Do countable Hausdorff connected topological spaces exist?
Are there any countable Hausdorff connected spaces?
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$\begingroup$
general-topology
examples-counterexamples
3 Answers
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Yes, see here (MathOverflow) for references to some non-trivial examples.
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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.
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0My mistake!I didn' t mean the trivial example – 2011-01-07
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2I assume "countable" here means countably infinite. – 2011-01-07
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0I 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. – 2011-01-07
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0Yes indeed .Is there another meaning that I'm not aware of? – 2011-01-07
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0I believe Qiaochu is only implying that my 'singleton' response was not a valid example. – 2011-01-07
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2@t.spero: some people use "countable" to mean "at most countable." – 2011-01-07
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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