:: TSP_1 semantic presentation

definition
let c1 be TopStruct ;
redefine mode SubSpace as SubSpace of c1 -> TopStruct means :Def1: :: TSP_1:def 1
( the carrier of a2 c= the carrier of a1 & ( for b1 being Subset of a2 holds
( b1 is open iff ex b2 being Subset of a1 st
( b2 is open & b1 = b2 /\ the carrier of a2 ) ) ) );
compatibility
for b1 being TopStruct holds
( b1 is SubSpace of c1 iff ( the carrier of b1 c= the carrier of c1 & ( for b2 being Subset of b1 holds
( b2 is open iff ex b3 being Subset of c1 st
( b3 is open & b2 = b3 /\ the carrier of b1 ) ) ) ) )
proof end;
coherence
for b1 being SubSpace of c1 holds
b1 is TopStruct
;
end;

:: deftheorem Def1 defines SubSpace TSP_1:def 1 :
for b1, b2 being TopStruct holds
( b2 is SubSpace of b1 iff ( the carrier of b2 c= the carrier of b1 & ( for b3 being Subset of b2 holds
( b3 is open iff ex b4 being Subset of b1 st
( b4 is open & b3 = b4 /\ the carrier of b2 ) ) ) ) );

theorem Th1: :: TSP_1:1
canceled;

theorem Th2: :: TSP_1:2
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Subset of b1 holds
not ( b3 is open & ( for b4 being Subset of b2 holds
not ( b4 is open & b4 = b3 /\ the carrier of b2 ) ) )
proof end;

definition
let c1 be TopStruct ;
redefine mode SubSpace as SubSpace of c1 -> TopStruct means :Def2: :: TSP_1:def 2
( the carrier of a2 c= the carrier of a1 & ( for b1 being Subset of a2 holds
( b1 is closed iff ex b2 being Subset of a1 st
( b2 is closed & b1 = b2 /\ the carrier of a2 ) ) ) );
compatibility
for b1 being TopStruct holds
( b1 is SubSpace of c1 iff ( the carrier of b1 c= the carrier of c1 & ( for b2 being Subset of b1 holds
( b2 is closed iff ex b3 being Subset of c1 st
( b3 is closed & b2 = b3 /\ the carrier of b1 ) ) ) ) )
proof end;
coherence
for b1 being SubSpace of c1 holds
b1 is TopStruct
;
end;

:: deftheorem Def2 defines SubSpace TSP_1:def 2 :
for b1, b2 being TopStruct holds
( b2 is SubSpace of b1 iff ( the carrier of b2 c= the carrier of b1 & ( for b3 being Subset of b2 holds
( b3 is closed iff ex b4 being Subset of b1 st
( b4 is closed & b3 = b4 /\ the carrier of b2 ) ) ) ) );

theorem Th3: :: TSP_1:3
canceled;

theorem Th4: :: TSP_1:4
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Subset of b1 holds
not ( b3 is closed & ( for b4 being Subset of b2 holds
not ( b4 is closed & b4 = b3 /\ the carrier of b2 ) ) )
proof end;

notation
let c1 be TopStruct ;
synonym T_0 c1 for discerning c1;
end;

definition
let c1 be TopStruct ;
redefine attr a1 is discerning means :Def3: :: TSP_1:def 3
( a1 is empty or for b1, b2 being Point of a1 holds
not ( b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is open & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is open & not b1 in b3 & b2 in b3 ) ) ) );
compatibility
( c1 is T_0 iff ( c1 is empty or for b1, b2 being Point of c1 holds
not ( b1 <> b2 & ( for b3 being Subset of c1 holds
not ( b3 is open & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of c1 holds
not ( b3 is open & not b1 in b3 & b2 in b3 ) ) ) ) )
proof end;
end;

:: deftheorem Def3 defines T_0 TSP_1:def 3 :
for b1 being TopStruct holds
( b1 is T_0 iff ( b1 is empty or for b2, b3 being Point of b1 holds
not ( b2 <> b3 & ( for b4 being Subset of b1 holds
not ( b4 is open & b2 in b4 & not b3 in b4 ) ) & ( for b4 being Subset of b1 holds
not ( b4 is open & not b2 in b4 & b3 in b4 ) ) ) ) );

definition
let c1 be TopStruct ;
redefine attr a1 is discerning means :Def4: :: TSP_1:def 4
( a1 is empty or for b1, b2 being Point of a1 holds
not ( b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) ) );
compatibility
( c1 is T_0 iff ( c1 is empty or for b1, b2 being Point of c1 holds
not ( b1 <> b2 & ( for b3 being Subset of c1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of c1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) ) ) )
proof end;
end;

:: deftheorem Def4 defines T_0 TSP_1:def 4 :
for b1 being TopStruct holds
( b1 is T_0 iff ( b1 is empty or for b2, b3 being Point of b1 holds
not ( b2 <> b3 & ( for b4 being Subset of b1 holds
not ( b4 is closed & b2 in b4 & not b3 in b4 ) ) & ( for b4 being Subset of b1 holds
not ( b4 is closed & not b2 in b4 & b3 in b4 ) ) ) ) );

registration
cluster non empty trivial -> non empty T_0 TopStruct ;
coherence
for b1 being non empty TopStruct holds
( b1 is trivial implies b1 is T_0 )
proof end;
cluster non empty non T_0 -> non empty non trivial TopStruct ;
coherence
for b1 being non empty TopStruct holds
not ( not b1 is T_0 & b1 is trivial )
proof end;
end;

Lemma7: for b1 being non empty non trivial anti-discrete TopStruct holds
not b1 is T_0
proof end;

registration
cluster non empty strict T_0 TopStruct ;
existence
ex b1 being TopSpace st
( b1 is strict & b1 is T_0 & not b1 is empty )
proof end;
cluster non empty non trivial strict non T_0 TopStruct ;
existence
ex b1 being TopSpace st
( b1 is strict & not b1 is T_0 & not b1 is empty )
proof end;
end;

registration
cluster non empty discrete -> non empty T_0 TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is discrete implies b1 is T_0 )
proof end;
cluster non empty non T_0 -> non empty non discrete TopStruct ;
coherence
for b1 being non empty TopSpace holds
not ( not b1 is T_0 & b1 is discrete )
proof end;
cluster non empty non trivial anti-discrete -> non empty non trivial non T_0 TopStruct ;
coherence
for b1 being non empty TopSpace holds
not ( b1 is anti-discrete & not b1 is trivial & b1 is T_0 )
by Lemma7;
cluster non empty anti-discrete T_0 -> non empty trivial T_0 TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is anti-discrete & b1 is T_0 implies b1 is trivial )
by Lemma7;
cluster non empty non trivial T_0 -> non empty non anti-discrete TopStruct ;
coherence
for b1 being non empty TopSpace holds
not ( b1 is T_0 & not b1 is trivial & b1 is anti-discrete )
by Lemma7;
end;

Lemma8: for b1 being non empty TopSpace
for b2 being Point of b1 holds b2 in Cl {b2}
proof end;

Lemma9: for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( b3 c= Cl b2 implies Cl b3 c= Cl b2 )
by TOPS_1:31;

definition
let c1 be non empty TopSpace;
redefine attr a1 is discerning means :Def5: :: TSP_1:def 5
for b1, b2 being Point of a1 holds
not ( b1 <> b2 & not Cl {b1} <> Cl {b2} );
compatibility
( c1 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 <> b2 & not Cl {b1} <> Cl {b2} ) )
proof end;
end;

:: deftheorem Def5 defines T_0 TSP_1:def 5 :
for b1 being non empty TopSpace holds
( b1 is T_0 iff for b2, b3 being Point of b1 holds
not ( b2 <> b3 & not Cl {b2} <> Cl {b3} ) );

definition
let c1 be non empty TopSpace;
redefine attr a1 is discerning means :Def6: :: TSP_1:def 6
for b1, b2 being Point of a1 holds
not ( b1 <> b2 & b1 in Cl {b2} & b2 in Cl {b1} );
compatibility
( c1 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 <> b2 & b1 in Cl {b2} & b2 in Cl {b1} ) )
proof end;
end;

:: deftheorem Def6 defines T_0 TSP_1:def 6 :
for b1 being non empty TopSpace holds
( b1 is T_0 iff for b2, b3 being Point of b1 holds
not ( b2 <> b3 & b2 in Cl {b3} & b3 in Cl {b2} ) );

definition
let c1 be non empty TopSpace;
redefine attr a1 is discerning means :: TSP_1:def 7
for b1, b2 being Point of a1 holds
not ( b1 <> b2 & b1 in Cl {b2} & Cl {b2} c= Cl {b1} );
compatibility
( c1 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 <> b2 & b1 in Cl {b2} & Cl {b2} c= Cl {b1} ) )
proof end;
end;

:: deftheorem Def7 defines T_0 TSP_1:def 7 :
for b1 being non empty TopSpace holds
( b1 is T_0 iff for b2, b3 being Point of b1 holds
not ( b2 <> b3 & b2 in Cl {b3} & Cl {b3} c= Cl {b2} ) );

registration
cluster non empty almost_discrete T_0 -> non empty discrete T_0 TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is almost_discrete & b1 is T_0 implies b1 is discrete )
proof end;
cluster non empty non discrete almost_discrete -> non empty non discrete non T_0 TopStruct ;
coherence
for b1 being non empty TopSpace holds
not ( b1 is almost_discrete & not b1 is discrete & b1 is T_0 )
proof end;
cluster non empty non discrete T_0 -> non empty non almost_discrete TopStruct ;
coherence
for b1 being non empty TopSpace holds
not ( not b1 is discrete & b1 is T_0 & b1 is almost_discrete )
proof end;
end;

definition
mode Kolmogorov_space is non empty T_0 TopSpace;
mode non-Kolmogorov_space is non empty non T_0 TopSpace;
end;

registration
cluster non trivial strict non anti-discrete TopStruct ;
existence
ex b1 being Kolmogorov_space st
( not b1 is trivial & b1 is strict )
proof end;
cluster non trivial strict TopStruct ;
existence
ex b1 being non-Kolmogorov_space st
( not b1 is trivial & b1 is strict )
proof end;
end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
attr a2 is T_0 means :Def8: :: TSP_1:def 8
for b1, b2 being Point of a1 holds
not ( b1 in a2 & b2 in a2 & b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is open & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is open & not b1 in b3 & b2 in b3 ) ) );
end;

:: deftheorem Def8 defines T_0 TSP_1:def 8 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is T_0 iff for b3, b4 being Point of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 & ( for b5 being Subset of b1 holds
not ( b5 is open & b3 in b5 & not b4 in b5 ) ) & ( for b5 being Subset of b1 holds
not ( b5 is open & not b3 in b5 & b4 in b5 ) ) ) );

definition
let c1 be non empty TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is T_0 means :Def9: :: TSP_1:def 9
for b1, b2 being Point of a1 holds
not ( b1 in a2 & b2 in a2 & b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) );
compatibility
( c2 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 in c2 & b2 in c2 & b1 <> b2 & ( for b3 being Subset of c1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of c1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) ) )
proof end;
end;

:: deftheorem Def9 defines T_0 TSP_1:def 9 :
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( b2 is T_0 iff for b3, b4 being Point of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 & ( for b5 being Subset of b1 holds
not ( b5 is closed & b3 in b5 & not b4 in b5 ) ) & ( for b5 being Subset of b1 holds
not ( b5 is closed & not b3 in b5 & b4 in b5 ) ) ) );

theorem Th5: :: TSP_1:5
for b1, b2 being TopStruct
for b3 being Subset of b1
for b4 being Subset of b2 holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 = b4 & b3 is T_0 implies b4 is T_0 )
proof end;

theorem Th6: :: TSP_1:6
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( b2 = the carrier of b1 implies ( b2 is T_0 iff b1 is T_0 ) )
proof end;

theorem Th7: :: TSP_1:7
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b3 c= b2 & b2 is T_0 implies b3 is T_0 )
proof end;

theorem Th8: :: TSP_1:8
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( ( b2 is T_0 or b3 is T_0 ) implies b2 /\ b3 is T_0 )
proof end;

theorem Th9: :: TSP_1:9
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( ( b2 is open or b3 is open ) & b2 is T_0 & b3 is T_0 implies b2 \/ b3 is T_0 )
proof end;

theorem Th10: :: TSP_1:10
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( ( b2 is closed or b3 is closed ) & b2 is T_0 & b3 is T_0 implies b2 \/ b3 is T_0 )
proof end;

theorem Th11: :: TSP_1:11
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( b2 is discrete implies b2 is T_0 )
proof end;

theorem Th12: :: TSP_1:12
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
not ( b2 is anti-discrete & not b2 is trivial & b2 is T_0 )
proof end;

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is T_0 means :Def10: :: TSP_1:def 10
for b1, b2 being Point of a1 holds
not ( b1 in a2 & b2 in a2 & b1 <> b2 & not Cl {b1} <> Cl {b2} );
compatibility
( c2 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 in c2 & b2 in c2 & b1 <> b2 & not Cl {b1} <> Cl {b2} ) )
proof end;
end;

:: deftheorem Def10 defines T_0 TSP_1:def 10 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is T_0 iff for b3, b4 being Point of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 & not Cl {b3} <> Cl {b4} ) );

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is T_0 means :Def11: :: TSP_1:def 11
for b1, b2 being Point of a1 holds
not ( b1 in a2 & b2 in a2 & b1 <> b2 & b1 in Cl {b2} & b2 in Cl {b1} );
compatibility
( c2 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 in c2 & b2 in c2 & b1 <> b2 & b1 in Cl {b2} & b2 in Cl {b1} ) )
proof end;
end;

:: deftheorem Def11 defines T_0 TSP_1:def 11 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is T_0 iff for b3, b4 being Point of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 & b3 in Cl {b4} & b4 in Cl {b3} ) );

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is T_0 means :: TSP_1:def 12
for b1, b2 being Point of a1 holds
not ( b1 in a2 & b2 in a2 & b1 <> b2 & b1 in Cl {b2} & Cl {b2} c= Cl {b1} );
compatibility
( c2 is T_0 iff for b1, b2 being Point of c1 holds
not ( b1 in c2 & b2 in c2 & b1 <> b2 & b1 in Cl {b2} & Cl {b2} c= Cl {b1} ) )
proof end;
end;

:: deftheorem Def12 defines T_0 TSP_1:def 12 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is T_0 iff for b3, b4 being Point of b1 holds
not ( b3 in b2 & b4 in b2 & b3 <> b4 & b3 in Cl {b4} & Cl {b4} c= Cl {b3} ) );

theorem Th13: :: TSP_1:13
for b1 being non empty TopSpace
for b2 being empty Subset of b1 holds b2 is T_0
proof end;

theorem Th14: :: TSP_1:14
for b1 being non empty TopSpace
for b2 being Point of b1 holds {b2} is T_0
proof end;

registration
let c1 be non empty TopStruct ;
cluster non empty strict T_0 SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is strict & b1 is T_0 & not b1 is empty )
proof end;
end;

E23: now
let c1 be TopStruct ;
let c2 be SubSpace of c1;
( [#] c2 c= [#] c1 & [#] c2 = the carrier of c2 ) by PRE_TOPC:def 9;
hence the carrier of c2 is Subset of c1 ;
end;

definition
let c1 be TopStruct ;
let c2 be SubSpace of c1;
redefine attr T_0 as a2 is T_0 means :: TSP_1:def 13
( a2 is empty or for b1, b2 being Point of a1 holds
not ( b1 is Point of a2 & b2 is Point of a2 & b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is open & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is open & not b1 in b3 & b2 in b3 ) ) ) );
compatibility
( c2 is T_0 iff ( c2 is empty or for b1, b2 being Point of c1 holds
not ( b1 is Point of c2 & b2 is Point of c2 & b1 <> b2 & ( for b3 being Subset of c1 holds
not ( b3 is open & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of c1 holds
not ( b3 is open & not b1 in b3 & b2 in b3 ) ) ) ) )
proof end;
end;

:: deftheorem Def13 defines T_0 TSP_1:def 13 :
for b1 being TopStruct
for b2 being SubSpace of b1 holds
( b2 is T_0 iff ( b2 is empty or for b3, b4 being Point of b1 holds
not ( b3 is Point of b2 & b4 is Point of b2 & b3 <> b4 & ( for b5 being Subset of b1 holds
not ( b5 is open & b3 in b5 & not b4 in b5 ) ) & ( for b5 being Subset of b1 holds
not ( b5 is open & not b3 in b5 & b4 in b5 ) ) ) ) );

definition
let c1 be TopStruct ;
let c2 be SubSpace of c1;
redefine attr T_0 as a2 is T_0 means :Def14: :: TSP_1:def 14
( a2 is empty or for b1, b2 being Point of a1 holds
not ( b1 is Point of a2 & b2 is Point of a2 & b1 <> b2 & ( for b3 being Subset of a1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of a1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) ) );
compatibility
( c2 is T_0 iff ( c2 is empty or for b1, b2 being Point of c1 holds
not ( b1 is Point of c2 & b2 is Point of c2 & b1 <> b2 & ( for b3 being Subset of c1 holds
not ( b3 is closed & b1 in b3 & not b2 in b3 ) ) & ( for b3 being Subset of c1 holds
not ( b3 is closed & not b1 in b3 & b2 in b3 ) ) ) ) )
proof end;
end;

:: deftheorem Def14 defines T_0 TSP_1:def 14 :
for b1 being TopStruct
for b2 being SubSpace of b1 holds
( b2 is T_0 iff ( b2 is empty or for b3, b4 being Point of b1 holds
not ( b3 is Point of b2 & b4 is Point of b2 & b3 <> b4 & ( for b5 being Subset of b1 holds
not ( b5 is closed & b3 in b5 & not b4 in b5 ) ) & ( for b5 being Subset of b1 holds
not ( b5 is closed & not b3 in b5 & b4 in b5 ) ) ) ) );

theorem Th15: :: TSP_1:15
for b1 being non empty TopStruct
for b2 being non empty SubSpace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies ( b3 is T_0 iff b2 is T_0 ) )
proof end;

theorem Th16: :: TSP_1:16
for b1 being non empty TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty V117 SubSpace of b1 holds
( b2 is SubSpace of b3 implies b2 is T_0 )
proof end;

theorem Th17: :: TSP_1:17
for b1 being non empty TopSpace
for b2 being non empty V117 SubSpace of b1
for b3 being non empty SubSpace of b1 holds
( b2 meets b3 implies b2 meet b3 is T_0 )
proof end;

theorem Th18: :: TSP_1:18
for b1 being non empty TopSpace
for b2, b3 being non empty V117 SubSpace of b1 holds
( ( b2 is open or b3 is open ) implies b2 union b3 is T_0 )
proof end;

theorem Th19: :: TSP_1:19
for b1 being non empty TopSpace
for b2, b3 being non empty V117 SubSpace of b1 holds
( ( b2 is closed or b3 is closed ) implies b2 union b3 is T_0 )
proof end;

definition
let c1 be non empty TopSpace;
mode Kolmogorov_subspace is non empty V117 SubSpace of a1;
end;

theorem Th20: :: TSP_1:20
for b1 being non empty TopSpace
for b2 being non empty Subset of b1 holds
not ( b2 is T_0 & ( for b3 being strict Kolmogorov_subspace of b1 holds
not b2 = the carrier of b3 ) )
proof end;

registration
let c1 be non empty non trivial TopSpace;
cluster strict proper SubSpace of a1;
existence
ex b1 being Kolmogorov_subspace of c1 st
( b1 is proper & b1 is strict )
proof end;
end;

registration
let c1 be Kolmogorov_space;
cluster non empty -> non empty V117 SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds b1 is T_0
proof end;
end;

registration
let c1 be non-Kolmogorov_space;
cluster non empty non proper -> non empty non trivial V117 SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
not ( not b1 is proper & b1 is T_0 )
proof end;
cluster non empty V117 -> non empty proper SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
( b1 is T_0 implies b1 is proper )
proof end;
end;

registration
let c1 be non-Kolmogorov_space;
cluster strict V117 SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is strict & not b1 is T_0 )
proof end;
end;

definition
let c1 be non-Kolmogorov_space;
mode non-Kolmogorov_subspace is V117 SubSpace of a1;
end;

theorem Th21: :: TSP_1:21
for b1 being non empty non-Kolmogorov_space
for b2 being Subset of b1 holds
not ( not b2 is T_0 & ( for b3 being strict non-Kolmogorov_subspace of b1 holds
not b2 = the carrier of b3 ) )
proof end;