799:
687:
280:
891:
816:, 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a
922:
958:
852:
345:
152:
119:
1004:
706:
808:
Now, by the first topological property, the set on the left-hand side cannot be closed. On the other hand, by the second topological property, the sets
592:
195:
857:
829:
122:
1201:
75:
542:
Since any non-empty open set contains an infinite sequence, a finite non-empty set cannot be open; put another way, the
61:. When examined closely, the proof is less a statement about topology than a statement about certain properties of
17:
1355:
1350:
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39:
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318:
96:
1295:
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1210:
1136:
1080:
1049:
696:
The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.
1230:
1226:
1188:
936:
1084:
158:
1321:
1344:
1307:
1176:
1158:
817:
794:{\displaystyle \mathbb {Z} \setminus \{-1,+1\}=\bigcup _{p\mathrm {\,prime} }S(p,0).}
35:
1239:
1108:
1024:
932:
58:
54:
1325:
31:
1334:
566:
547:
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1092:
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965:
182:
1283:
1141:
1124:
682:{\displaystyle S(a,b)=\mathbb {Z} \setminus \bigcup _{j=1}^{a-1}S(a,b+j).}
430:
The intersection of two (and hence finitely many) open sets is open: let
155:
43:
1222:
50:
275:{\displaystyle S(a,b)=\{an+b\mid n\in \mathbb {Z} \}=a\mathbb {Z} +b.}
1299:
126:
1277:
https://kconrad.math.uconn.edu/blurbs/ugradnumthy/primestopology.pdf
1214:
1113:
See discussion immediately prior to Lemma 3.2 or see
Section 3.5.
1322:
Furstenberg's proof that there are infinitely many prime numbers
354:
Any union of open sets is open: for any collection of open sets
968:, which makes it clear that any finite subset of it, such as
886:{\displaystyle \mathbb {Z} \subset {\hat {\mathbb {Z} }}}
1284:"Some observations on the Furstenberg topological space"
974:
944:
899:
860:
838:
709:
595:
331:
198:
138:
105:
1240:"On Furstenberg's Proof of the Infinitude of Primes"
1025:"On Furstenberg's Proof of the Infinitude of Primes"
18:
Fürstenberg's proof of the infinitude of primes
998:
952:
916:
885:
846:
820:, so there must be infinitely many prime numbers.
793:
681:
339:
274:
146:
113:
569:: they are open by definition, and we can write
1331:Fürstenberg's proof of the infinitude of primes
581:) as the complement of an open set as follows:
189:(empty union) of arithmetic sequences), where
8:
993:
975:
736:
718:
249:
220:
1125:"Adic Topologies for the Rational Integers"
1069:"The Euclidean Criterion for Irreducibles"
538:This topology has two notable properties:
73:. The proof was published in 1955 in the
42:'s proof of the infinitude of primes is a
1258:
1140:
1043:
973:
946:
945:
943:
904:
903:
901:
900:
898:
873:
872:
870:
869:
862:
861:
859:
854:is the topology induced by the inclusion
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839:
837:
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259:
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245:
244:
197:
140:
139:
137:
107:
106:
104:
1199:(1955). "On the infinitude of primes".
1015:
715:
622:
546:of a finite non-empty set cannot be a
917:{\displaystyle {\hat {\mathbb {Z} }}}
351:(1, 0), and so is open as well.
7:
1101:10.4169/amer.math.monthly.124.3.198
1085:10.4169/amer.math.monthly.124.3.198
1123:Broughan, Kevin A. (August 2003).
928:ring with its profinite topology.
764:
761:
758:
755:
752:
25:
1073:The American Mathematical Monthly
289:is open if and only if for every
27:Proof of the infinitude of primes
1129:Canadian Journal of Mathematics
297:there is some non-zero integer
79:while Furstenberg was still an
1187:(Document). Berlin, New York:
908:
877:
830:evenly spaced integer topology
785:
773:
673:
655:
611:
599:
476:establishing membership). Set
214:
202:
123:evenly spaced integer topology
1:
1247:American Mathematical Monthly
1202:American Mathematical Monthly
1032:American Mathematical Monthly
325:∅ is open by definition, and
76:American Mathematical Monthly
1282:Lovas, R.; Mező, I. (2015).
953:{\displaystyle \mathbb {Q} }
847:{\displaystyle \mathbb {Z} }
340:{\displaystyle \mathbb {Z} }
147:{\displaystyle \mathbb {Z} }
114:{\displaystyle \mathbb {Z} }
69:, Furstenberg's proof is a
1377:
1238:Mercer, Idris D. (2009).
1023:Mercer, Idris D. (2009).
999:{\displaystyle \{-1,+1\}}
1269:10.4169/193009709X470218
1054:10.4169/193009709X470218
185:(which can be seen as a
165:of arithmetic sequences
67:Euclid's classical proof
1288:Elemente der Mathematik
1067:Clark, Pete L. (2017).
1185:"Proofs from The Book"
1142:10.4153/CJM-2003-030-3
1000:
954:
918:
887:
848:
824:Topological properties
795:
683:
651:
341:
276:
181: ≠ 0, or is
148:
115:
71:proof by contradiction
1001:
955:
919:
888:
849:
796:
684:
625:
482:least common multiple
444:be open sets and let
371:, any of the numbers
347:is just the sequence
342:
321:are easily verified:
319:axioms for a topology
277:
149:
116:
90:
81:undergraduate student
972:
942:
897:
858:
836:
707:
593:
567:both open and closed
329:
196:
136:
103:
63:arithmetic sequences
964:inherited from the
91:Furstenberg's proof
1197:Furstenberg, Harry
1181:Ziegler, Günter M.
1006:, cannot be open.
996:
950:
914:
883:
844:
791:
769:
679:
337:
272:
144:
111:
85:Yeshiva University
40:Hillel Furstenberg
34:, particularly in
962:subspace topology
926:profinite integer
911:
880:
742:
125:, by declaring a
16:(Redirected from
1368:
1356:General topology
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1234:
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1047:
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1005:
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959:
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949:
937:rational numbers
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685:
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621:
406:also shows that
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99:on the integers
21:
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1318:
1281:
1260:10.1.1.559.9528
1242:
1237:
1215:10.2307/2307043
1195:
1189:Springer-Verlag
1175:
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1045:10.1.1.559.9528
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894:
856:
855:
834:
833:
826:
705:
704:
591:
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553:The basis sets
533:
522:
497:
490:
475:
468:
461:
454:
443:
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418:
405:
392:
379:
367:in their union
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327:
326:
194:
193:
134:
133:
101:
100:
93:
28:
23:
22:
15:
12:
11:
5:
1374:
1372:
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1363:
1358:
1353:
1351:Article proofs
1343:
1342:
1339:
1338:
1328:
1317:
1316:External links
1314:
1313:
1312:
1300:10.4171/EM/283
1294:(3): 103–116.
1279:
1273:
1253:(4): 355–356.
1235:
1193:
1177:Aigner, Martin
1171:
1168:
1165:
1164:
1135:(4): 711–723.
1115:
1079:(3): 198–216.
1059:
1038:(4): 355–356.
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582:
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531:
527:) ⊆
518:
510:) ⊆
495:
488:
473:
466:
462:(with numbers
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452:
441:
434:
428:
423:) ⊆
414:
401:
397:) ⊆
388:
375:
358:
352:
335:
313:) ⊆
285:Equivalently,
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1361:Prime numbers
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818:contradiction
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455: ∩
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448: ∈
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217:
211:
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199:
192:
191:
190:
188:
187:nullary union
184:
180:
176:
172:
168:
164:
160:
157:
132: ⊆
131:
128:
124:
121:, called the
98:
88:
86:
82:
78:
77:
72:
68:
64:
60:
59:prime numbers
56:
52:
48:
45:
41:
37:
36:number theory
33:
19:
1291:
1287:
1250:
1246:
1206:
1200:
1132:
1128:
1118:
1076:
1072:
1062:
1035:
1031:
1018:
933:homeomorphic
930:
827:
813:
809:
807:
695:
578:
574:
570:
562:
558:
554:
537:
528:
524:
519:
515:
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492:
485:
477:
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407:
402:
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364:
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314:
310:
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302:
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294:
290:
286:
284:
178:
174:
170:
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94:
74:
29:
1326:Everything2
44:topological
32:mathematics
1345:Categories
1335:PlanetMath
1209:(5): 353.
1170:References
548:closed set
544:complement
480:to be the
380:for which
301:such that
65:. Unlike
55:infinitely
1308:126337479
1255:CiteSeerX
1159:121286344
1151:0008-414X
1093:0002-9890
1040:CiteSeerX
979:−
966:real line
960:with the
909:^
878:^
867:⊂
744:⋃
722:−
716:∖
645:−
627:⋃
623:∖
242:∈
236:∣
154:to be an
95:Define a
49:that the
1183:(1998).
1109:92986609
893:, where
577:,
561:,
523:,
506:,
419:,
393:,
309:,
173:,
161:it is a
156:open set
97:topology
53:contain
51:integers
1231:0068566
1223:2307043
935:to the
924:is the
498:. Then
1306:
1257:
1229:
1221:
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1099:
1091:
1042:
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565:) are
317:. The
177:) for
127:subset
1304:S2CID
1243:(PDF)
1219:JSTOR
1155:S2CID
1105:S2CID
1097:JSTOR
1028:(PDF)
1010:Notes
183:empty
163:union
57:many
47:proof
1147:ISSN
1089:ISSN
828:The
491:and
469:and
437:and
363:and
1333:at
1324:at
1296:doi
1265:doi
1251:116
1211:doi
1137:doi
1081:doi
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1036:116
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484:of
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