reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;

theorem Th1:
  for F st (ex A st F={A}) holds F is discrete
proof
  let F;
  assume ex A st F={A};
  then consider A such that
A1: F={A};
  now
    let p;
    take O= [#]T;
    thus p in O;
    now
      let B,C;
      assume that
A2:   B in F and
A3:   C in F;
      {B}c={A} by A1,A2,ZFMISC_1:31;
      then
A4:   B=A by ZFMISC_1:3;
      {C}c={A} by A1,A3,ZFMISC_1:31;
      hence B=C by A4,ZFMISC_1:3;
    end;
    hence for B,C st B in F & C in F holds O meets B & O meets C implies B=C;
  end;
  hence thesis;
end;
