reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;

theorem
  for X0 being non empty SubSpace of X holds X0 is nowhere_dense iff ex
  X1 being closed strict non empty SubSpace of X st X1 is boundary & X0 is
  SubSpace of X1
proof
  let X0 be non empty SubSpace of X;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  thus X0 is nowhere_dense implies ex X1 being closed strict non empty
  SubSpace of X st X1 is boundary & X0 is SubSpace of X1
  proof
    assume X0 is nowhere_dense;
    then A is nowhere_dense;
    then consider C being Subset of X such that
A1: A c= C and
A2: C is closed & C is boundary by TOPS_3:27;
    C is non empty by A1,XBOOLE_1:3;
    then consider X1 being closed strict non empty SubSpace of X such that
A3: X1 is boundary & C = the carrier of X1 by A2,Th41;
    take X1;
    thus thesis by A1,A3,TSEP_1:4;
  end;
  given X1 being closed strict non empty SubSpace of X such that
A4: X1 is boundary & X0 is SubSpace of X1;
  reconsider C = the carrier of X1 as Subset of X by TSEP_1:1;
  now
    take C;
    thus A c= C & C is boundary & C is closed by A4,TSEP_1:4,11;
  end;
  then for A be Subset of X st A = the carrier of X0 holds A is nowhere_dense
  by TOPS_3:27;
  hence thesis;
end;
