
theorem Th17:
  for T being non empty TopSpace, p being Point of T holds Cl{p} is irreducible
proof
  let T be non empty TopSpace, p be Point of T;
  assume
A1: not thesis;
  Cl{p} is non empty by PCOMPS_1:2;
  then consider S1,S2 being Subset of T such that
A2: S1 is closed & S2 is closed and
A3: Cl{p} = S1 \/ S2 and
A4: S1 <> Cl{p} & S2 <> Cl{p} by A1;
  {p} c= Cl{p} & p in {p} by PRE_TOPC:18,TARSKI:def 1;
  then p in S1 or p in S2 by A3,XBOOLE_0:def 3;
  then {p} c= S1 or {p} c= S2 by ZFMISC_1:31;
  then
A5: Cl{p} c= S1 or Cl{p} c= S2 by A2,TOPS_1:5;
  S1 c= Cl{p} & S2 c= Cl{p} by A3,XBOOLE_1:7;
  hence contradiction by A4,A5;
end;
