Article TOPGEN_3, MML version 4.99.1005
:: TOPGEN_3:attrnot 1 => TOPGEN_3:attr 1
definition
let a1 be set;
let a2 be Element of bool bool a1;
attr a2 is point-filtered means
for b1, b2, b3 being set
st b2 in a2 & b3 in a2 & b1 in b2 /\ b3
holds ex b4 being Element of bool a1 st
b4 in a2 & b1 in b4 & b4 c= b2 /\ b3;
end;
:: TOPGEN_3:dfs 1
definiens
let a1 be set;
let a2 be Element of bool bool a1;
To prove
a2 is point-filtered
it is sufficient to prove
thus for b1, b2, b3 being set
st b2 in a2 & b3 in a2 & b1 in b2 /\ b3
holds ex b4 being Element of bool a1 st
b4 in a2 & b1 in b4 & b4 c= b2 /\ b3;
:: TOPGEN_3:def 1
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is point-filtered(b1)
iff
for b3, b4, b5 being set
st b4 in b2 & b5 in b2 & b3 in b4 /\ b5
holds ex b6 being Element of bool b1 st
b6 in b2 & b3 in b6 & b6 c= b4 /\ b5;
:: TOPGEN_3:th 1
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is covering(bool bool b1)
iff
for b3 being set
st b3 in b1
holds ex b4 being Element of bool b1 st
b4 in b2 & b3 in b4;
:: TOPGEN_3:th 2
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 is point-filtered(b1) & b2 is covering(bool bool b1)
for b3 being TopStruct
st the carrier of b3 = b1 & the topology of b3 = UniCl b2
holds b3 is TopSpace-like TopStruct & b2 is Basis of b3;
:: TOPGEN_3:th 3
theorem
for b1 being set
for b2 being non-empty ManySortedSet of b1
st proj2 b2 c= bool bool b1 &
(for b3, b4 being set
st b3 in b1 & b4 in b2 . b3
holds b3 in b4) &
(for b3, b4, b5 being set
st b3 in b5 & b5 in b2 . b4 & b4 in b1
holds ex b6 being set st
b6 in b2 . b3 & b6 c= b5) &
(for b3, b4, b5 being set
st b3 in b1 & b4 in b2 . b3 & b5 in b2 . b3
holds ex b6 being set st
b6 in b2 . b3 & b6 c= b4 /\ b5)
holds ex b3 being Element of bool bool b1 st
b3 = Union b2 &
(for b4 being TopStruct
st the carrier of b4 = b1 & the topology of b4 = UniCl b3
holds b4 is TopSpace-like TopStruct &
(for b5 being non empty TopSpace-like TopStruct
st b5 = b4
holds b2 is Neighborhood_System of b5));
:: TOPGEN_3:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1
st {} in b2 &
b1 in b2 &
(for b3, b4 being set
st b3 in b2 & b4 in b2
holds b3 \/ b4 in b2) &
(for b3 being Element of bool bool b1
st b3 c= b2
holds Intersect b3 in b2)
for b3 being TopStruct
st the carrier of b3 = b1 & the topology of b3 = COMPLEMENT b2
holds b3 is TopSpace-like TopStruct &
(for b4 being Element of bool the carrier of b3 holds
b4 is closed(b3)
iff
b4 in b2);
:: TOPGEN_3:sch 1
scheme TOPGEN_3:sch 1
{F1 -> set}:
ex b1 being strict TopSpace-like TopStruct st
the carrier of b1 = F1() &
(for b2 being Element of bool the carrier of b1 holds
b2 is closed(b1)
iff
P1[b2])
provided
P1[{}] & P1[F1()]
and
for b1, b2 being set
st P1[b1] & P1[b2]
holds P1[b1 \/ b2]
and
for b1 being Element of bool bool F1()
st for b2 being set
st b2 in b1
holds P1[b2]
holds P1[Intersect b1];
:: TOPGEN_3:th 5
theorem
for b1, b2 being TopSpace-like TopStruct
st for b3 being set holds
b3 is open Element of bool the carrier of b1
iff
b3 is open Element of bool the carrier of b2
holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);
:: TOPGEN_3:th 6
theorem
for b1, b2 being TopSpace-like TopStruct
st for b3 being set holds
b3 is closed Element of bool the carrier of b1
iff
b3 is closed Element of bool the carrier of b2
holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);
:: TOPGEN_3:funcnot 1 => TOPGEN_3:func 1
definition
let a1, a2 be set;
let a3 be Element of bool [:a1,bool a2:];
redefine func rng a3 -> Element of bool bool a2;
end;
:: TOPGEN_3:th 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of bool b1,bool b1
st b2 . {} = {} &
(for b3 being Element of bool b1 holds
b3 c= b2 . b3) &
(for b3, b4 being Element of bool b1 holds
b2 . (b3 \/ b4) = (b2 . b3) \/ (b2 . b4)) &
(for b3 being Element of bool b1 holds
b2 . (b2 . b3) = b2 . b3)
for b3 being TopStruct
st the carrier of b3 = b1 & the topology of b3 = COMPLEMENT rng b2
holds b3 is TopSpace-like TopStruct &
(for b4 being Element of bool the carrier of b3 holds
Cl b4 = b2 . b4);
:: TOPGEN_3:sch 2
scheme TOPGEN_3:sch 2
{F1 -> set,
F2 -> set}:
ex b1 being strict TopSpace-like TopStruct st
the carrier of b1 = F1() &
(for b2 being Element of bool the carrier of b1 holds
Cl b2 = F2(b2))
provided
F2({}) = {}
and
for b1 being Element of bool F1() holds
b1 c= F2(b1) & F2(b1) c= F1()
and
for b1, b2 being Element of bool F1() holds
F2(b1 \/ b2) = F2(b1) \/ F2(b2)
and
for b1 being Element of bool F1() holds
F2(F2(b1)) = F2(b1);
:: TOPGEN_3:th 8
theorem
for b1, b2 being TopSpace-like TopStruct
st the carrier of b1 = the carrier of b2 &
(for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds Cl b3 = Cl b4)
holds the topology of b1 = the topology of b2;
:: TOPGEN_3:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of bool b1,bool b1
st b2 . b1 = b1 &
(for b3 being Element of bool b1 holds
b2 . b3 c= b3) &
(for b3, b4 being Element of bool b1 holds
b2 . (b3 /\ b4) = (b2 . b3) /\ (b2 . b4)) &
(for b3 being Element of bool b1 holds
b2 . (b2 . b3) = b2 . b3)
for b3 being TopStruct
st the carrier of b3 = b1 & the topology of b3 = rng b2
holds b3 is TopSpace-like TopStruct &
(for b4 being Element of bool the carrier of b3 holds
Int b4 = b2 . b4);
:: TOPGEN_3:sch 3
scheme TOPGEN_3:sch 3
{F1 -> set,
F2 -> set}:
ex b1 being strict TopSpace-like TopStruct st
the carrier of b1 = F1() &
(for b2 being Element of bool the carrier of b1 holds
Int b2 = F2(b2))
provided
F2(F1()) = F1()
and
for b1 being Element of bool F1() holds
F2(b1) c= b1
and
for b1, b2 being Element of bool F1() holds
F2(b1 /\ b2) = F2(b1) /\ F2(b2)
and
for b1 being Element of bool F1() holds
F2(F2(b1)) = F2(b1);
:: TOPGEN_3:th 10
theorem
for b1, b2 being TopSpace-like TopStruct
st the carrier of b1 = the carrier of b2 &
(for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds Int b3 = Int b4)
holds the topology of b1 = the topology of b2;
:: TOPGEN_3:funcnot 2 => TOPGEN_3:func 2
definition
func Sorgenfrey-line -> non empty strict TopSpace-like TopStruct means
the carrier of it = REAL &
(ex b1 being Element of bool bool REAL st
the topology of it = UniCl b1 &
b1 = {[.b2,b3.[ where b2 is Element of REAL, b3 is Element of REAL: b2 < b3 & b3 is rational});
end;
:: TOPGEN_3:def 2
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
b1 = Sorgenfrey-line
iff
the carrier of b1 = REAL &
(ex b2 being Element of bool bool REAL st
the topology of b1 = UniCl b2 &
b2 = {[.b3,b4.[ where b3 is Element of REAL, b4 is Element of REAL: b3 < b4 & b4 is rational});
:: TOPGEN_3:th 11
theorem
for b1, b2 being real set holds
[.b1,b2.[ is open Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_3:th 12
theorem
for b1, b2 being real set holds
].b1,b2.[ is open Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_3:th 13
theorem
for b1 being real set holds
halfline b1 is open Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_3:th 14
theorem
for b1 being real set holds
right_open_halfline b1 is open Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_3:th 15
theorem
for b1 being real set holds
right_closed_halfline b1 is open Element of bool the carrier of Sorgenfrey-line;
:: TOPGEN_3:th 16
theorem
Card INT = alef 0;
:: TOPGEN_3:th 17
theorem
Card RAT = alef 0;
:: TOPGEN_3:th 18
theorem
for b1 being set
st b1 is mutually-disjoint &
(for b2 being set
st b2 in b1
holds ex b3, b4 being real set st
b3 < b4 & ].b3,b4.[ c= b2)
holds b1 is countable;
:: TOPGEN_3:prednot 1 => TOPGEN_3:pred 1
definition
let a1 be set;
let a2 be real set;
pred A2 is_local_minimum_of A1 means
a2 in a1 &
(ex b1 being real set st
b1 < a2 & ].b1,a2.[ misses a1);
end;
:: TOPGEN_3:dfs 3
definiens
let a1 be set;
let a2 be real set;
To prove
a2 is_local_minimum_of a1
it is sufficient to prove
thus a2 in a1 &
(ex b1 being real set st
b1 < a2 & ].b1,a2.[ misses a1);
:: TOPGEN_3:def 3
theorem
for b1 being set
for b2 being real set holds
b2 is_local_minimum_of b1
iff
b2 in b1 &
(ex b3 being real set st
b3 < b2 & ].b3,b2.[ misses b1);
:: TOPGEN_3:th 19
theorem
for b1 being Element of bool REAL holds
{b2 where b2 is Element of REAL: b2 is_local_minimum_of b1} is countable;
:: TOPGEN_3:funcreg 1
registration
let a1 be infinite cardinal set;
cluster exp(2,a1) -> infinite cardinal;
end;
:: TOPGEN_3:funcnot 3 => TOPGEN_3:func 3
definition
func continuum -> infinite cardinal set equals
Card REAL;
end;
:: TOPGEN_3:def 4
theorem
continuum = Card REAL;
:: TOPGEN_3:th 20
theorem
Card {[.b1,b2.[ where b1 is Element of REAL, b2 is Element of REAL: b1 < b2 & b2 is rational} = continuum;
:: TOPGEN_3:funcnot 4 => TOPGEN_3:func 4
definition
let a1 be set;
let a2 be real set;
func A1 -powers A2 -> Function-like quasi_total Relation of NAT,REAL means
for b1 being natural set
st (b1 in a1 implies it . b1 <> a2 |^ b1)
holds not b1 in a1 & it . b1 = 0;
end;
:: TOPGEN_3:def 5
theorem
for b1 being set
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = b1 -powers b2
iff
for b4 being natural set
st (b4 in b1 implies b3 . b4 <> b2 |^ b4)
holds not b4 in b1 & b3 . b4 = 0;
:: TOPGEN_3:th 21
theorem
for b1 being set
for b2 being real set
st 0 < b2 & b2 < 1
holds b1 -powers b2 is summable;
:: TOPGEN_3:th 22
theorem
for b1 being real set
for b2 being Element of NAT
st 0 < b1 & b1 < 1
holds Sum (b1 GeoSeq ^\ b2) = (b1 |^ b2) / (1 - b1);
:: TOPGEN_3:th 23
theorem
for b1 being Element of NAT holds
Sum ((1 / 2) GeoSeq ^\ (b1 + 1)) = (1 / 2) |^ b1;
:: TOPGEN_3:th 24
theorem
for b1 being real set
for b2 being set
st 0 < b1 & b1 < 1
holds Sum (b2 -powers b1) <= Sum (b1 GeoSeq);
:: TOPGEN_3:th 25
theorem
for b1 being set
for b2 being Element of NAT holds
Sum ((b1 -powers (1 / 2)) ^\ (b2 + 1)) <= (1 / 2) |^ b2;
:: TOPGEN_3:th 26
theorem
for b1 being infinite Element of bool NAT
for b2 being natural set holds
(Partial_Sums (b1 -powers (1 / 2))) . b2 < Sum (b1 -powers (1 / 2));
:: TOPGEN_3:th 27
theorem
for b1, b2 being infinite Element of bool NAT
st Sum (b1 -powers (1 / 2)) = Sum (b2 -powers (1 / 2))
holds b1 = b2;
:: TOPGEN_3:th 28
theorem
for b1 being set
st b1 is countable
holds Fin b1 is countable;
:: TOPGEN_3:th 29
theorem
continuum = exp(2,alef 0);
:: TOPGEN_3:th 30
theorem
alef 0 in continuum;
:: TOPGEN_3:th 31
theorem
for b1 being Element of bool bool REAL
st Card b1 in continuum
holds Card {b2 where b2 is Element of REAL: ex b3 being set st
b3 in UniCl b1 & b2 is_local_minimum_of b3} in continuum;
:: TOPGEN_3:th 32
theorem
for b1 being Element of bool bool REAL
st Card b1 in continuum
holds ex b2 being real set st
ex b3 being rational set st
b2 < b3 & not [.b2,b3.[ in UniCl b1;
:: TOPGEN_3:th 33
theorem
weight Sorgenfrey-line = continuum;
:: TOPGEN_3:funcnot 5 => TOPGEN_3:func 5
definition
let a1 be set;
func ClFinTop A1 -> strict TopSpace-like TopStruct means
the carrier of it = a1 &
(for b1 being Element of bool the carrier of it holds
b1 is closed(it)
iff
(b1 is finite or b1 = a1));
end;
:: TOPGEN_3:def 6
theorem
for b1 being set
for b2 being strict TopSpace-like TopStruct holds
b2 = ClFinTop b1
iff
the carrier of b2 = b1 &
(for b3 being Element of bool the carrier of b2 holds
b3 is closed(b2)
iff
(b3 is finite or b3 = b1));
:: TOPGEN_3:th 34
theorem
for b1 being set
for b2 being Element of bool the carrier of ClFinTop b1 holds
b2 is open(ClFinTop b1)
iff
(b2 = {} or b2 ` is finite);
:: TOPGEN_3:th 35
theorem
for b1 being infinite set
for b2 being Element of bool b1
st b2 is finite
holds b2 ` is infinite;
:: TOPGEN_3:funcreg 2
registration
let a1 be non empty set;
cluster ClFinTop a1 -> non empty strict TopSpace-like;
end;
:: TOPGEN_3:th 36
theorem
for b1 being infinite set
for b2, b3 being non empty open Element of bool the carrier of ClFinTop b1 holds
b2 meets b3;
:: TOPGEN_3:funcnot 6 => TOPGEN_3:func 6
definition
let a1, a2 be set;
func A2 -PointClTop A1 -> strict TopSpace-like TopStruct means
the carrier of it = a1 &
(for b1 being Element of bool the carrier of it holds
Cl b1 = IFEQ(b1,{},b1,b1 \/ ({a2} /\ a1)));
end;
:: TOPGEN_3:def 7
theorem
for b1, b2 being set
for b3 being strict TopSpace-like TopStruct holds
b3 = b2 -PointClTop b1
iff
the carrier of b3 = b1 &
(for b4 being Element of bool the carrier of b3 holds
Cl b4 = IFEQ(b4,{},b4,b4 \/ ({b2} /\ b1)));
:: TOPGEN_3:funcreg 3
registration
let a1 be non empty set;
let a2 be set;
cluster a2 -PointClTop a1 -> non empty strict TopSpace-like;
end;
:: TOPGEN_3:th 37
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty Element of bool the carrier of b2 -PointClTop b1 holds
Cl b3 = b3 \/ {b2};
:: TOPGEN_3:th 38
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty Element of bool the carrier of b2 -PointClTop b1 holds
b3 is closed(b2 -PointClTop b1)
iff
b2 in b3;
:: TOPGEN_3:th 39
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being proper Element of bool the carrier of b2 -PointClTop b1 holds
b3 is open(b2 -PointClTop b1)
iff
not b2 in b3;
:: TOPGEN_3:th 40
theorem
for b1, b2, b3 being set
st b2 in b1
holds {b3} is closed Element of bool the carrier of b2 -PointClTop b1
iff
b3 = b2;
:: TOPGEN_3:th 41
theorem
for b1, b2, b3 being set
st {b2} c< b1
holds {b3} is open Element of bool the carrier of b2 -PointClTop b1
iff
b3 in b1 & b3 <> b2;
:: TOPGEN_3:funcnot 7 => TOPGEN_3:func 7
definition
let a1, a2 be set;
func A2 -DiscreteTop A1 -> strict TopSpace-like TopStruct means
the carrier of it = a1 &
(for b1 being Element of bool the carrier of it holds
Int b1 = IFEQ(b1,a1,b1,b1 /\ a2));
end;
:: TOPGEN_3:def 8
theorem
for b1, b2 being set
for b3 being strict TopSpace-like TopStruct holds
b3 = b2 -DiscreteTop b1
iff
the carrier of b3 = b1 &
(for b4 being Element of bool the carrier of b3 holds
Int b4 = IFEQ(b4,b1,b4,b4 /\ b2));
:: TOPGEN_3:funcreg 4
registration
let a1 be non empty set;
let a2 be set;
cluster a2 -DiscreteTop a1 -> non empty strict TopSpace-like;
end;
:: TOPGEN_3:th 42
theorem
for b1 being non empty set
for b2 being set
for b3 being proper Element of bool the carrier of b2 -DiscreteTop b1 holds
Int b3 = b3 /\ b2;
:: TOPGEN_3:th 43
theorem
for b1 being non empty set
for b2 being set
for b3 being proper Element of bool the carrier of b2 -DiscreteTop b1 holds
b3 is open(b2 -DiscreteTop b1)
iff
b3 c= b2;
:: TOPGEN_3:th 44
theorem
for b1 being set
for b2 being Element of bool b1 holds
the topology of b2 -DiscreteTop b1 = {b1} \/ bool b2;
:: TOPGEN_3:th 45
theorem
for b1 being set holds
ADTS b1 = {} -DiscreteTop b1;
:: TOPGEN_3:th 46
theorem
for b1 being set holds
1TopSp b1 = b1 -DiscreteTop b1;