Article PCOMPS_1, MML version 4.99.1005
:: PCOMPS_1:th 1
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1
for b3, b4 being real set
st b3 <= b4
holds Ball(b2,b3) c= Ball(b2,b4);
:: PCOMPS_1:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
Cl b2 <> {}
iff
b2 <> {};
:: PCOMPS_1:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1
for b3 being Element of the carrier of b1 holds
ex b4 being Element of bool the carrier of b1 st
b3 in b4 & b4 in b2;
:: PCOMPS_1:funcnot 1 => PCOMPS_1:func 1
definition
let a1 be set;
redefine func bool a1 -> Element of bool bool a1;
end;
:: PCOMPS_1:th 7
theorem
for b1 being set holds
the topology of 1TopSp b1 = bool b1;
:: PCOMPS_1:th 8
theorem
for b1 being set holds
the carrier of 1TopSp b1 = b1;
:: PCOMPS_1:th 9
theorem
for b1 being set holds
1TopSp {b1} is compact;
:: PCOMPS_1:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
st b1 is being_T2
holds {b2} is closed(b1);
:: PCOMPS_1:attrnot 1 => PCOMPS_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is locally_finite means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool the carrier of a1 st
b1 in b2 &
b2 is open(a1) &
{b3 where b3 is Element of bool the carrier of a1: b3 in a2 & b3 meets b2} is finite;
end;
:: PCOMPS_1:dfs 1
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is locally_finite
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of bool the carrier of a1 st
b1 in b2 &
b2 is open(a1) &
{b3 where b3 is Element of bool the carrier of a1: b3 in a2 & b3 meets b2} is finite;
:: PCOMPS_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is locally_finite(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being Element of bool the carrier of b1 st
b3 in b4 &
b4 is open(b1) &
{b5 where b5 is Element of bool the carrier of b1: b5 in b2 & b5 meets b4} is finite;
:: PCOMPS_1:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
{b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b4 meets b3} c= b2;
:: PCOMPS_1:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3 & b3 is locally_finite(b1)
holds b2 is locally_finite(b1);
:: PCOMPS_1:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is finite
holds b2 is locally_finite(b1);
:: PCOMPS_1:funcnot 2 => PCOMPS_1:func 2
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
func clf A2 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
ex b2 being Element of bool the carrier of a1 st
b1 = Cl b2 & b2 in a2;
end;
:: PCOMPS_1:def 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
b3 = clf b2
iff
for b4 being Element of bool the carrier of b1 holds
b4 in b3
iff
ex b5 being Element of bool the carrier of b1 st
b4 = Cl b5 & b5 in b2;
:: PCOMPS_1:th 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
clf b2 is closed(b1);
:: PCOMPS_1:th 15
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 = {}
holds clf b2 = {};
:: PCOMPS_1:th 16
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 clf b3 = {Cl b2};
:: PCOMPS_1:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3
holds clf b2 c= clf b3;
:: PCOMPS_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1 holds
clf (b2 \/ b3) = (clf b2) \/ clf b3;
:: PCOMPS_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is finite
holds Cl union b2 = union clf b2;
:: PCOMPS_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is_finer_than clf b2;
:: PCOMPS_1:sch 1
scheme PCOMPS_1:sch 1
{F1 -> TopSpace-like TopStruct,
F2 -> Element of bool bool the carrier of F1(),
F3 -> Element of bool bool the carrier of F1(),
F4 -> Element of bool the carrier of F1()}:
ex b1 being Function-like quasi_total Relation of F2(),F3() st
for b2 being Element of bool the carrier of F1()
st b2 in F2()
holds b1 . b2 = F4(b2)
provided
for b1 being Element of bool the carrier of F1()
st b1 in F2()
holds F4(b1) in F3();
:: PCOMPS_1:th 21
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 clf b2 is locally_finite(b1);
:: PCOMPS_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
union b2 c= union clf b2;
:: PCOMPS_1:th 23
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 Cl union b2 = union clf b2;
:: PCOMPS_1:th 24
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) & b2 is closed(b1)
holds union b2 is closed(b1);
:: PCOMPS_1:attrnot 2 => PCOMPS_1:attr 2
definition
let a1 be TopStruct;
attr a1 is paracompact means
for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 is open(a1) & b2 is_a_cover_of a1 & b2 is_finer_than b1 & b2 is locally_finite(a1);
end;
:: PCOMPS_1:dfs 3
definiens
let a1 be TopStruct;
To prove
a1 is paracompact
it is sufficient to prove
thus for b1 being Element of bool bool the carrier of a1
st b1 is_a_cover_of a1 & b1 is open(a1)
holds ex b2 being Element of bool bool the carrier of a1 st
b2 is open(a1) & b2 is_a_cover_of a1 & b2 is_finer_than b1 & b2 is locally_finite(a1);
:: PCOMPS_1:def 4
theorem
for b1 being TopStruct holds
b1 is paracompact
iff
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_1:exreg 1
registration
cluster non empty TopSpace-like paracompact TopStruct;
end;
:: PCOMPS_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is compact
holds b1 is paracompact;
:: PCOMPS_1:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b1 is paracompact &
b2 is closed(b1) &
b3 is closed(b1) &
b2 misses b3 &
(for b4 being Element of the carrier of b1
st b4 in b3
holds ex b5, b6 being Element of bool the carrier of b1 st
b5 is open(b1) & b6 is open(b1) & b2 c= b5 & b4 in b6 & b5 misses b6)
holds ex b4, b5 being Element of bool the carrier of b1 st
b4 is open(b1) & b5 is open(b1) & b2 c= b4 & b3 c= b5 & b4 misses b5;
:: PCOMPS_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2 & b1 is paracompact
holds b1 is being_T3;
:: PCOMPS_1:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T2 & b1 is paracompact
holds b1 is being_T4;
:: PCOMPS_1:funcnot 3 => PCOMPS_1:func 3
definition
let a1 be MetrStruct;
func Family_open_set A1 -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
for b2 being Element of the carrier of a1
st b2 in b1
holds ex b3 being Element of REAL st
0 < b3 & Ball(b2,b3) c= b1;
end;
:: PCOMPS_1:def 5
theorem
for b1 being MetrStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 = Family_open_set b1
iff
for b3 being Element of bool the carrier of b1 holds
b3 in b2
iff
for b4 being Element of the carrier of b1
st b4 in b3
holds ex b5 being Element of REAL st
0 < b5 & Ball(b4,b5) c= b3;
:: PCOMPS_1:th 29
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1 holds
ex b3 being Element of REAL st
0 < b3 & Ball(b2,b3) c= the carrier of b1;
:: PCOMPS_1:th 30
theorem
for b1 being MetrStruct
for b2, b3 being Element of the carrier of b1
for b4 being real set
st b1 is triangle & b2 in Ball(b3,b4)
holds ex b5 being Element of REAL st
0 < b5 & Ball(b2,b5) c= Ball(b3,b4);
:: PCOMPS_1:th 31
theorem
for b1 being MetrStruct
for b2, b3 being Element of REAL
for b4, b5, b6 being Element of the carrier of b1
st b1 is triangle & b4 in (Ball(b5,b2)) /\ Ball(b6,b3)
holds ex b7 being Element of REAL st
Ball(b4,b7) c= Ball(b5,b2) & Ball(b4,b7) c= Ball(b6,b3);
:: PCOMPS_1:th 33
theorem
for b1 being MetrStruct
for b2 being Element of the carrier of b1
for b3 being real set
st b1 is triangle
holds Ball(b2,b3) in Family_open_set b1;
:: PCOMPS_1:th 34
theorem
for b1 being MetrStruct holds
the carrier of b1 in Family_open_set b1;
:: PCOMPS_1:th 35
theorem
for b1 being MetrStruct
for b2, b3 being Element of bool the carrier of b1
st b2 in Family_open_set b1 & b3 in Family_open_set b1
holds b2 /\ b3 in Family_open_set b1;
:: PCOMPS_1:th 36
theorem
for b1 being MetrStruct
for b2 being Element of bool bool the carrier of b1
st b2 c= Family_open_set b1
holds union b2 in Family_open_set b1;
:: PCOMPS_1:th 37
theorem
for b1 being MetrStruct holds
TopStruct(#the carrier of b1,Family_open_set b1#) is TopSpace-like TopStruct;
:: PCOMPS_1:funcnot 4 => PCOMPS_1:func 4
definition
let a1 be MetrStruct;
func TopSpaceMetr A1 -> TopStruct equals
TopStruct(#the carrier of a1,Family_open_set a1#);
end;
:: PCOMPS_1:def 6
theorem
for b1 being MetrStruct holds
TopSpaceMetr b1 = TopStruct(#the carrier of b1,Family_open_set b1#);
:: PCOMPS_1:funcreg 1
registration
let a1 be MetrStruct;
cluster TopSpaceMetr a1 -> strict TopSpace-like;
end;
:: PCOMPS_1:funcreg 2
registration
let a1 be non empty MetrStruct;
cluster TopSpaceMetr a1 -> non empty;
end;
:: PCOMPS_1:th 38
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
TopSpaceMetr b1 is being_T2;
:: PCOMPS_1:exreg 2
registration
cluster non empty strict TopSpace-like being_T2 TopStruct;
end;
:: PCOMPS_1:prednot 1 => PCOMPS_1:pred 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
pred A2 is_metric_of A1 means
for b1, b2, b3 being Element of a1 holds
(a2 .(b1,b2) = 0 implies b1 = b2) &
(b1 = b2 implies a2 .(b1,b2) = 0) &
a2 .(b1,b2) = a2 .(b2,b1) &
a2 .(b1,b3) <= (a2 .(b1,b2)) + (a2 .(b2,b3));
end;
:: PCOMPS_1:dfs 6
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
To prove
a2 is_metric_of a1
it is sufficient to prove
thus for b1, b2, b3 being Element of a1 holds
(a2 .(b1,b2) = 0 implies b1 = b2) &
(b1 = b2 implies a2 .(b1,b2) = 0) &
a2 .(b1,b2) = a2 .(b2,b1) &
a2 .(b1,b3) <= (a2 .(b1,b2)) + (a2 .(b2,b3));
:: PCOMPS_1:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_metric_of b1
iff
for b3, b4, b5 being Element of b1 holds
(b2 .(b3,b4) = 0 implies b3 = b4) &
(b3 = b4 implies b2 .(b3,b4) = 0) &
b2 .(b3,b4) = b2 .(b4,b3) &
b2 .(b3,b5) <= (b2 .(b3,b4)) + (b2 .(b4,b5));
:: PCOMPS_1:th 39
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_metric_of b1
iff
MetrStruct(#b1,b2#) is Reflexive discerning symmetric triangle MetrStruct;
:: PCOMPS_1:funcnot 5 => PCOMPS_1:func 5
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
assume a2 is_metric_of a1;
func SpaceMetr(A1,A2) -> non empty strict Reflexive discerning symmetric triangle MetrStruct equals
MetrStruct(#a1,a2#);
end;
:: PCOMPS_1:def 8
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
st b2 is_metric_of b1
holds SpaceMetr(b1,b2) = MetrStruct(#b1,b2#);
:: PCOMPS_1:attrnot 3 => PCOMPS_1:attr 3
definition
let a1 be non empty TopStruct;
attr a1 is metrizable means
ex b1 being Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],REAL st
b1 is_metric_of the carrier of a1 &
Family_open_set SpaceMetr(the carrier of a1,b1) = the topology of a1;
end;
:: PCOMPS_1:dfs 8
definiens
let a1 be non empty TopStruct;
To prove
a1 is metrizable
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],REAL st
b1 is_metric_of the carrier of a1 &
Family_open_set SpaceMetr(the carrier of a1,b1) = the topology of a1;
:: PCOMPS_1:def 9
theorem
for b1 being non empty TopStruct holds
b1 is metrizable
iff
ex b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL st
b2 is_metric_of the carrier of b1 &
Family_open_set SpaceMetr(the carrier of b1,b2) = the topology of b1;
:: PCOMPS_1:exreg 3
registration
cluster non empty strict TopSpace-like metrizable TopStruct;
end;