A Connectedness Puzzle

(This is a point-set topology puzzle.  For definitions of the terms
involved, see any elementary topology text.)

Our puzzle takes place in the unit square S = [0,1] x [0,1] in the
plane.  Take the midpoints of the edges P1 = (1/2,0) , P2 = (1/2,1) , 
Q1 = (0,1/2) and Q2 = (1,1/2).  Can you find two disjoint connected
subsets K and L of S, such that K contains P1 and P2, and L contains Q1
and Q2?  If not, prove it can't be done.