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Do countable Hausdorff connected topological spaces exist?

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3 Answers 3

up vote 6 down vote accepted

Yes, see here (MathOverflow) for references to some non-trivial examples.

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Since I haven't got enough reputation to comment, here's my answer to your question to Glen's answer: Some people use "countable" to mean "finite or countably infinite" (you can count the elements of a finite set, can't you?). –  t.b. Jan 7 '11 at 13:48
    
I changed the link from https to http, since I got an error message about an invalid security certificate. –  Hans Lundmark Jan 7 '11 at 15:51
    
@Hans: Thank you. –  t.b. Jan 7 '11 at 15:53

Trivially yes, a singleton for example ;). Non-trivial examples abound, for example ''A countable connected Hausdorff space'' by Brown, in Bull. Amer. Math. Soc., 59 (1953) p. 367.

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My mistake!I didn' t mean the trivial example –  t.k Jan 7 '11 at 13:36
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I assume "countable" here means countably infinite. –  Qiaochu Yuan Jan 7 '11 at 13:41
    
I thought so :). I added a reference that you might find interesting. In general these things are quite strange. But I guess you can get more out of Theo's link. –  Glen Wheeler Jan 7 '11 at 13:41
    
Yes indeed .Is there another meaning that I'm not aware of? –  t.k Jan 7 '11 at 13:42
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@t.spero: some people use "countable" to mean "at most countable." –  Qiaochu Yuan Jan 7 '11 at 19:04

$\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following countable, connected, Hausdorff spaces. You can learn more about any of them by visiting the search result.

Gustin's Sequence Space

Irrational Slope Topology

Prime Integer Topology

Relatively Prime Integer Topology

Roy's Lattice Space

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