Article TOPGEN_4, MML version 4.99.1005
:: TOPGEN_4:funcnot 1 => TOPGEN_4:func 1
definition
let a1 be 1-sorted;
func TotFam A1 -> Element of bool bool the carrier of a1 equals
bool the carrier of a1;
end;
:: TOPGEN_4:def 1
theorem
for b1 being 1-sorted holds
TotFam b1 = bool the carrier of b1;
:: TOPGEN_4:th 1
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is countable
iff
COMPLEMENT b2 is countable;
:: TOPGEN_4:funcreg 1
registration
cluster RAT -> countable;
end;
:: TOPGEN_4:sch 1
scheme TOPGEN_4:sch 1
{{b1} where b1 is Element of RAT: P1[b1]} is countable
:: TOPGEN_4:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Der b2 = {b3 where b3 is Element of the carrier of b1: b3 in Cl (b2 \ {b3})};
:: TOPGEN_4:condreg 1
registration
cluster finite -> second-countable (TopStruct);
end;
:: TOPGEN_4:funcreg 2
registration
cluster REAL -> non countable;
end;
:: TOPGEN_4:condreg 2
registration
cluster non countable -> infinite (set);
end;
:: TOPGEN_4:condreg 3
registration
cluster infinite -> non trivial (set);
end;
:: TOPGEN_4:exreg 1
registration
cluster non empty non countable set;
end;
:: TOPGEN_4:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
Der b2 c= b2;
:: TOPGEN_4:th 4
theorem
for b1 being non empty TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1
st b3 is open(b1) & b3 <> {}
holds ex b4 being Element of bool the carrier of b1 st
b4 in b2 & b4 c= b3 & b4 <> {};
:: TOPGEN_4:th 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
density b1 c= weight b1;
:: TOPGEN_4:th 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is separable
iff
ex b2 being Element of bool the carrier of b1 st
b2 is dense(b1) & b2 is countable;
:: TOPGEN_4:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is second-countable
holds b1 is separable;
:: TOPGEN_4:condreg 4
registration
cluster non empty TopSpace-like second-countable -> separable (TopStruct);
end;
:: TOPGEN_4:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_separated
holds Fr (b2 \/ b3) = (Fr b2) \/ Fr b3;
:: TOPGEN_4:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is locally_finite(b1)
holds Fr union b2 c= union Fr b2;
:: TOPGEN_4:th 10
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is separable
iff
Card [#] b1 c= alef 0;
:: TOPGEN_4:th 11
theorem
for b1 being non empty TopSpace-like discrete TopStruct holds
b1 is separable
iff
b1 is countable;
:: TOPGEN_4:attrnot 1 => TOPGEN_4:attr 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is all-open-containing means
for b1 being Element of bool the carrier of a1
st b1 is open(a1)
holds b1 in a2;
end;
:: TOPGEN_4:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is all-open-containing
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is open(a1)
holds b1 in a2;
:: TOPGEN_4:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is all-open-containing(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 is open(b1)
holds b3 in b2;
:: TOPGEN_4:attrnot 2 => TOPGEN_4:attr 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is all-closed-containing means
for b1 being Element of bool the carrier of a1
st b1 is closed(a1)
holds b1 in a2;
end;
:: TOPGEN_4:dfs 3
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is all-closed-containing
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is closed(a1)
holds b1 in a2;
:: TOPGEN_4:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is all-closed-containing(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 is closed(b1)
holds b3 in b2;
:: TOPGEN_4:attrnot 3 => TOPGEN_4:attr 3
definition
let a1 be set;
let a2 be Element of bool bool a1;
attr a2 is closed_for_countable_unions means
for b1 being countable Element of bool bool a1
st b1 c= a2
holds union b1 in a2;
end;
:: TOPGEN_4:dfs 4
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is closed_for_countable_unions
it is sufficient to prove
thus for b1 being countable Element of bool bool a1
st b1 c= a2
holds union b1 in a2;
:: TOPGEN_4:def 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is closed_for_countable_unions(b1)
iff
for b3 being countable Element of bool bool b1
st b3 c= b2
holds union b3 in b2;
:: TOPGEN_4:condreg 5
registration
let a1 be set;
cluster non empty compl-closed sigma-multiplicative -> closed_for_countable_unions (Element of bool bool a1);
end;
:: TOPGEN_4:th 12
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 is closed_for_countable_unions(b1)
holds {} in b2;
:: TOPGEN_4:condreg 6
registration
let a1 be set;
cluster closed_for_countable_unions -> non empty (Element of bool bool a1);
end;
:: TOPGEN_4:th 13
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is non empty compl-closed sigma-multiplicative Element of bool bool b1
iff
b2 is compl-closed(b1) & b2 is closed_for_countable_unions(b1);
:: TOPGEN_4:attrnot 4 => TOPGEN_4:attr 4
definition
let a1 be set;
let a2 be Element of bool bool a1;
attr a2 is closed_for_countable_meets means
for b1 being countable Element of bool bool a1
st b1 c= a2
holds meet b1 in a2;
end;
:: TOPGEN_4:dfs 5
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is closed_for_countable_meets
it is sufficient to prove
thus for b1 being countable Element of bool bool a1
st b1 c= a2
holds meet b1 in a2;
:: TOPGEN_4:def 5
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is closed_for_countable_meets(b1)
iff
for b3 being countable Element of bool bool b1
st b3 c= b2
holds meet b3 in b2;
:: TOPGEN_4:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is all-closed-containing(b1) & b2 is compl-closed(the carrier of b1)
iff
b2 is all-open-containing(b1) & b2 is compl-closed(the carrier of b1);
:: TOPGEN_4:th 15
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 is compl-closed(b1)
holds b2 = COMPLEMENT b2;
:: TOPGEN_4:th 16
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
st b2 c= b3 & b3 is compl-closed(b1)
holds COMPLEMENT b2 c= b3;
:: TOPGEN_4:th 17
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is closed_for_countable_meets(b1) & b2 is compl-closed(b1)
iff
b2 is closed_for_countable_unions(b1) & b2 is compl-closed(b1);
:: TOPGEN_4:condreg 7
registration
let a1 be non empty TopSpace-like TopStruct;
cluster compl-closed all-open-containing closed_for_countable_unions -> all-closed-containing closed_for_countable_meets (Element of bool bool the carrier of a1);
end;
:: TOPGEN_4:condreg 8
registration
let a1 be non empty TopSpace-like TopStruct;
cluster compl-closed all-closed-containing closed_for_countable_meets -> all-open-containing closed_for_countable_unions (Element of bool bool the carrier of a1);
end;
:: TOPGEN_4:funcreg 3
registration
let a1 be set;
let a2 be countable Element of bool bool a1;
cluster COMPLEMENT a2 -> countable;
end;
:: TOPGEN_4:condreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
cluster empty -> open closed (Element of bool bool the carrier of a1);
end;
:: TOPGEN_4:exreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
cluster countable open closed Element of bool bool the carrier of a1;
end;
:: TOPGEN_4:th 18
theorem
for b1 being set holds
{} is empty Element of bool bool b1;
:: TOPGEN_4:condreg 10
registration
cluster empty -> countable (set);
end;
:: TOPGEN_4:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 = {b2}
holds b2 is open(b1)
iff
b3 is open(b1);
:: TOPGEN_4:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 = {b2}
holds b2 is closed(b1)
iff
b3 is closed(b1);
:: TOPGEN_4:funcnot 2 => TOPGEN_4:func 2
definition
let a1 be set;
let a2, a3 be Element of bool bool a1;
redefine func INTERSECTION(a2,a3) -> Element of bool bool a1;
commutativity;
:: for a1 being set
:: for a2, a3 being Element of bool bool a1 holds
:: INTERSECTION(a2,a3) = INTERSECTION(a3,a2);
end;
:: TOPGEN_4:funcnot 3 => TOPGEN_4:func 3
definition
let a1 be set;
let a2, a3 be Element of bool bool a1;
redefine func UNION(a2,a3) -> Element of bool bool a1;
commutativity;
:: for a1 being set
:: for a2, a3 being Element of bool bool a1 holds
:: UNION(a2,a3) = UNION(a3,a2);
end;
:: TOPGEN_4:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is closed(b1) & b3 is closed(b1)
holds INTERSECTION(b2,b3) is closed(b1);
:: TOPGEN_4:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is closed(b1) & b3 is closed(b1)
holds UNION(b2,b3) is closed(b1);
:: TOPGEN_4:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is open(b1) & b3 is open(b1)
holds INTERSECTION(b2,b3) is open(b1);
:: TOPGEN_4:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is open(b1) & b3 is open(b1)
holds UNION(b2,b3) is open(b1);
:: TOPGEN_4:th 25
theorem
for b1 being set
for b2, b3 being Element of bool bool b1 holds
Card INTERSECTION(b2,b3) c= Card [:b2,b3:];
:: TOPGEN_4:th 26
theorem
for b1 being set
for b2, b3 being Element of bool bool b1 holds
Card UNION(b2,b3) c= Card [:b2,b3:];
:: TOPGEN_4:th 27
theorem
for b1, b2 being set holds
union UNION(b1,b2) c= (union b1) \/ union b2;
:: TOPGEN_4:th 28
theorem
for b1, b2 being set
st b1 <> {} & b2 <> {}
holds (union b1) \/ union b2 = union UNION(b1,b2);
:: TOPGEN_4:th 29
theorem
for b1 being set holds
UNION({},b1) = {};
:: TOPGEN_4:th 30
theorem
for b1, b2 being set
st UNION(b1,b2) = {} & b1 <> {}
holds b2 = {};
:: TOPGEN_4:th 31
theorem
for b1, b2 being set
st INTERSECTION(b1,b2) = {} & b1 <> {}
holds b2 = {};
:: TOPGEN_4:th 32
theorem
for b1, b2 being set holds
meet UNION(b1,b2) c= (meet b1) \/ meet b2;
:: TOPGEN_4:th 33
theorem
for b1, b2 being set
st b1 <> {} & b2 <> {}
holds meet UNION(b1,b2) = (meet b1) \/ meet b2;
:: TOPGEN_4:th 34
theorem
for b1, b2 being set
st b1 <> {} & b2 <> {}
holds (meet b1) /\ meet b2 = meet INTERSECTION(b1,b2);
:: TOPGEN_4:attrnot 5 => TOPGEN_4:attr 5
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is F_sigma means
ex b1 being countable closed Element of bool bool the carrier of a1 st
a2 = union b1;
end;
:: TOPGEN_4: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 F_sigma
it is sufficient to prove
thus ex b1 being countable closed Element of bool bool the carrier of a1 st
a2 = union b1;
:: TOPGEN_4:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is F_sigma(b1)
iff
ex b3 being countable closed Element of bool bool the carrier of b1 st
b2 = union b3;
:: TOPGEN_4:attrnot 6 => TOPGEN_4:attr 6
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is G_delta means
ex b1 being countable open Element of bool bool the carrier of a1 st
a2 = meet b1;
end;
:: TOPGEN_4:dfs 7
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is G_delta
it is sufficient to prove
thus ex b1 being countable open Element of bool bool the carrier of a1 st
a2 = meet b1;
:: TOPGEN_4:def 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is G_delta(b1)
iff
ex b3 being countable open Element of bool bool the carrier of b1 st
b2 = meet b3;
:: TOPGEN_4:th 35
theorem
for b1 being non empty TopSpace-like TopStruct holds
{} b1 is F_sigma(b1);
:: TOPGEN_4:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
{} b1 is G_delta(b1);
:: TOPGEN_4:funcreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster {} a1 -> F_sigma G_delta;
end;
:: TOPGEN_4:th 37
theorem
for b1 being non empty TopSpace-like TopStruct holds
[#] b1 is F_sigma(b1);
:: TOPGEN_4:th 38
theorem
for b1 being non empty TopSpace-like TopStruct holds
[#] b1 is G_delta(b1);
:: TOPGEN_4:funcreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster [#] a1 -> F_sigma G_delta;
end;
:: TOPGEN_4:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is F_sigma(b1)
holds b2 ` is G_delta(b1);
:: TOPGEN_4:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is G_delta(b1)
holds b2 ` is F_sigma(b1);
:: TOPGEN_4:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is F_sigma(b1) & b3 is F_sigma(b1)
holds b2 /\ b3 is F_sigma(b1);
:: TOPGEN_4:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is F_sigma(b1) & b3 is F_sigma(b1)
holds b2 \/ b3 is F_sigma(b1);
:: TOPGEN_4:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is G_delta(b1) & b3 is G_delta(b1)
holds b2 \/ b3 is G_delta(b1);
:: TOPGEN_4:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is G_delta(b1) & b3 is G_delta(b1)
holds b2 /\ b3 is G_delta(b1);
:: TOPGEN_4:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
holds b2 is F_sigma(b1);
:: TOPGEN_4:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is open(b1)
holds b2 is G_delta(b1);
:: TOPGEN_4:th 47
theorem
for b1 being Element of bool the carrier of R^1
st b1 = RAT
holds b1 is F_sigma(R^1);
:: TOPGEN_4:attrnot 7 => TOPGEN_4:attr 7
definition
let a1 be TopSpace-like TopStruct;
attr a1 is T_1/2 means
for b1 being Element of bool the carrier of a1 holds
Der b1 is closed(a1);
end;
:: TOPGEN_4:dfs 8
definiens
let a1 be TopSpace-like TopStruct;
To prove
a1 is T_1/2
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1 holds
Der b1 is closed(a1);
:: TOPGEN_4:def 8
theorem
for b1 being TopSpace-like TopStruct holds
b1 is T_1/2
iff
for b2 being Element of bool the carrier of b1 holds
Der b2 is closed(b1);
:: TOPGEN_4:th 48
theorem
for b1 being TopSpace-like TopStruct
st b1 is being_T1
holds b1 is T_1/2;
:: TOPGEN_4:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is T_1/2
holds b1 is discerning;
:: TOPGEN_4:condreg 11
registration
cluster TopSpace-like T_1/2 -> discerning (TopStruct);
end;
:: TOPGEN_4:condreg 12
registration
cluster TopSpace-like being_T1 -> T_1/2 (TopStruct);
end;
:: TOPGEN_4:prednot 1 => TOPGEN_4:pred 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
pred A3 is_a_condensation_point_of A2 means
for b1 being a_neighborhood of a3 holds
b1 /\ a2 is not countable;
end;
:: TOPGEN_4:dfs 9
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
To prove
a3 is_a_condensation_point_of a2
it is sufficient to prove
thus for b1 being a_neighborhood of a3 holds
b1 /\ a2 is not countable;
:: TOPGEN_4:def 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 is_a_condensation_point_of b2
iff
for b4 being a_neighborhood of b3 holds
b4 /\ b2 is not countable;
:: TOPGEN_4:th 51
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b4 is_a_condensation_point_of b2 & b2 c= b3
holds b4 is_a_condensation_point_of b3;
:: TOPGEN_4:funcnot 4 => TOPGEN_4:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
func A2 ^0 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
b1 is_a_condensation_point_of a2;
end;
:: TOPGEN_4:def 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b3 = b2 ^0
iff
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
b4 is_a_condensation_point_of b2;
:: TOPGEN_4:th 52
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 is_a_condensation_point_of b2
holds b3 is_an_accumulation_point_of b2;
:: TOPGEN_4:th 53
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 ^0 c= Der b2;
:: TOPGEN_4:th 54
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 ^0 = Cl (b2 ^0);
:: TOPGEN_4:th 55
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds b2 ^0 c= b3 ^0;
:: TOPGEN_4:th 56
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b4 is_a_condensation_point_of b2 \/ b3 & not b4 is_a_condensation_point_of b2
holds b4 is_a_condensation_point_of b3;
:: TOPGEN_4:th 57
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(b2 \/ b3) ^0 = b2 ^0 \/ (b3 ^0);
:: TOPGEN_4:th 58
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is countable
for b3 being Element of the carrier of b1 holds
not b3 is_a_condensation_point_of b2;
:: TOPGEN_4:th 59
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is countable
holds b2 ^0 = {};
:: TOPGEN_4:funcreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be countable Element of bool the carrier of a1;
cluster a2 ^0 -> empty;
end;
:: TOPGEN_4:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is second-countable
holds ex b2 being Basis of b1 st
b2 is countable;
:: TOPGEN_4:exreg 3
registration
cluster non empty TopSpace-like second-countable TopStruct;
end;
:: TOPGEN_4:funcreg 7
registration
let a1 be non empty TopSpace-like TopStruct;
cluster TotFam a1 -> non empty compl-closed all-open-containing closed_for_countable_unions;
end;
:: TOPGEN_4:th 61
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,bool b1 holds
rng b2 is non empty countable Element of bool bool b1;
:: TOPGEN_4:th 63
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is all-open-containing(b1) & b2 c= b3
holds b3 is all-open-containing(b1);
:: TOPGEN_4:th 64
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is all-closed-containing(b1) & b2 c= b3
holds b3 is all-closed-containing(b1);
:: TOPGEN_4:modenot 1
definition
let a1 be 1-sorted;
mode SigmaField of a1 is non empty compl-closed sigma-multiplicative Element of bool bool the carrier of a1;
end;
:: TOPGEN_4:exreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster compl-closed all-open-containing all-closed-containing closed_for_countable_unions closed_for_countable_meets Element of bool bool the carrier of a1;
end;
:: TOPGEN_4:th 65
theorem
for b1 being non empty TopSpace-like TopStruct holds
sigma TotFam b1 is all-open-containing(b1) & sigma TotFam b1 is compl-closed(the carrier of b1) & sigma TotFam b1 is closed_for_countable_unions(the carrier of b1);
:: TOPGEN_4:funcreg 8
registration
let a1 be non empty TopSpace-like TopStruct;
cluster sigma TotFam a1 -> non empty compl-closed sigma-multiplicative all-open-containing closed_for_countable_unions;
end;
:: TOPGEN_4:exreg 5
registration
let a1 be non empty 1-sorted;
cluster non empty compl-closed sigma-additive closed_for_countable_unions Element of bool bool the carrier of a1;
end;
:: TOPGEN_4:condreg 13
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty compl-closed sigma-multiplicative -> closed_for_countable_unions (Element of bool bool the carrier of a1);
end;
:: TOPGEN_4:th 66
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is compl-closed(the carrier of b1) & b2 is closed_for_countable_unions(the carrier of b1)
holds b2 is non empty compl-closed sigma-multiplicative Element of bool bool the carrier of b1;
:: TOPGEN_4:exreg 6
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty cup-closed cap-closed compl-closed sigma-multiplicative sigma-additive all-open-containing closed_for_countable_unions Element of bool bool the carrier of a1;
end;
:: TOPGEN_4:funcreg 9
registration
let a1 be non empty TopSpace-like TopStruct;
cluster Topology_of a1 -> open all-open-containing;
end;
:: TOPGEN_4:th 67
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
ex b3 being compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of b1 st
b2 c= b3 &
(for b4 being compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of b1
st b2 c= b4
holds b3 c= b4);
:: TOPGEN_4:funcnot 5 => TOPGEN_4:func 5
definition
let a1 be non empty TopSpace-like TopStruct;
func BorelSets A1 -> compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of a1 means
for b1 being compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of a1 holds
it c= b1;
end;
:: TOPGEN_4:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of b1 holds
b2 = BorelSets b1
iff
for b3 being compl-closed all-open-containing closed_for_countable_unions Element of bool bool the carrier of b1 holds
b2 c= b3;
:: TOPGEN_4:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being closed Element of bool bool the carrier of b1 holds
b2 c= BorelSets b1;
:: TOPGEN_4:th 69
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being open Element of bool bool the carrier of b1 holds
b2 c= BorelSets b1;
:: TOPGEN_4:th 70
theorem
for b1 being non empty TopSpace-like TopStruct holds
BorelSets b1 = sigma Topology_of b1;
:: TOPGEN_4:attrnot 8 => TOPGEN_4:attr 8
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is Borel means
a2 in BorelSets a1;
end;
:: TOPGEN_4:dfs 12
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is Borel
it is sufficient to prove
thus a2 in BorelSets a1;
:: TOPGEN_4:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is Borel(b1)
iff
b2 in BorelSets b1;
:: TOPGEN_4:condreg 14
registration
let a1 be non empty TopSpace-like TopStruct;
cluster F_sigma -> Borel (Element of bool the carrier of a1);
end;
:: TOPGEN_4:condreg 15
registration
let a1 be non empty TopSpace-like TopStruct;
cluster G_delta -> Borel (Element of bool the carrier of a1);
end;