Article TOPGEN_2, MML version 4.99.1005
:: TOPGEN_2:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Element of the carrier of b1 holds
{b4 where b4 is Element of bool the carrier of b1: b3 in b4 & b4 in b2} is Basis of b3;
:: TOPGEN_2:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being ManySortedSet of the carrier of b1
st for b3 being Element of the carrier of b1 holds
b2 . b3 is Basis of b3
holds Union b2 is Basis of b1;
:: TOPGEN_2:funcnot 1 => TOPGEN_2:func 1
definition
let a1 be non empty TopStruct;
let a2 be Element of the carrier of a1;
func Chi(A2,A1) -> cardinal set means
(ex b1 being Basis of a2 st
it = Card b1) &
(for b1 being Basis of a2 holds
it c= Card b1);
end;
:: TOPGEN_2:def 1
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being cardinal set holds
b3 = Chi(b2,b1)
iff
(ex b4 being Basis of b2 st
b3 = Card b4) &
(for b4 being Basis of b2 holds
b3 c= Card b4);
:: TOPGEN_2:th 3
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is cardinal set
holds union b1 is cardinal set;
:: TOPGEN_2:funcnot 2 => TOPGEN_2:func 2
definition
let a1 be non empty TopStruct;
func Chi A1 -> cardinal set means
(for b1 being Element of the carrier of a1 holds
Chi(b1,a1) c= it) &
(for b1 being cardinal set
st for b2 being Element of the carrier of a1 holds
Chi(b2,a1) c= b1
holds it c= b1);
end;
:: TOPGEN_2:def 2
theorem
for b1 being non empty TopStruct
for b2 being cardinal set holds
b2 = Chi b1
iff
(for b3 being Element of the carrier of b1 holds
Chi(b3,b1) c= b2) &
(for b3 being cardinal set
st for b4 being Element of the carrier of b1 holds
Chi(b4,b1) c= b3
holds b2 c= b3);
:: TOPGEN_2:th 4
theorem
for b1 being non empty TopStruct holds
Chi b1 = union {Chi(b2,b1) where b2 is Element of the carrier of b1: TRUE};
:: TOPGEN_2:th 5
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1 holds
Chi(b2,b1) c= Chi b1;
:: TOPGEN_2:th 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
b1 is first-countable
iff
Chi b1 c= alef {};
:: TOPGEN_2:modenot 1 => TOPGEN_2:mode 1
definition
let a1 be non empty TopSpace-like TopStruct;
mode Neighborhood_System of A1 -> ManySortedSet of the carrier of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 is Basis of b1;
end;
:: TOPGEN_2:dfs 3
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be ManySortedSet of the carrier of a1;
To prove
a2 is Neighborhood_System of a1
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 . b1 is Basis of b1;
:: TOPGEN_2:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being ManySortedSet of the carrier of b1 holds
b2 is Neighborhood_System of b1
iff
for b3 being Element of the carrier of b1 holds
b2 . b3 is Basis of b3;
:: TOPGEN_2:funcnot 3 => TOPGEN_2:func 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Neighborhood_System of a1;
redefine func Union a2 -> Basis of a1;
end;
:: TOPGEN_2:funcnot 4 => TOPGEN_2:func 4
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be Neighborhood_System of a1;
let a3 be Element of the carrier of a1;
redefine func a2 . a3 -> Basis of a3;
end;
:: TOPGEN_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Basis of b2
for b5 being Basis of b3
for b6 being set
st b2 in b6 & b6 in b5
holds ex b7 being open Element of bool the carrier of b1 st
b7 in b4 & b7 c= b6;
:: TOPGEN_2:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2
for b4, b5 being set
st b4 in b3 & b5 in b3
holds ex b6 being open Element of bool the carrier of b1 st
b6 in b3 & b6 c= b4 /\ b5;
:: TOPGEN_2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in Cl b2
iff
for b4 being Basis of b3
for b5 being set
st b5 in b4
holds b5 meets b2;
:: TOPGEN_2:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in Cl b2
iff
ex b4 being Basis of b3 st
for b5 being set
st b5 in b4
holds b5 meets b2;
:: TOPGEN_2:exreg 1
registration
let a1 be TopSpace-like TopStruct;
cluster non empty open Element of bool bool the carrier of a1;
end;
:: TOPGEN_2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being open Element of bool bool the carrier of b1 holds
ex b3 being open Element of bool bool the carrier of b1 st
b3 c= b2 & union b3 = union b2 & Card b3 c= weight b1;
:: TOPGEN_2:attrnot 1 => TOPGEN_2:attr 1
definition
let a1 be TopStruct;
attr a1 is finite-weight means
weight a1 is finite;
end;
:: TOPGEN_2:dfs 4
definiens
let a1 be TopStruct;
To prove
a1 is finite-weight
it is sufficient to prove
thus weight a1 is finite;
:: TOPGEN_2:def 4
theorem
for b1 being TopStruct holds
b1 is finite-weight
iff
weight b1 is finite;
:: TOPGEN_2:attrnot 2 => TOPGEN_2:attr 1
notation
let a1 be TopStruct;
antonym infinite-weight for finite-weight;
end;
:: TOPGEN_2:condreg 1
registration
cluster finite -> finite-weight (TopStruct);
end;
:: TOPGEN_2:condreg 2
registration
cluster infinite-weight -> infinite (TopStruct);
end;
:: TOPGEN_2:exreg 2
registration
cluster non empty finite TopSpace-like TopStruct;
end;
:: TOPGEN_2:th 13
theorem
for b1 being set holds
Card SmallestPartition b1 = Card b1;
:: TOPGEN_2:th 14
theorem
for b1 being non empty discrete TopStruct holds
SmallestPartition the carrier of b1 is Basis of b1 &
(for b2 being Basis of b1 holds
SmallestPartition the carrier of b1 c= b2);
:: TOPGEN_2:th 15
theorem
for b1 being non empty discrete TopStruct holds
weight b1 = Card the carrier of b1;
:: TOPGEN_2:exreg 3
registration
cluster TopSpace-like infinite-weight TopStruct;
end;
:: TOPGEN_2:th 17
theorem
for b1 being non empty TopSpace-like finite-weight TopStruct holds
ex b2 being Basis of b1 st
ex b3 being Function-like quasi_total Relation of the carrier of b1,the topology of b1 st
b2 = rng b3 &
(for b4 being Element of the carrier of b1 holds
b4 in b3 . b4 &
(for b5 being open Element of bool the carrier of b1
st b4 in b5
holds b3 . b4 c= b5));
:: TOPGEN_2:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Basis of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the topology of b1
st b2 = rng b4 &
(for b5 being Element of the carrier of b1 holds
b5 in b4 . b5 &
(for b6 being open Element of bool the carrier of b1
st b5 in b6
holds b4 . b5 c= b6))
holds b2 c= b3;
:: TOPGEN_2:th 19
theorem
for b1 being TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the topology of b1
st b2 = rng b3 &
(for b4 being Element of the carrier of b1 holds
b4 in b3 . b4 &
(for b5 being open Element of bool the carrier of b1
st b4 in b5
holds b3 . b4 c= b5))
holds weight b1 = Card b2;
:: TOPGEN_2:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1 holds
ex b3 being Basis of b1 st
b3 c= b2 & Card b3 = weight b1;
:: TOPGEN_2:funcnot 5 => TOPGEN_2:func 5
definition
let a1, a2 be set;
func DiscrWithInfin(A1,A2) -> strict TopStruct means
the carrier of it = a1 &
the topology of it = {b1 where b1 is Element of bool a1: not a2 in b1} \/ {b1 ` where b1 is Element of bool a1: b1 is finite};
end;
:: TOPGEN_2:def 5
theorem
for b1, b2 being set
for b3 being strict TopStruct holds
b3 = DiscrWithInfin(b1,b2)
iff
the carrier of b3 = b1 &
the topology of b3 = {b4 where b4 is Element of bool b1: not b2 in b4} \/ {b4 ` where b4 is Element of bool b1: b4 is finite};
:: TOPGEN_2:funcreg 1
registration
let a1, a2 be set;
cluster DiscrWithInfin(a1,a2) -> strict TopSpace-like;
end;
:: TOPGEN_2:funcreg 2
registration
let a1 be non empty set;
let a2 be set;
cluster DiscrWithInfin(a1,a2) -> non empty strict;
end;
:: TOPGEN_2:th 21
theorem
for b1, b2 being set
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2) holds
b3 is open(DiscrWithInfin(b1,b2))
iff
(b2 in b3 implies b3 ` is finite);
:: TOPGEN_2:th 22
theorem
for b1, b2 being set
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2) holds
b3 is closed(DiscrWithInfin(b1,b2))
iff
(b2 in b1 & not b2 in b3 implies b3 is finite);
:: TOPGEN_2:th 23
theorem
for b1, b2, b3 being set
st b3 in b1
holds {b3} is closed Element of bool the carrier of DiscrWithInfin(b1,b2);
:: TOPGEN_2:th 24
theorem
for b1, b2, b3 being set
st b3 in b1 & b3 <> b2
holds {b3} is open Element of bool the carrier of DiscrWithInfin(b1,b2);
:: TOPGEN_2:th 25
theorem
for b1, b2 being set
st b1 is infinite
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2)
st b3 = {b2}
holds b3 is open(not DiscrWithInfin(b1,b2));
:: TOPGEN_2:th 26
theorem
for b1, b2 being set
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2)
st b3 is finite
holds Cl b3 = b3;
:: TOPGEN_2:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 is not closed(b1)
for b3 being Element of the carrier of b1
st b2 \/ {b3} is closed(b1)
holds Cl b2 = b2 \/ {b3};
:: TOPGEN_2:th 28
theorem
for b1, b2 being set
st b2 in b1
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2)
st b3 is infinite
holds Cl b3 = b3 \/ {b2};
:: TOPGEN_2:th 29
theorem
for b1, b2 being set
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2)
st b3 ` is finite
holds Int b3 = b3;
:: TOPGEN_2:th 30
theorem
for b1, b2 being set
st b2 in b1
for b3 being Element of bool the carrier of DiscrWithInfin(b1,b2)
st b3 ` is infinite
holds Int b3 = b3 \ {b2};
:: TOPGEN_2:th 31
theorem
for b1, b2 being set holds
ex b3 being Basis of DiscrWithInfin(b1,b2) st
b3 = ((SmallestPartition b1) \ {{b2}}) \/ {b4 ` where b4 is Element of bool b1: b4 is finite};
:: TOPGEN_2:th 32
theorem
for b1 being infinite set holds
Card Fin b1 = Card b1;
:: TOPGEN_2:th 33
theorem
for b1 being infinite set holds
Card {b2 ` where b2 is Element of bool b1: b2 is finite} = Card b1;
:: TOPGEN_2:th 34
theorem
for b1 being infinite set
for b2 being set
for b3 being Basis of DiscrWithInfin(b1,b2)
st b3 = ((SmallestPartition b1) \ {{b2}}) \/ {b4 ` where b4 is Element of bool b1: b4 is finite}
holds Card b3 = Card b1;
:: TOPGEN_2:th 35
theorem
for b1 being infinite set
for b2 being set
for b3 being Basis of DiscrWithInfin(b1,b2) holds
Card b1 c= Card b3;
:: TOPGEN_2:th 36
theorem
for b1 being infinite set
for b2 being set holds
weight DiscrWithInfin(b1,b2) = Card b1;
:: TOPGEN_2:th 37
theorem
for b1 being non empty set
for b2 being set holds
ex b3 being prebasis of DiscrWithInfin(b1,b2) st
b3 = ((SmallestPartition b1) \ {{b2}}) \/ {{b4} ` where b4 is Element of b1: TRUE};
:: TOPGEN_2:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being non empty Element of bool bool b2
st for b4 being set
st b4 in b3
holds b2 \ b4 is finite
holds Cl union b2 = (union clf b2) \/ meet {Cl union b4 where b4 is Element of bool bool the carrier of b1: b4 in b3};
:: TOPGEN_2:th 39
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
Cl union b2 = (union clf b2) \/ meet {Cl union b3 where b3 is Element of bool bool the carrier of b1: b3 c= b2 & b2 \ b3 is finite};
:: TOPGEN_2:th 40
theorem
for b1 being set
for b2 being Element of bool bool bool b1
st for b3 being Element of bool bool b1
st b3 in b2
holds TopStruct(#b1,b3#) is TopSpace-like TopStruct
holds ex b3 being Element of bool bool b1 st
b3 = Intersect b2 &
TopStruct(#b1,b3#) is TopSpace-like TopStruct &
(for b4 being Element of bool bool b1
st b4 in b2
holds TopStruct(#b1,b4#) is TopExtension of TopStruct(#b1,b3#)) &
(for b4 being TopSpace-like TopStruct
st the carrier of b4 = b1 &
(for b5 being Element of bool bool b1
st b5 in b2
holds TopStruct(#b1,b5#) is TopExtension of b4)
holds TopStruct(#b1,b3#) is TopExtension of b4);
:: TOPGEN_2:th 41
theorem
for b1 being set
for b2 being Element of bool bool bool b1 holds
ex b3 being Element of bool bool b1 st
b3 = UniCl FinMeetCl union b2 &
TopStruct(#b1,b3#) is TopSpace-like TopStruct &
(for b4 being Element of bool bool b1
st b4 in b2
holds TopStruct(#b1,b3#) is TopExtension of TopStruct(#b1,b4#)) &
(for b4 being TopSpace-like TopStruct
st the carrier of b4 = b1 &
(for b5 being Element of bool bool b1
st b5 in b2
holds b4 is TopExtension of TopStruct(#b1,b5#))
holds b4 is TopExtension of TopStruct(#b1,b3#));