reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem
  R is closed implies (R is nowhere_dense iff R = Fr R)
proof
  assume R is closed;
  then
A1: R = Cl R by PRE_TOPC:22;
  hereby
    assume R is nowhere_dense;
    then Cl R is boundary;
    then (Cl R)` is dense;
    then Fr R = R /\ [#] GX by A1;
    hence R = Fr R by XBOOLE_1:28;
  end;
  assume R = Fr R;
  then R = R \ Int R by A1,Lm2;
  then Int(Cl R) = {} by A1,Th16,XBOOLE_1:38;
  then Cl R is boundary by Th48;
  hence thesis;
end;
