Slight hints and then you show some work:
Draw the set S_0> can you see a rather obvious set of points in converging to a point NOT in S0 (look closely at S2)? this is (1)
If you get a good grip of the set S0 then you msut have a "feeling" what point there can't be joined with other points there by an arc (and this is (2))
S0 is connected: assume S0 = A \/ B with A,B open . Some of these sets must contain the point in S3. prove this set is all of S0 and you're done.