Do countable Hausdorff connected topological spaces exist?
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$\pi$-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
Steven Clontz
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Austin Mohr
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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.
Glen Wheeler
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My mistake!I didn' t mean the trivial example – t.k Jan 07 '11 at 13:36
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2I assume "countable" here means countably infinite. – Qiaochu Yuan Jan 07 '11 at 13:41
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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 07 '11 at 13:41
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Yes indeed .Is there another meaning that I'm not aware of? – t.k Jan 07 '11 at 13:42
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I believe Qiaochu is only implying that my 'singleton' response was not a valid example. – Glen Wheeler Jan 07 '11 at 13:46
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2@t.spero: some people use "countable" to mean "at most countable." – Qiaochu Yuan Jan 07 '11 at 19:04