🚀 go-pugleaf

A couple of weeks before Christmas the following nice problem was posed on 
this net:

  Does it exist a countable set of disjoint line segments of length 1 such 
  that no line segment of length 1 can be added without touching one of the 
  line segments in the set?
  The problem is to be solved for
      i) open line segments
     ii) half open line segments
    iii) closed line segments
 
Now I should like to see the complete solution of this exiting problem.

For open intervals the answer i clearly "yes"; there are several simple 
patterns here.  For half open and closed intervals I think the answers must 
be "no", but the solution is not so trivial as proposed in an earlier post 
message.

If the problem is solved, please let us have the solution.  If the problem 
is still open, it deserves to be given to the sci.math conference.

 o            o
Havard Johnsbraten, Telemark College of Education, Notodden, Norway
E-mail  haavard.johnsbraaten@tmlh.no  or
G=haavard;S=johnsbraaten;O=tmlh;P=uninett;A= ;C=no