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 countable "tree" of $X$'s can be topologized so that it is connected, Hausdorff, and removing any point disconnects the space.