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 Th28:
  A is nowhere_dense iff for G being Subset of X st G <> {} & G is
  open ex H being Subset of X st H c= G & H <> {} & H is open & A misses H
proof
  thus A is nowhere_dense implies for G being Subset of X st G <> {} & G is
  open ex H being Subset of X st H c= G & H <> {} & H is open & A misses H
  proof
    assume A is nowhere_dense;
    then
A1: Cl A is boundary;
    let G be Subset of X;
    assume G <> {} & G is open;
    then consider H being Subset of X such that
A2: H c= G & H <> {} & H is open & Cl A misses H by A1,TOPS_1:51;
    take H;
    thus thesis by A2,PRE_TOPC:18,XBOOLE_1:63;
  end;
  assume
A3: for G being Subset of X st G <> {} & G is open ex H being Subset of
  X st H c= G & H <> {} & H is open & A misses H;
  for G being Subset of X st G <> {} & G is open ex H being Subset of X st
  H c= G & H <> {} & H is open & Cl A misses H
  proof
    let G be Subset of X;
    assume G <> {} & G is open;
    then consider H being Subset of X such that
A4: H c= G & H <> {} & H is open & A misses H by A3;
    take H;
    thus thesis by A4,TSEP_1:36;
  end;
  then Cl A is boundary by TOPS_1:51;
  hence thesis;
end;
