Do countable Hausdorff connected topological spaces exist?
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Yes, see here (MathOverflow) for references to some non-trivial examples.
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.
$\pi$-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