# Are there any countable Hausdorff connected spaces?

Do countable Hausdorff connected topological spaces exist?

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It's Hausdorff. –  Rasmus Jan 7 '11 at 14:16
Why not put the entire question the title as well as in the body? –  Jason DeVito Jan 7 '11 at 15:19

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
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
@t.spero: some people use "countable" to mean "at most countable." –  Qiaochu Yuan Jan 7 '11 at 19:04