let T be non empty TopSpace; for p being Point of T holds Cl {p} is irreducible
let p be Point of T; Cl {p} is irreducible
assume A1:
not Cl {p} is irreducible
; contradiction
not Cl {p} is 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; verum