Construct a countable, connected topological space with at least two points, satisfying the highest separability axiom you can.

Notice that if two points can be separated by a continuous function, then a connected space with more than one point is uncountable, so your space cannot be (or higher).

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The Sierpiński space is a example, and this is as good as finite examples can get, since a finite space is discrete and therefore not connected if it has more than one point.

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This is a example.

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