reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;

theorem Th27:
  A is nowhere_dense iff ex C being Subset of X st A c= C & C is
  closed & C is boundary
proof
  thus A is nowhere_dense implies ex C being Subset of X st A c= C & C is
  closed & C is boundary
  proof
    assume
A1: A is nowhere_dense;
    take Cl A;
    thus thesis by A1,PRE_TOPC:18;
  end;
  given C being Subset of X such that
A2: A c= C & C is closed & C is boundary;
  Cl A is boundary by A2,Th11,TOPS_1:5;
  hence thesis;
end;
