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 Th39:
  for X0 being SubSpace of X st X0 is nowhere_dense for A being
  Subset of X st A c= the carrier of X0 holds A is nowhere_dense
proof
  let X0 be SubSpace of X;
  reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
  assume X0 is nowhere_dense;
  then
A1: C is nowhere_dense;
  let A be Subset of X;
  assume A c= the carrier of X0;
  hence thesis by A1,TOPS_3:26;
end;
