Article PCOMPS_2, MML version 4.99.1005
:: PCOMPS_2:th 5
theorem
for b1 being Relation-like set
for b2 being set
st b1 well_orders b2
holds b1 |_2 b2 well_orders b2 & b2 = field (b1 |_2 b2);
:: PCOMPS_2:sch 1
scheme PCOMPS_2:sch 1
{F1 -> set,
F2 -> Relation-like set}:
ex b1 being set st
b1 in F1() &
P1[b1] &
(for b2 being set
st b2 in F1() & P1[b2]
holds [b1,b2] in F2())
provided
F2() well_orders F1()
and
ex b1 being set st
b1 in F1() & P1[b1];
:: PCOMPS_2:funcnot 1 => PCOMPS_2:func 1
definition
let a1 be set;
let a2 be Relation-like set;
let a3 be Element of a1;
func PartUnion(A3,A2) -> set equals
union (a2 -Seg a3);
end;
:: PCOMPS_2:def 1
theorem
for b1 being set
for b2 being Relation-like set
for b3 being Element of b1 holds
PartUnion(b3,b2) = union (b2 -Seg b3);
:: PCOMPS_2:funcnot 2 => PCOMPS_2:func 2
definition
let a1 be set;
let a2 be Relation-like set;
func DisjointFam(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Element of a1 st
b2 in a1 & b1 = b2 \ PartUnion(b2,a2);
end;
:: PCOMPS_2:def 2
theorem
for b1 being set
for b2 being Relation-like set
for b3 being set holds
b3 = DisjointFam(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Element of b1 st
b5 in b1 & b4 = b5 \ PartUnion(b5,b2);
:: PCOMPS_2:funcnot 3 => PCOMPS_2:func 3
definition
let a1 be set;
let a2 be Element of NAT;
let a3 be Function-like quasi_total Relation of NAT,bool a1;
func PartUnionNat(A2,A3) -> set equals
union (a3 .: ((Seg a2) \ {a2}));
end;
:: PCOMPS_2:def 3
theorem
for b1 being set
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,bool b1 holds
PartUnionNat(b2,b3) = union (b3 .: ((Seg b2) \ {b2}));
:: PCOMPS_2:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T3
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1 & b2 is open(b1)
holds ex b3 being Element of bool bool the carrier of b1 st
b3 is open(b1) &
b3 is_a_cover_of b1 &
(for b4 being Element of bool the carrier of b1
st b4 in b3
holds ex b5 being Element of bool the carrier of b1 st
b5 in b2 & Cl b4 c= b5);
:: PCOMPS_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b1 is being_T2 & b1 is paracompact & b2 is_a_cover_of b1 & b2 is open(b1)
holds ex b3 being Element of bool bool the carrier of b1 st
b3 is open(b1) & b3 is_a_cover_of b1 & clf b3 is_finer_than b2 & b3 is locally_finite(b1);
:: PCOMPS_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
st b3 is_metric_of the carrier of b1 & b2 = SpaceMetr(the carrier of b1,b3)
holds the carrier of b2 = the carrier of b1;
:: PCOMPS_2:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Reflexive discerning symmetric triangle MetrStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
st b4 is_metric_of the carrier of b1 & b2 = SpaceMetr(the carrier of b1,b4)
holds b3 is Element of bool bool the carrier of b1
iff
b3 is Element of bool bool the carrier of b2;
:: PCOMPS_2:funcnot 4 => PCOMPS_2:func 4
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of NAT,(bool bool a1) *;
let a3 be Element of NAT;
redefine func a2 . a3 -> FinSequence of bool bool a1;
end;
:: PCOMPS_2:sch 2
scheme PCOMPS_2:sch 2
{F1 -> non empty set,
F2 -> Element of bool bool F1(),
F3 -> Element of bool F1()}:
ex b1 being Function-like quasi_total Relation of NAT,bool bool F1() st
b1 . 0 = F2() &
(for b2 being Element of NAT holds
b1 . (b2 + 1) = {union {F3(b4, b2) where b4 is Element of F1(): for b5 being Element of bool bool F1()
for b6 being Element of NAT
st b6 <= b2 & b5 = b1 . b6
holds P2[b4, b3, b2, b5]} where b3 is Element of bool F1(): P1[b3]})
:: PCOMPS_2:sch 3
scheme PCOMPS_2:sch 3
{F1 -> non empty set,
F2 -> Element of bool bool F1(),
F3 -> Element of bool F1()}:
ex b1 being Function-like quasi_total Relation of NAT,bool bool F1() st
b1 . 0 = F2() &
(for b2 being Element of NAT holds
b1 . (b2 + 1) = {union {F3(b4, b2) where b4 is Element of F1(): P2[b4, b3, b2] &
not b4 in union {union (b1 . b5) where b5 is Element of NAT: b5 <= b2}} where b3 is Element of bool F1(): P1[b3]})
:: PCOMPS_2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is metrizable
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1 & b2 is open(b1)
holds ex b3 being Element of bool bool the carrier of b1 st
b3 is open(b1) & b3 is_a_cover_of b1 & b3 is_finer_than b2 & b3 is locally_finite(b1);
:: PCOMPS_2:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is metrizable
holds b1 is paracompact;