated sets
1113:
1109:
1100:
1095:
1088:
1058:
1053:
1046:
1016:
1011:
993:
990:if it is both R
976:Urysohn's lemma
957:
948:
941:
908:and closed set
891:
887:
882:
864:
837:and closed set
820:
811:
793:
779:
766:
762:
758:
741:
723:
720:space is also R
719:
705:
680:
676:
672:
668:
646:
623:
610:
592:
563:
559:
543:
538:
468:from the space
310:
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226:
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165:
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147:
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129:
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112:
108:
106:
93:
91:
88:
75:
73:
70:
57:
55:
52:
32:
24:
17:
12:
11:
5:
1935:
1933:
1925:
1924:
1919:
1909:
1908:
1905:
1904:
1903:from Schechter
1898:
1891:
1890:External links
1888:
1887:
1886:
1880:
1873:
1858:
1852:
1845:
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1817:Schechter 1997
1809:
1805:Schechter 1997
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1793:Schechter 1997
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1781:Schechter 1997
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1769:Schechter 1997
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1564:weak Hausdorff
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1245:discrete space
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1191:
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995:
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987:normal regular
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739:
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678:
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635:
634:are separated.
621:
612:
608:
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561:
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542:
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531:Separated sets
376:neighbourhoods
314:separated sets
309:
306:
295:Trennungsaxiom
286:like those of
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1620:quasi-regular
1616:
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1598:
1597:
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1589:
1582:
1579:is closed in
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860:
856:
852:
848:
844:
840:
836:
832:
831:
826:
823:
816:
812:
805:
804:
799:
796:
789:
785:
781:
780:
772:
769:
754:
750:
746:
742:
735:
734:
729:
726:
715:
711:
707:
706:
698:
695:
692:
688:
687:Fréchet space
684:
664:
660:
656:
652:
648:
647:
639:
636:
633:
629:
625:
624:
616:
613:
606:
602:
598:
594:
593:
585:
582:
581:
580:
578:
574:
569:
565:
555:
550:
548:
540:
536:
532:
527:
523:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
482:
478:
475:
471:
467:
464:
460:
456:
452:
448:
444:
439:
437:
433:
429:
425:
420:
416:
412:
408:
404:
400:
396:
392:
387:
385:
381:
377:
373:
369:
365:
361:
357:
352:
350:
346:
342:
338:
334:
330:
329:disjoint sets
325:
323:
319:
315:
307:
305:
303:
298:
296:
293:
289:
285:
280:
278:
274:
270:
266:
262:
258:
248:
237:
234:
233:
230:
224:
219:
210:
204:
199:
190:
186:
181:
172:
168:
166:
155:
151:
146:
137:
133:
128:
119:
115:
113:
101:
97:
92:
83:
79:
74:
65:
61:
56:
47:
43:
39:
36:
31:
27:
22:
1881:
1863:
1853:
1835:
1812:
1800:
1788:
1776:
1764:
1739:(Hausdorff).
1700:
1673:
1664:
1660:
1655:has an open
1648:fully normal
1646:
1642:
1636:
1632:
1628:
1624:
1618:
1614:
1608:
1594:
1590:
1580:
1576:
1572:
1568:
1562:
1558:
1548:
1544:
1540:
1536:
1530:
1526:
1520:
1513:
1497:
1474:
1469:abbreviation
1454:
1384:Tychonoff (T
1338:Hausdorff (T
1257:
1243:
1233:
1218:
1104:
1091:
1084:
1078:
1074:
1066:
1062:
1050:completely T
1049:
1042:
1036:
1032:
1024:
1020:
1007:
1001:
997:
985:
981:
971:
965:
961:
951:
945:completely T
944:
937:
931:
927:
921:
917:
913:
909:
905:
899:
895:
878:
872:
868:
858:
854:
850:
846:
842:
838:
834:
828:
824:
814:
808:completely T
807:
801:
797:
787:
783:
774:
770:
752:
748:
744:
737:
731:
727:
713:
709:
700:
696:
682:
662:
658:
654:
650:
641:
637:
631:
627:
618:
614:
600:
596:
587:
583:
572:
570:
566:
551:
544:
524:
519:
515:
511:
507:
503:
499:
495:
491:
487:
483:
476:
469:
465:
458:
454:
450:
446:
442:
440:
435:
431:
427:
423:
418:
414:
410:
406:
398:
394:
390:
388:
371:
367:
363:
359:
355:
353:
348:
340:
326:
321:
311:
299:
294:
281:
272:
268:
254:
122:completely T
62:(Kolmogorov)
29:
1733:uncountably
1694:paracompact
1596:semiregular
1092:perfectly T
575:is again a
389:Two points
261:mathematics
169:(Tychonoff)
98:(Hausdorff)
1911:Categories
1846:0126227608
1827:References
1684:space is T
1653:open cover
1148:Preregular
1122:Separated
916:such that
845:such that
710:preregular
651:accessible
597:Kolmogorov
514:({0}) and
506:such that
288:set theory
42:Kolmogorov
1771:, p. 441.
1692:space is
1651:if every
1547:} equals
1194:Symmetric
1169:Hausdorff
1143:Symmetric
933:Tychonoff
745:separated
733:Hausdorff
628:symmetric
474:real line
419:separated
399:separated
349:separated
116:(Urysohn)
80:(Fréchet)
1922:Topology
1744:See also
1396:Normal T
1379:Regular
1308:version
481:preimage
441:Subsets
380:open set
333:distinct
275:, after
257:topology
1666:fully T
1603:form a
1599:if the
1402:Normal
1301:version
1199:Regular
1174:Urysohn
1164:Fréchet
830:regular
784:Urysohn
655:Fréchet
549:above.
522:({1}).
518:equals
472:to the
434:} and {
403:closure
384:closure
345:subsets
337:unequal
251:points.
236:History
162:3½
1871:
1851:(has R
1843:
1234:always
1224:Normal
1219:always
967:normal
292:German
284:axioms
221:
201:
183:
148:
130:
109:½
94:
76:
58:
1756:Notes
1707:and T
1672:, or
1532:sober
1304:Non-T
1258:The T
1083:, or
1041:, or
1006:, or
950:, or
936:, or
877:, or
806:, or
782:, or
759:and R
736:, or
708:, or
673:and R
649:, or
626:, or
595:, or
498:from
1869:ISBN
1841:ISBN
1605:base
1270:is R
857:and
665:is T
603:are
533:and
526:on.
449:are
445:and
426:and
417:are
409:and
397:are
393:and
370:are
362:and
354:Let
331:and
1731:on
1663:is
1645:is
1617:is
1593:is
1571:to
1561:is
1529:is
1090:or
1077:is
1065:is
1048:or
1035:is
1023:is
1000:is
984:is
964:is
930:is
912:in
898:is
871:is
841:in
827:is
800:is
773:is
743:or
730:is
699:is
689:in
653:or
640:is
617:is
586:is
502:to
413:of
366:in
255:In
33:in
1913::
1659:.
1386:3½
1355:2½
978:.)
943:,
940:3½
890:2½
819:2½
792:2½
778:2½
693:.)
579:.
386:.
279:.
1882:i
1877:.
1854:i
1849:.
1737:2
1725:2
1721:1
1713:1
1709:2
1705:1
1690:4
1686:4
1682:4
1678:1
1668:4
1661:X
1643:X
1639:.
1637:G
1633:H
1629:H
1625:G
1615:X
1609:X
1591:X
1587:.
1585:1
1581:X
1577:f
1573:X
1569:f
1559:X
1555:.
1553:0
1549:C
1545:p
1541:p
1537:C
1527:X
1509:1
1505:0
1503:R
1500:0
1493:5
1489:0
1485:0
1481:5
1477:2
1465:0
1461:0
1457:0
1446:)
1444:6
1434:)
1432:5
1421:0
1411:)
1409:4
1398:0
1388:)
1376:)
1374:3
1353:T
1347:1
1345:R
1342:)
1340:2
1332:0
1330:R
1326:1
1324:T
1315:0
1313:T
1306:0
1299:0
1297:T
1281:0
1279:T
1272:1
1268:0
1264:0
1260:0
1112:1
1108:1
1106:T
1099:0
1094:4
1087:6
1085:T
1075:X
1063:X
1057:1
1052:4
1045:5
1043:T
1033:X
1021:X
1015:1
1010:4
1008:T
998:X
992:0
982:X
972:X
962:X
956:0
947:3
938:T
928:X
922:F
918:x
914:X
910:F
906:x
896:X
892:.
886:0
881:3
879:T
869:X
865:.
863:1
859:F
855:x
851:F
847:x
843:X
839:F
835:x
825:X
821:.
815:X
810:2
798:X
788:X
776:T
771:X
767:.
765:1
761:1
757:0
753:X
749:X
740:2
738:T
728:X
724:.
722:0
718:1
714:X
704:1
702:R
697:X
683:X
679:1
675:0
671:0
667:1
663:X
659:X
645:1
643:T
638:X
632:X
622:0
620:R
615:X
609:0
601:X
591:0
589:T
584:X
573:X
562:3
558:4
537:.
520:f
516:B
512:f
508:A
504:R
500:X
496:f
488:f
484:f
477:R
470:X
466:f
447:B
443:A
436:y
432:x
428:y
424:x
415:X
411:B
407:A
395:y
391:x
368:X
364:y
360:x
356:X
216:6
213:T
196:5
193:T
178:4
175:T
158:T
143:3
140:T
125:2
107:2
104:T
89:2
86:T
71:1
68:T
53:0
50:T
23:.
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