Article NAGATA_1, MML version 4.99.1005
:: NAGATA_1:attrnot 1 => NAGATA_1:attr 1
definition
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is discrete means
for b1 being Element of the carrier of a1 holds
ex b2 being open Element of bool the carrier of a1 st
b1 in b2 &
(for b3, b4 being Element of bool the carrier of a1
st b3 in a2 & b4 in a2 & b2 meets b3 & b2 meets b4
holds b3 = b4);
end;
:: NAGATA_1:dfs 1
definiens
let a1 be TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is discrete
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being open Element of bool the carrier of a1 st
b1 in b2 &
(for b3, b4 being Element of bool the carrier of a1
st b3 in a2 & b4 in a2 & b2 meets b3 & b2 meets b4
holds b3 = b4);
:: NAGATA_1:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is discrete(b1)
iff
for b3 being Element of the carrier of b1 holds
ex b4 being open Element of bool the carrier of b1 st
b3 in b4 &
(for b5, b6 being Element of bool the carrier of b1
st b5 in b2 & b6 in b2 & b4 meets b5 & b4 meets b6
holds b5 = b6);
:: NAGATA_1:exreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
cluster discrete Element of bool bool the carrier of a1;
end;
:: NAGATA_1:exreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
cluster empty discrete Element of bool bool the carrier of a1;
end;
:: NAGATA_1:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st ex b3 being Element of bool the carrier of b1 st
b2 = {b3}
holds b2 is discrete(b1);
:: NAGATA_1:th 2
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 discrete(b1)
holds b2 is discrete(b1);
:: NAGATA_1:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is discrete(b1)
holds b2 /\ b3 is discrete(b1);
:: NAGATA_1:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is discrete(b1)
holds b2 \ b3 is discrete(b1);
:: NAGATA_1:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool bool the carrier of b1
st b2 is discrete(b1) & b3 is discrete(b1) & INTERSECTION(b2,b3) = b4
holds b4 is discrete(b1);
:: NAGATA_1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b2 is discrete(b1) & b3 in b2 & b4 in b2 & b3 <> b4
holds b3 misses b4;
:: NAGATA_1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is discrete(b1)
for b3 being Element of the carrier of b1 holds
ex b4 being open Element of bool the carrier of b1 st
b3 in b4 &
(INTERSECTION({b4},b2)) \ {{}} is trivial;
:: NAGATA_1:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is discrete(b1)
iff
(for b3 being Element of the carrier of b1 holds
ex b4 being open Element of bool the carrier of b1 st
b3 in b4 &
(INTERSECTION({b4},b2)) \ {{}} is trivial) &
(for b3, b4 being Element of bool the carrier of b1
st b3 in b2 & b4 in b2 & b3 <> b4
holds b3 misses b4);
:: NAGATA_1:funcreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be discrete Element of bool bool the carrier of a1;
cluster clf a2 -> discrete;
end;
:: NAGATA_1: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 discrete(b1)
for b3, b4 being Element of bool the carrier of b1
st b3 in b2 & b4 in b2
holds Cl (b3 /\ b4) = (Cl b3) /\ Cl b4;
:: NAGATA_1:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is discrete(b1)
holds Cl union b2 = union clf b2;
:: NAGATA_1:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is discrete(b1)
holds b2 is locally_finite(b1);
:: NAGATA_1:modenot 1
definition
let a1 be TopSpace-like TopStruct;
mode FamilySequence of a1 is Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
end;
:: NAGATA_1:funcnot 1 => NAGATA_1:func 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
let a3 be Element of NAT;
redefine func a2 . a3 -> Element of bool bool the carrier of a1;
end;
:: NAGATA_1:funcnot 2 => NAGATA_1:func 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
redefine func Union a2 -> Element of bool bool the carrier of a1;
end;
:: NAGATA_1:attrnot 2 => NAGATA_1:attr 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
attr a2 is sigma_discrete means
for b1 being Element of NAT holds
a2 . b1 is discrete(a1);
end;
:: NAGATA_1:dfs 2
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
a2 is sigma_discrete
it is sufficient to prove
thus for b1 being Element of NAT holds
a2 . b1 is discrete(a1);
:: NAGATA_1:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
b2 is sigma_discrete(b1)
iff
for b3 being Element of NAT holds
b2 . b3 is discrete(b1);
:: NAGATA_1:exreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty Relation-like Function-like quasi_total total sigma_discrete Relation of NAT,bool bool the carrier of a1;
end;
:: NAGATA_1:attrnot 3 => NAGATA_1:attr 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
attr a2 is sigma_locally_finite means
for b1 being Element of NAT holds
a2 . b1 is locally_finite(a1);
end;
:: NAGATA_1:dfs 3
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
a2 is sigma_locally_finite
it is sufficient to prove
thus for b1 being Element of NAT holds
a2 . b1 is locally_finite(a1);
:: NAGATA_1:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
b2 is sigma_locally_finite(b1)
iff
for b3 being Element of NAT holds
b2 . b3 is locally_finite(b1);
:: NAGATA_1:attrnot 4 => NAGATA_1:attr 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is sigma_discrete means
ex b1 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of a1 st
a2 = Union b1;
end;
:: NAGATA_1:dfs 4
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is sigma_discrete
it is sufficient to prove
thus ex b1 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of a1 st
a2 = Union b1;
:: NAGATA_1:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is sigma_discrete(b1)
iff
ex b3 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of b1 st
b2 = Union b3;
:: NAGATA_1:attrnot 5 => CARD_4:attr 1
notation
let a1 be set;
antonym uncountable for countable;
end;
:: NAGATA_1:condreg 1
registration
cluster non countable -> non empty (set);
end;
:: NAGATA_1:exreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty Relation-like Function-like quasi_total total sigma_locally_finite Relation of NAT,bool bool the carrier of a1;
end;
:: NAGATA_1:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1
st b2 is sigma_discrete(b1)
holds b2 is sigma_locally_finite(b1);
:: NAGATA_1:th 13
theorem
for b1 being non countable set holds
ex b2 being Element of bool bool the carrier of 1TopSp [:b1,b1:] st
b2 is locally_finite(1TopSp [:b1,b1:]) & b2 is sigma_discrete(not 1TopSp [:b1,b1:]);
:: NAGATA_1:attrnot 6 => NAGATA_1:attr 5
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
attr a2 is Basis_sigma_discrete means
a2 is sigma_discrete(a1) & Union a2 is Basis of a1;
end;
:: NAGATA_1:dfs 5
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
a2 is Basis_sigma_discrete
it is sufficient to prove
thus a2 is sigma_discrete(a1) & Union a2 is Basis of a1;
:: NAGATA_1:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
b2 is Basis_sigma_discrete(b1)
iff
b2 is sigma_discrete(b1) & Union b2 is Basis of b1;
:: NAGATA_1:attrnot 7 => NAGATA_1:attr 6
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
attr a2 is Basis_sigma_locally_finite means
a2 is sigma_locally_finite(a1) & Union a2 is Basis of a1;
end;
:: NAGATA_1:dfs 6
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
a2 is Basis_sigma_locally_finite
it is sufficient to prove
thus a2 is sigma_locally_finite(a1) & Union a2 is Basis of a1;
:: NAGATA_1:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
b2 is Basis_sigma_locally_finite(b1)
iff
b2 is sigma_locally_finite(b1) & Union b2 is Basis of b1;
:: NAGATA_1:th 14
theorem
for b1 being non empty MetrStruct
for b2 being real set
st b1 is non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Element of the carrier of b1 holds
([#] b1) \ cl_Ball(b3,b2) in Family_open_set b1;
:: NAGATA_1:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is metrizable
holds b1 is being_T3 & b1 is being_T1;
:: NAGATA_1:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is metrizable
holds ex b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 st
b2 is Basis_sigma_locally_finite(b1);
:: NAGATA_1:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
st for b3 being Element of NAT holds
b2 . b3 is open(b1)
holds Union b2 is open(b1);
:: NAGATA_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being Element of bool the carrier of b1
for b3 being open Element of bool the carrier of b1
st b2 is closed(b1) & b3 is open(b1) & b2 c= b3
holds ex b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1 st
b2 c= Union b4 &
Union b4 c= b3 &
(for b5 being Element of NAT holds
Cl (b4 . b5) c= b3 & b4 . b5 is open(b1))
holds b1 is being_T4;
:: NAGATA_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T3
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1
st Union b2 is Basis of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
st b3 is open(b1) & b4 in b3
holds ex b5 being Element of bool the carrier of b1 st
b4 in b5 & Cl b5 c= b3 & b5 in Union b2;
:: NAGATA_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T3 &
b1 is being_T1 &
(ex b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 st
b2 is Basis_sigma_locally_finite(b1))
holds b1 is being_T4;
:: NAGATA_1:funcnot 3 => NAGATA_1:func 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
redefine func A2 + A3 -> Function-like quasi_total Relation of the carrier of a1,REAL means
for b1 being Element of the carrier of a1 holds
it . b1 = (a2 . b1) + (a3 . b1);
commutativity;
:: for a1 being non empty TopSpace-like TopStruct
:: for a2, a3 being Function-like quasi_total Relation of the carrier of a1,REAL holds
:: a2 + a3 = a3 + a2;
end;
:: NAGATA_1:def 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b4 = b2 + b3
iff
for b5 being Element of the carrier of b1 holds
b4 . b5 = (b2 . b5) + (b3 . b5);
:: NAGATA_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
st b2 is continuous(b1)
for b3 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
st for b4, b5 being Element of the carrier of b1 holds
b3 .(b4,b5) = abs ((b2 . b4) - (b2 . b5))
holds b3 is continuous([:b1,b1:]);
:: NAGATA_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,REAL
st b2 is continuous(b1) & b3 is continuous(b1)
holds b2 + b3 is continuous(b1);
:: NAGATA_1:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 .(b3,b4) = b3 + b4
holds b2 is having_a_unity(Funcs(the carrier of b1,REAL)) & b2 is commutative(Funcs(the carrier of b1,REAL)) & b2 is associative(Funcs(the carrier of b1,REAL));
:: NAGATA_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 .(b3,b4) = b3 + b4
for b3 being Element of Funcs(the carrier of b1,REAL)
st b3 is_a_unity_wrt b2
holds b3 is continuous(b1);
:: NAGATA_1:funcnot 4 => NAGATA_1:func 4
definition
let a1, a2 be non empty set;
let a3 be Function-like quasi_total Relation of a1,Funcs(a1,a2);
func A3 Toler -> Function-like quasi_total Relation of a1,a2 means
for b1 being Element of a1 holds
it . b1 = (a3 . b1) . b1;
end;
:: NAGATA_1:def 8
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,Funcs(b1,b2)
for b4 being Function-like quasi_total Relation of b1,b2 holds
b4 = b3 Toler
iff
for b5 being Element of b1 holds
b4 . b5 = (b3 . b5) . b5;
:: NAGATA_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b2 .(b3,b4) = b3 + b4
for b3 being FinSequence of Funcs(the carrier of b1,REAL)
st for b4 being Element of NAT
st 0 <> b4 & b4 <= len b3
holds b3 . b4 is Function-like quasi_total continuous Relation of the carrier of b1,REAL
holds b2 "**" b3 is Function-like quasi_total continuous Relation of the carrier of b1,REAL;
:: NAGATA_1:th 26
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,Funcs(the carrier of b1,the carrier of b2)
st for b4 being Element of the carrier of b1 holds
b3 . b4 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,bool the carrier of b1
st for b5 being Element of the carrier of b1 holds
b5 in b4 . b5 &
b4 . b5 is open(b1) &
(for b6, b7 being Element of the carrier of b1
st b6 in b4 . b7
holds (b3 . b6) . b6 = (b3 . b7) . b6)
holds b3 Toler is continuous(b1, b2);
:: NAGATA_1:funcnot 5 => NAGATA_1:func 5
definition
let a1 be set;
let a2 be Element of REAL;
let a3 be Function-like quasi_total Relation of a1,REAL;
func min(A2,A3) -> Function-like quasi_total Relation of a1,REAL means
for b1 being set
st b1 in a1
holds it . b1 = min(a2,a3 . b1);
end;
:: NAGATA_1:def 9
theorem
for b1 being set
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of b1,REAL holds
b4 = min(b2,b3)
iff
for b5 being set
st b5 in b1
holds b4 . b5 = min(b2,b3 . b5);
:: NAGATA_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
st b3 is continuous(b1)
holds min(b2,b3) is continuous(b1);
:: NAGATA_1:prednot 1 => NAGATA_1:pred 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
pred A2 is_a_pseudometric_of A1 means
a2 is Reflexive(a1) & a2 is symmetric(a1) & a2 is triangle(a1);
end;
:: NAGATA_1:dfs 10
definiens
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
To prove
a2 is_a_pseudometric_of a1
it is sufficient to prove
thus a2 is Reflexive(a1) & a2 is symmetric(a1) & a2 is triangle(a1);
:: NAGATA_1:def 10
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_a_pseudometric_of b1
iff
b2 is Reflexive(b1) & b2 is symmetric(b1) & b2 is triangle(b1);
:: NAGATA_1:th 28
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
b2 is_a_pseudometric_of b1
iff
for b3, b4, b5 being Element of b1 holds
b2 .(b3,b3) = 0 &
b2 .(b3,b5) <= (b2 .(b3,b4)) + (b2 .(b5,b4));
:: NAGATA_1:th 29
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
st b2 is_a_pseudometric_of b1
for b3, b4 being Element of b1 holds
0 <= b2 .(b3,b4);
:: NAGATA_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
st 0 < b2 & b3 is_a_pseudometric_of the carrier of b1
holds min(b2,b3) is_a_pseudometric_of the carrier of b1;
:: NAGATA_1:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
st 0 < b2 & b3 is_metric_of the carrier of b1
holds min(b2,b3) is_metric_of the carrier of b1;