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May 15, 2013 at 19:11 comment added The User In fact there is the notion of a “Stone space” (compact, totally disconnected, Hausdorff). Thus it can be reduced to “every non-empty, second countable, perfect Stone space is homeomorphic to the Cantor space”. Or using Stone duality: There exists one and only one (up to isomorphism) countable, atomless Boolean algebra. Thus you could argue that “non-empty, compact, totally disconnected, Hausdorff” is just the setting related to Boolean algebras.
Oct 22, 2010 at 15:11 comment added gowers Also exactly the kind of thing I was after!
Oct 22, 2010 at 10:52 history answered Bruno Martelli CC BY-SA 2.5