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 Th7:
  F is discrete implies for p ex O st p in O & INTERSECTION({O},F)\
  {{}} is trivial
proof
  assume
A1: F is discrete;
  let p;
  consider O such that
A2: p in O and
A3: for C,D st C in F & D in F holds O meets C & O meets D implies C=D
  by A1;
  set I=INTERSECTION({O},F);
A4: for f,g being set st f in I & g in I holds f=g or f={} or g={}
  proof
    let f,g be set;
    assume that
A5: f in I and
A6: g in I;
    consider o1,f1 being set such that
A7: o1 in {O} and
A8: f1 in F and
A9: f= o1 /\ f1 by A5,SETFAM_1:def 5;
    consider o2,g1 being set such that
A10: o2 in {O} and
A11: g1 in F and
A12: g= o2 /\ g1 by A6,SETFAM_1:def 5;
    {o2}c={O} by A10,ZFMISC_1:31;
    then
A13: o2=O by ZFMISC_1:3;
    {o1}c={O} by A7,ZFMISC_1:31;
    then
A14: o1=O by ZFMISC_1:3;
    then O meets f1 & O meets g1 or f={} or g={} by A9,A12,A13,XBOOLE_0:def 7;
    hence thesis by A3,A8,A9,A11,A12,A14,A13;
  end;
A15: for f being set st f<>{} & f in I holds I c={{},f}
  proof
    let f be set;
    assume
A16: f<>{} & f in I;
    now
      let g be object;
      assume g in I;
      then f=g or g={} by A4,A16;
      hence g in {{},f} by TARSKI:def 2;
    end;
    hence thesis;
  end;
  now
    per cases;
    suppose
      I\{{}}<>{};
      then consider f being object such that
A17:  f in I\{{}} by XBOOLE_0:def 1;
      f <>{} by A17,ZFMISC_1:56;
      then
A18:  I c={{},f} by A15,A17;
      now
        let g be object;
        assume g in I;
        then {g} c={{},f} by A18,ZFMISC_1:31;
        then g={} or g =f by ZFMISC_1:19;
        then g in {{}} or g in {f} by ZFMISC_1:31;
        hence g in {{}}\/{f} by XBOOLE_0:def 3;
      end;
      then I c= {{}}\/{f};
      then I\{{}}c={f} by XBOOLE_1:43;
      then I\{{}}={} or I\{{}}={f} by ZFMISC_1:33;
      hence I\{{}} is trivial;
    end;
    suppose
      I\{{}}={};
      hence I\{{}} is trivial;
    end;
  end;
  hence thesis by A2;
end;
