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

theorem Th35:
  for C being non empty set, c0 being Element of C holds C \ {c0}
  is non empty iff STS(C,c0) is non almost_discrete
proof
  let C be non empty set, c0 be Element of C;
  hereby
    assume
A1: C \ {c0} is non empty;
    now
      reconsider A = {c0} as non empty Subset of STS(C,c0);
      take A;
A2:   A is boundary by A1,Th21;
      A is closed by A1,Th21;
      hence A is nowhere_dense by A2;
    end;
    hence STS(C,c0) is non almost_discrete;
  end;
  assume STS(C,c0) is non almost_discrete;
  then consider A being non empty Subset of STS(C,c0) such that
A3: A is nowhere_dense;
  assume C \ {c0} is empty;
  then C c= {c0} by XBOOLE_1:37;
  then C = {c0} by XBOOLE_0:def 10;
  then A = C by ZFMISC_1:33;
  hence contradiction by A3,TOPS_3:23;
end;
