Article FRECHET, MML version 4.99.1005
:: FRECHET:th 1
theorem
for b1 being non empty 1-sorted
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
rng b2 is Element of bool the carrier of b1;
:: FRECHET:th 2
theorem
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st rng b3 c= the carrier of b2
holds b3 is Function-like quasi_total Relation of NAT,the carrier of b2;
:: FRECHET:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2 holds
b3 <> {};
:: FRECHET:condreg 1
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster -> non empty (Basis of a2);
end;
:: FRECHET:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2 is open(b1) & b3 is closed(b1)
holds b2 \ b3 is open(b1);
:: FRECHET:th 5
theorem
for b1 being TopStruct
st {} b1 is closed(b1) &
[#] b1 is closed(b1) &
(for b2, b3 being Element of bool the carrier of b1
st b2 is closed(b1) & b3 is closed(b1)
holds b2 \/ b3 is closed(b1)) &
(for b2 being Element of bool bool the carrier of b1
st b2 is closed(b1)
holds meet b2 is closed(b1))
holds b1 is TopSpace-like TopStruct;
:: FRECHET:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st for b4 being Element of bool the carrier of b2 holds
b4 is closed(b2)
iff
b3 " b4 is closed(b1)
holds b2 is TopSpace-like TopStruct;
:: FRECHET:th 7
theorem
for b1 being Element of the carrier of RealSpace
for b2, b3 being Element of REAL
st b2 = b1 & 0 < b3
holds Ball(b1,b3) = ].b2 - b3,b2 + b3.[;
:: FRECHET:th 8
theorem
for b1 being Element of bool the carrier of R^1 holds
b1 is open(R^1)
iff
for b2 being Element of REAL
st b2 in b1
holds ex b3 being Element of REAL st
0 < b3 & ].b2 - b3,b2 + b3.[ c= b1;
:: FRECHET:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,the carrier of R^1
st for b2 being Element of NAT holds
b1 . b2 in ].b2 - (1 / 4),b2 + (1 / 4).[
holds rng b1 is closed(R^1);
:: FRECHET:th 10
theorem
for b1 being Element of bool the carrier of R^1
st b1 = NAT
holds b1 is closed(R^1);
:: FRECHET:th 11
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of the carrier of TopSpaceMetr b1
for b3 being Element of the carrier of b1
st b2 = b3
holds ex b4 being Basis of b2 st
b4 = {Ball(b3,1 / b5) where b5 is Element of NAT: b5 <> 0} &
b4 is countable &
(ex b5 being Function-like quasi_total Relation of NAT,b4 st
for b6 being set
st b6 in NAT
holds ex b7 being Element of NAT st
b6 = b7 &
b5 . b6 = Ball(b3,1 / (b7 + 1)));
:: FRECHET:th 12
theorem
for b1, b2 being Relation-like Function-like set holds
proj2 (b1 +* b2) = (b1 .: ((proj1 b1) \ proj1 b2)) \/ proj2 b2;
:: FRECHET:th 13
theorem
for b1, b2 being set
st b2 c= b1
holds (id b1) .: b2 = b2;
:: FRECHET:th 15
theorem
for b1, b2, b3 being set holds
proj1 ((id b1) +* (b2 --> b3)) = b1 \/ b2;
:: FRECHET:th 16
theorem
for b1, b2, b3 being set
st b2 <> {}
holds proj2 ((id b1) +* (b2 --> b3)) = (b1 \ b2) \/ {b3};
:: FRECHET:th 17
theorem
for b1, b2, b3, b4 being set
st b3 c= b1
holds ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = (b3 \ b2) \ {b4};
:: FRECHET:th 18
theorem
for b1, b2, b3 being set
st not b3 in b1
holds ((id b1) +* (b2 --> b3)) " {b3} = b2;
:: FRECHET:th 19
theorem
for b1, b2, b3, b4 being set
st b3 c= b1 & not b4 in b1
holds ((id b1) +* (b2 --> b4)) " (b3 \/ {b4}) = b3 \/ b2;
:: FRECHET:th 20
theorem
for b1, b2, b3, b4 being set
st b3 c= b1 & not b4 in b1
holds ((id b1) +* (b2 --> b4)) " (b3 \ {b4}) = b3 \ b2;
:: FRECHET:attrnot 1 => FRECHET:attr 1
definition
let a1 be non empty TopStruct;
attr a1 is first-countable means
for b1 being Element of the carrier of a1 holds
ex b2 being Basis of b1 st
b2 is countable;
end;
:: FRECHET:dfs 1
definiens
let a1 be non empty TopStruct;
To prove
a1 is first-countable
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Basis of b1 st
b2 is countable;
:: FRECHET:def 1
theorem
for b1 being non empty TopStruct holds
b1 is first-countable
iff
for b2 being Element of the carrier of b1 holds
ex b3 being Basis of b2 st
b3 is countable;
:: FRECHET:th 21
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
TopSpaceMetr b1 is first-countable;
:: FRECHET:th 22
theorem
R^1 is first-countable;
:: FRECHET:funcreg 1
registration
cluster R^1 -> strict TopSpace-like first-countable;
end;
:: FRECHET:prednot 1 => FRECHET:pred 1
definition
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
pred A2 is_convergent_to A3 means
for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds a2 . b3 in b1;
end;
:: FRECHET:dfs 2
definiens
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
let a3 be Element of the carrier of a1;
To prove
a2 is_convergent_to a3
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
holds ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds a2 . b3 in b1;
:: FRECHET:def 2
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of the carrier of b1 holds
b2 is_convergent_to b3
iff
for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
holds ex b5 being Element of NAT st
for b6 being Element of NAT
st b5 <= b6
holds b2 . b6 in b4;
:: FRECHET:th 23
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 = NAT --> b2
holds b3 is_convergent_to b2;
:: FRECHET:attrnot 2 => FRECHET:attr 2
definition
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
attr a2 is convergent means
ex b1 being Element of the carrier of a1 st
a2 is_convergent_to b1;
end;
:: FRECHET:dfs 3
definiens
let a1 be TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
a2 is convergent
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a2 is_convergent_to b1;
:: FRECHET:def 3
theorem
for b1 being TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 is convergent(b1)
iff
ex b3 being Element of the carrier of b1 st
b2 is_convergent_to b3;
:: FRECHET:funcnot 1 => FRECHET:func 1
definition
let a1 be non empty TopStruct;
let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
func Lim A2 -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
a2 is_convergent_to b1;
end;
:: FRECHET:def 4
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 = Lim b2
iff
for b4 being Element of the carrier of b1 holds
b4 in b3
iff
b2 is_convergent_to b4;
:: FRECHET:attrnot 3 => FRECHET:attr 3
definition
let a1 be non empty TopStruct;
attr a1 is Frechet means
for b1 being Element of bool the carrier of a1
for b2 being Element of the carrier of a1
st b2 in Cl b1
holds ex b3 being Function-like quasi_total Relation of NAT,the carrier of a1 st
rng b3 c= b1 & b2 in Lim b3;
end;
:: FRECHET:dfs 5
definiens
let a1 be non empty TopStruct;
To prove
a1 is Frechet
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
for b2 being Element of the carrier of a1
st b2 in Cl b1
holds ex b3 being Function-like quasi_total Relation of NAT,the carrier of a1 st
rng b3 c= b1 & b2 in Lim b3;
:: FRECHET:def 5
theorem
for b1 being non empty TopStruct holds
b1 is Frechet
iff
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b3 in Cl b2
holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
rng b4 c= b2 & b3 in Lim b4;
:: FRECHET:attrnot 4 => FRECHET:attr 4
definition
let a1 be non empty TopStruct;
attr a1 is sequential means
for b1 being Element of bool the carrier of a1 holds
b1 is closed(a1)
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of a1
st b2 is convergent(a1) & rng b2 c= b1
holds Lim b2 c= b1;
end;
:: FRECHET:dfs 6
definiens
let a1 be non empty TopStruct;
To prove
a1 is sequential
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1 holds
b1 is closed(a1)
iff
for b2 being Function-like quasi_total Relation of NAT,the carrier of a1
st b2 is convergent(a1) & rng b2 c= b1
holds Lim b2 c= b1;
:: FRECHET:def 6
theorem
for b1 being non empty TopStruct holds
b1 is sequential
iff
for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & rng b3 c= b2
holds Lim b3 c= b2;
:: FRECHET:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is first-countable
holds b1 is Frechet;
:: FRECHET:condreg 2
registration
cluster non empty TopSpace-like first-countable -> Frechet (TopStruct);
end;
:: FRECHET:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed(b1)
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & rng b3 c= b2
holds Lim b3 c= b2;
:: FRECHET:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being Element of bool the carrier of b1
st for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
st b3 is convergent(b1) & rng b3 c= b2
holds Lim b3 c= b2
holds b2 is closed(b1)
holds b1 is sequential;
:: FRECHET:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is Frechet
holds b1 is sequential;
:: FRECHET:condreg 3
registration
cluster non empty TopSpace-like Frechet -> sequential (TopStruct);
end;
:: FRECHET:funcnot 2 => FRECHET:func 2
definition
func REAL? -> non empty strict TopSpace-like TopStruct means
the carrier of it = (REAL \ NAT) \/ {REAL} &
(ex b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of it st
b1 = (id REAL) +* (NAT --> REAL) &
(for b2 being Element of bool the carrier of it holds
b2 is closed(it)
iff
b1 " b2 is closed(R^1)));
end;
:: FRECHET:def 7
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
b1 = REAL?
iff
the carrier of b1 = (REAL \ NAT) \/ {REAL} &
(ex b2 being Function-like quasi_total Relation of the carrier of R^1,the carrier of b1 st
b2 = (id REAL) +* (NAT --> REAL) &
(for b3 being Element of bool the carrier of b1 holds
b3 is closed(b1)
iff
b2 " b3 is closed(R^1)));
:: FRECHET:th 30
theorem
REAL is Element of the carrier of REAL?;
:: FRECHET:th 31
theorem
for b1 being Element of bool the carrier of REAL? holds
b1 is open(REAL?) & REAL in b1
iff
ex b2 being Element of bool the carrier of R^1 st
b2 is open(R^1) &
NAT c= b2 &
b1 = (b2 \ NAT) \/ {REAL};
:: FRECHET:th 32
theorem
for b1 being set holds
b1 is Element of bool the carrier of REAL? & not REAL in b1
iff
b1 is Element of bool the carrier of R^1 & NAT /\ b1 = {};
:: FRECHET:th 33
theorem
for b1 being Element of bool the carrier of R^1
for b2 being Element of bool the carrier of REAL?
st b1 = b2
holds NAT /\ b1 = {} & b1 is open(R^1)
iff
not REAL in b2 & b2 is open(REAL?);
:: FRECHET:th 34
theorem
for b1 being Element of bool the carrier of REAL?
st b1 = {REAL}
holds b1 is closed(REAL?);
:: FRECHET:th 35
theorem
REAL? is not first-countable;
:: FRECHET:th 36
theorem
REAL? is Frechet;
:: FRECHET:th 37
theorem
ex b1 being non empty TopSpace-like TopStruct st
b1 is Frechet & b1 is not first-countable;
:: FRECHET:th 39
theorem
for b1, b2 being Relation-like Function-like set
st b1 tolerates b2
holds proj2 (b1 +* b2) = (proj2 b1) \/ proj2 b2;
:: FRECHET:th 40
theorem
for b1 being Element of REAL
st 0 < b1
holds ex b2 being Element of NAT st
1 / b2 < b1 & 0 < b2;