Article TSP_2, MML version 4.99.1005
:: TSP_2:attrnot 1 => TSP_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is T_0 means
for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2 & b1 <> b2
holds MaxADSet b1 misses MaxADSet b2;
end;
:: TSP_2:dfs 1
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is T_0
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2 & b1 <> b2
holds MaxADSet b1 misses MaxADSet b2;
:: TSP_2:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is T_0(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b3 <> b4
holds MaxADSet b3 misses MaxADSet b4;
:: TSP_2:attrnot 2 => TSP_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is T_0 means
for b1 being Element of the carrier of a1
st b1 in a2
holds a2 /\ MaxADSet b1 = {b1};
end;
:: TSP_2:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is T_0
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 in a2
holds a2 /\ MaxADSet b1 = {b1};
:: TSP_2:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is T_0(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds b2 /\ MaxADSet b3 = {b3};
:: TSP_2:attrnot 3 => TSP_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is T_0 means
for b1 being Element of the carrier of a1
st b1 in a2
holds ex b2 being Element of bool the carrier of a1 st
b2 is maximal_anti-discrete(a1) & a2 /\ b2 = {b1};
end;
:: TSP_2:dfs 3
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is T_0
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 in a2
holds ex b2 being Element of bool the carrier of a1 st
b2 is maximal_anti-discrete(a1) & a2 /\ b2 = {b1};
:: TSP_2:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is T_0(b1)
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds ex b4 being Element of bool the carrier of b1 st
b4 is maximal_anti-discrete(b1) & b2 /\ b4 = {b3};
:: TSP_2:attrnot 4 => TSP_2:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is maximal_T_0 means
a2 is T_0(a1) &
(for b1 being Element of bool the carrier of a1
st b1 is T_0(a1) & a2 c= b1
holds a2 = b1);
end;
:: TSP_2:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is maximal_T_0
it is sufficient to prove
thus a2 is T_0(a1) &
(for b1 being Element of bool the carrier of a1
st b1 is T_0(a1) & a2 c= b1
holds a2 = b1);
:: TSP_2:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is maximal_T_0(b1)
iff
b2 is T_0(b1) &
(for b3 being Element of bool the carrier of b1
st b3 is T_0(b1) & b2 c= b3
holds b2 = b3);
:: TSP_2:th 1
theorem
for b1, b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
b3 = b4 &
b3 is maximal_T_0(b1)
holds b4 is maximal_T_0(b2);
:: TSP_2:attrnot 5 => TSP_2:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is maximal_T_0 means
a2 is T_0(a1) & MaxADSet a2 = the carrier of a1;
end;
:: TSP_2:dfs 5
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is maximal_T_0
it is sufficient to prove
thus a2 is T_0(a1) & MaxADSet a2 = the carrier of a1;
:: TSP_2:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is maximal_T_0(b1)
iff
b2 is T_0(b1) & MaxADSet b2 = the carrier of b1;
:: TSP_2:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
holds b2 is dense(b1);
:: TSP_2:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st (b2 is open(b1) or b2 is closed(b1)) & b2 is maximal_T_0(b1)
holds b2 is proper(not bool the carrier of b1);
:: TSP_2:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being empty Element of bool the carrier of b1 holds
b2 is not maximal_T_0(b1);
:: TSP_2:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
for b3 being Element of bool the carrier of b1
st b3 is closed(b1)
holds b3 = MaxADSet (b2 /\ b3);
:: TSP_2:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
for b3 being Element of bool the carrier of b1
st b3 is open(b1)
holds b3 = MaxADSet (b2 /\ b3);
:: TSP_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
for b3 being Element of bool the carrier of b1 holds
Cl b3 = MaxADSet (b2 /\ Cl b3);
:: TSP_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
for b3 being Element of bool the carrier of b1 holds
Int b3 = MaxADSet (b2 /\ Int b3);
:: TSP_2:attrnot 6 => TSP_2:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is maximal_T_0 means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 in a2 & a2 /\ MaxADSet b1 = {b2};
end;
:: TSP_2:dfs 6
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is maximal_T_0
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
b2 in a2 & a2 /\ MaxADSet b1 = {b2};
:: TSP_2:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is maximal_T_0(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
b4 in b2 & b2 /\ MaxADSet b3 = {b4};
:: TSP_2:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is T_0(b1)
holds ex b3 being Element of bool the carrier of b1 st
b2 c= b3 & b3 is maximal_T_0(b1);
:: TSP_2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct holds
ex b2 being Element of bool the carrier of b1 st
b2 is maximal_T_0(b1);
:: TSP_2:attrnot 7 => TSP_2:attr 2
definition
let a1 be non empty TopStruct;
let a2 be SubSpace of a1;
attr a2 is maximal_T_0 means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is maximal_T_0(a1);
end;
:: TSP_2:dfs 7
definiens
let a1 be non empty TopStruct;
let a2 be SubSpace of a1;
To prove
a2 is maximal_T_0
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is maximal_T_0(a1);
:: TSP_2:def 7
theorem
for b1 being non empty TopStruct
for b2 being SubSpace of b1 holds
b2 is maximal_T_0(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is maximal_T_0(b1);
:: TSP_2:th 11
theorem
for b1 being non empty TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is maximal_T_0(b1)
iff
b2 is maximal_T_0(b1);
:: TSP_2:condreg 1
registration
let a1 be non empty TopStruct;
cluster non empty maximal_T_0 -> discerning (SubSpace of a1);
end;
:: TSP_2:condreg 2
registration
let a1 be non empty TopStruct;
cluster non empty non discerning -> non maximal_T_0 (SubSpace of a1);
end;
:: TSP_2:attrnot 8 => TSP_2:attr 2
definition
let a1 be non empty TopStruct;
let a2 be SubSpace of a1;
attr a2 is maximal_T_0 means
a2 is discerning &
(for b1 being non empty discerning SubSpace of a1
st a2 is SubSpace of b1
holds TopStruct(#the carrier of a2,the topology of a2#) = TopStruct(#the carrier of b1,the topology of b1#));
end;
:: TSP_2:dfs 8
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
To prove
a2 is maximal_T_0
it is sufficient to prove
thus a2 is discerning &
(for b1 being non empty discerning SubSpace of a1
st a2 is SubSpace of b1
holds TopStruct(#the carrier of a2,the topology of a2#) = TopStruct(#the carrier of b1,the topology of b1#));
:: TSP_2:def 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is maximal_T_0(b1)
iff
b2 is discerning &
(for b3 being non empty discerning SubSpace of b1
st b2 is SubSpace of b3
holds TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b3,the topology of b3#));
:: TSP_2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
st b2 is maximal_T_0(b1)
holds ex b3 being non empty strict SubSpace of b1 st
b3 is maximal_T_0(b1) & b2 = the carrier of b3;
:: TSP_2:condreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
cluster maximal_T_0 -> dense (SubSpace of a1);
end;
:: TSP_2:condreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non dense -> non maximal_T_0 (SubSpace of a1);
end;
:: TSP_2:condreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster open maximal_T_0 -> non proper (SubSpace of a1);
end;
:: TSP_2:condreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
cluster open proper -> non maximal_T_0 (SubSpace of a1);
end;
:: TSP_2:condreg 7
registration
let a1 be non empty TopSpace-like TopStruct;
cluster proper maximal_T_0 -> non open (SubSpace of a1);
end;
:: TSP_2:condreg 8
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed maximal_T_0 -> non proper (SubSpace of a1);
end;
:: TSP_2:condreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
cluster closed proper -> non maximal_T_0 (SubSpace of a1);
end;
:: TSP_2:condreg 10
registration
let a1 be non empty TopSpace-like TopStruct;
cluster proper maximal_T_0 -> non closed (SubSpace of a1);
end;
:: TSP_2:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty discerning SubSpace of b1 holds
ex b3 being strict SubSpace of b1 st
b2 is SubSpace of b3 & b3 is maximal_T_0(b1);
:: TSP_2:exreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty strict TopSpace-like maximal_T_0 SubSpace of a1;
end;
:: TSP_2:modenot 1
definition
let a1 be non empty TopSpace-like TopStruct;
mode maximal_Kolmogorov_subspace of a1 is maximal_T_0 SubSpace of a1;
end;
:: TSP_2:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 = b3
holds b4 is open(b2)
iff
MaxADSet b3 is open(b1) & b4 = (MaxADSet b3) /\ the carrier of b2;
:: TSP_2:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1 holds
b3 is open(b1)
iff
b3 = MaxADSet b3 &
(ex b4 being Element of bool the carrier of b2 st
b4 is open(b2) & b4 = b3 /\ the carrier of b2);
:: TSP_2:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 = b3
holds b4 is closed(b2)
iff
MaxADSet b3 is closed(b1) & b4 = (MaxADSet b3) /\ the carrier of b2;
:: TSP_2:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1 holds
b3 is closed(b1)
iff
b3 = MaxADSet b3 &
(ex b4 being Element of bool the carrier of b2 st
b4 is closed(b2) & b4 = b3 /\ the carrier of b2);
:: TSP_2:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2 &
(for b5 being Element of the carrier of b1 holds
b4 /\ MaxADSet b5 = {b3 . b5})
holds b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TSP_2:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st for b4 being Element of the carrier of b1 holds
b3 . b4 in MaxADSet b4
holds b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TSP_2:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2 &
(for b5 being Element of the carrier of b1 holds
b4 /\ MaxADSet b5 = {b3 . b5})
holds b3 is being_a_retraction(b1, b2);
:: TSP_2:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
st for b4 being Element of the carrier of b1 holds
b3 . b4 in MaxADSet b4
holds b3 is being_a_retraction(b1, b2);
:: TSP_2:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1 holds
ex b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
b3 is being_a_retraction(b1, b2);
:: TSP_2:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1 holds
b2 is_a_retract_of b1;
:: TSP_2:funcnot 1 => TSP_2:func 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty maximal_T_0 SubSpace of a1;
func Stone-retraction(A1,A2) -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 means
it is being_a_retraction(a1, a2);
end;
:: TSP_2:def 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
b3 = Stone-retraction(b1,b2)
iff
b3 is being_a_retraction(b1, b2);
:: TSP_2:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st b3 = b4
holds (Stone-retraction(b1,b2)) " Cl {b4} = Cl {b3};
:: TSP_2:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
st b3 = b4
holds (Stone-retraction(b1,b2)) " {b4} = MaxADSet b3;
:: TSP_2:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 = b3
holds (Stone-retraction(b1,b2)) " b4 = MaxADSet b3;
:: TSP_2:funcnot 2 => TSP_2:func 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty maximal_T_0 SubSpace of a1;
redefine func Stone-retraction(A1,A2) -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
for b2 being Element of the carrier of a1 holds
b1 /\ MaxADSet b2 = {it . b2};
end;
:: TSP_2:def 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
b3 = Stone-retraction(b1,b2)
iff
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2
for b5 being Element of the carrier of b1 holds
b4 /\ MaxADSet b5 = {b3 . b5};
:: TSP_2:funcnot 3 => TSP_2:func 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty maximal_T_0 SubSpace of a1;
redefine func Stone-retraction(A1,A2) -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 means
for b1 being Element of the carrier of a1 holds
it . b1 in MaxADSet b1;
end;
:: TSP_2:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
b3 = Stone-retraction(b1,b2)
iff
for b4 being Element of the carrier of b1 holds
b3 . b4 in MaxADSet b4;
:: TSP_2:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of the carrier of b1 holds
(Stone-retraction(b1,b2)) " {(Stone-retraction(b1,b2)) . b3} = MaxADSet b3;
:: TSP_2:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of the carrier of b1 holds
Im(Stone-retraction(b1,b2),b3) = (Stone-retraction(b1,b2)) .: MaxADSet b3;
:: TSP_2:funcnot 4 => TSP_2:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty maximal_T_0 SubSpace of a1;
redefine func Stone-retraction(A1,A2) -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
for b2 being Element of bool the carrier of a1 holds
b1 /\ MaxADSet b2 = it .: b2;
end;
:: TSP_2:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 holds
b3 = Stone-retraction(b1,b2)
iff
for b4 being Element of bool the carrier of b1
st b4 = the carrier of b2
for b5 being Element of bool the carrier of b1 holds
b4 /\ MaxADSet b5 = b3 .: b5;
:: TSP_2:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1 holds
(Stone-retraction(b1,b2)) " ((Stone-retraction(b1,b2)) .: b3) = MaxADSet b3;
:: TSP_2:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1 holds
(Stone-retraction(b1,b2)) .: b3 = (Stone-retraction(b1,b2)) .: MaxADSet b3;
:: TSP_2:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3, b4 being Element of bool the carrier of b1 holds
(Stone-retraction(b1,b2)) .: (b3 \/ b4) = ((Stone-retraction(b1,b2)) .: b3) \/ ((Stone-retraction(b1,b2)) .: b4);
:: TSP_2:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
st (b3 is open(b1) or b4 is open(b1))
holds (Stone-retraction(b1,b2)) .: (b3 /\ b4) = ((Stone-retraction(b1,b2)) .: b3) /\ ((Stone-retraction(b1,b2)) .: b4);
:: TSP_2:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
st (b3 is closed(b1) or b4 is closed(b1))
holds (Stone-retraction(b1,b2)) .: (b3 /\ b4) = ((Stone-retraction(b1,b2)) .: b3) /\ ((Stone-retraction(b1,b2)) .: b4);
:: TSP_2:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 is open(b1)
holds (Stone-retraction(b1,b2)) .: b3 is open(b2);
:: TSP_2:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty maximal_T_0 SubSpace of b1
for b3 being Element of bool the carrier of b1
st b3 is closed(b1)
holds (Stone-retraction(b1,b2)) .: b3 is closed(b2);