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From: William Elliot on 17 Jul 2010 05:03 Let (S,tau) be a T0 space with a finite topology. Show S is finite. How's this for a proof? Any suggestions for a more direct proof? Since S has a finite topology, it has isolated points, only finite many. Remove from S, the isolated points and from tau, the singletons. The new space has fewer open sets. Iteration of this construction will end in a finite number of steps, with the empty space. Lemma. A T0 space with a finite topology has an isolated point. By the Hausdorff maximality theorm, there's a maximal chain C of nonnul open sets. /\C is a nonnul minimal open set. By T0, /\C is a open singleton.
From: Tim Little on 17 Jul 2010 07:47 On 2010-07-17, William Elliot <marsh(a)rdrop.remove.com> wrote: > Let (S,tau) be a T0 space with a finite topology. Show S is finite. For any point x in S, let O_x be the intersection of all open sets containing x. This is a T0 space, so O_x = {x}. There are only finitely many open sets, so O_x is also open for all x, and consequently there are only finitely many such x. - Tim
From: Prof Lobster on 17 Jul 2010 15:24 William Elliot <marsh(a)rdrop.remove.com> wrote in news:20100717014001.Y32754(a)agora.rdrop.com: > Let (S,tau) be a T0 space with a finite topology. Show S is finite. > > How's this for a proof? Any suggestions for a more direct proof? > > Since S has a finite topology, it has isolated points, > only finite many. Remove from S, the isolated points > and from tau, the singletons. The new space has fewer > open sets. Iteration of this construction will end in > a finite number of steps, with the empty space. > > Lemma. A T0 space with a finite topology has an isolated point. > By the Hausdorff maximality theorm, there's a maximal > chain C of nonnul open sets. /\C is a nonnul minimal > open set. By T0, /\C is a open singleton. > > The map x |-> {U : U is open and x is an element of U} injects S into a finite set. PL
From: Butch Malahide on 17 Jul 2010 22:00 On Jul 17, 6:47 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-17, William Elliot <ma...(a)rdrop.remove.com> wrote: > > > Let (S,tau) be a T0 space with a finite topology. Show S is finite. > > For any point x in S, let O_x be the intersection of all open sets > containing x. This is a T0 space, so O_x = {x}. No. The space S = {0, 1} with open sets {1}, S, and the empty set is a T0 space. You must be thinking of T1 spaces.
From: Tim Little on 17 Jul 2010 23:46
On 2010-07-18, Butch Malahide <fred.galvin(a)gmail.com> wrote: > The space S = {0, 1} with open sets {1}, S, and the empty set is a > T0 space. You must be thinking of T1 spaces. Yes, dammit :-( - Tim |