Could someone please help me with how to do this problem?

Given:

S1 = {(x,0):0<x<=1}

S2 = {(1/n,y): n=1,2,3,.... and 0<=y<=1}

S3 = {(0,1/2)}

Consider the sets:

S = S1 in union with S2

S0= S1 in union with S2 in union with S3

1) Use the Bolzano-Weierstrass Theorem to prove that S0 is not compact.

2) Is the set S arcwise connected? (Justify)

3) Is the set S0 connected? (Justify)

Thanks a lot.