tag:blogger.com,1999:blog-264226589944705290.post4266569000502393962..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: Furstenberg's topological proof of the infinitude of primesMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-264226589944705290.post-89908200445351147772008-03-30T15:19:00.000-07:002008-03-30T15:19:00.000-07:00You write: "Now, there are no finite open sets in ...You write: "Now, there are no finite open sets in this topology, so there are no cofinite closed sets. Thus the union of the pZ isn't closed. So it can't be an intersection of finitely many closed sets, since finite intersections of closed sets are closed."<BR/><BR/>Three points: "no finite open sets" should read "no finite nonempty open sets"; "intersection of finitely many closed" should read "union of finitely many closed"; and "finite intersections of closed sets" should read "finite unions of closed sets."Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-57218042013758816102008-03-30T05:51:00.000-07:002008-03-30T05:51:00.000-07:00Why is that unfortunate? What was topology first ...Why is that unfortunate? What was topology first developed for by analysis? The analytic proof should more or less be a special case of the topological one, written out with the specific definitions of the topology on the real line.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-53204023174460221042008-03-29T21:50:00.000-07:002008-03-29T21:50:00.000-07:00Ahh, topology! That reminds me of a homework probl...Ahh, topology! That reminds me of a homework problem that one of my friends got a while ago: to prove that a continuous function that is zero at every rational number must be zero everywhere. The topological proof is about four sentences long. Unfortunately for my friend, the homework was for an analysis class. :)Aaronhttps://www.blogger.com/profile/18281785407407667986noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-47927072338773925312008-03-29T15:22:00.000-07:002008-03-29T15:22:00.000-07:00This is one of six proofs in "Proofs from the Book...This is one of six proofs in "Proofs from the Book". Since you had not heard of this proof before and it is given on page 5, I presume you don't own it. I think it would be very much your kind of book!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-65262104839785236342008-03-29T11:54:00.000-07:002008-03-29T11:54:00.000-07:00Wow, even I (almost) understood that. Makes me wan...Wow, even I (almost) understood that. Makes me want to read more. Thanks!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-55487885856176041862008-03-29T08:31:00.000-07:002008-03-29T08:31:00.000-07:00Aw man, I was going to write about this! You beat ...Aw man, I was going to write about this! You beat me to it.Anonymousnoreply@blogger.com