reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem Th33:
  X is non almost_discrete iff ex A being non empty Subset of X st
  A is boundary & A is closed
proof
  thus X is non almost_discrete implies ex A being non empty Subset of X st A
  is boundary & A is closed
  proof
    assume X is non almost_discrete;
    then consider A being non empty Subset of X such that
A1: A is nowhere_dense;
    consider C being Subset of X such that
A2: A c= C and
A3: C is closed and
A4: C is boundary by A1,TOPS_3:27;
    reconsider C as non empty Subset of X by A2;
    reconsider C as non empty Subset of X;
    take C;
    thus thesis by A3,A4;
  end;
  given A being non empty Subset of X such that
A5: A is boundary and
A6: A is closed;
  thus thesis by A5,A6;
end;
