The other answer describes the "Khalimsky line". It is not $T_1$, but it is possible to obtain Hausdorff examples by starting with a countable connected Hausdorff space $X$, blowing up its points into more copies of $X$, and continuing this process infinitely many times. This ever-branching "tree" of $X$'s can be topologized so that it is countable, connected, Hausdorff, and removing any point disconnected the space.
Forever Mozart
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