Article WAYBEL18, MML version 4.99.1005
:: WAYBEL18:th 2
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
st (b3 = b2 \ {{}} or b2 = b3 \/ {{}})
holds UniCl b2 = UniCl b3;
:: WAYBEL18:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is Basis of b1
iff
b2 \ {{}} is Basis of b1;
:: WAYBEL18:attrnot 1 => WAYBEL18:attr 1
definition
let a1 be Relation-like set;
attr a1 is TopSpace-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is TopStruct;
end;
:: WAYBEL18:dfs 1
definiens
let a1 be Relation-like set;
To prove
a1 is TopSpace-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is TopStruct;
:: WAYBEL18:def 1
theorem
for b1 being Relation-like set holds
b1 is TopSpace-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is TopStruct;
:: WAYBEL18:condreg 1
registration
cluster Relation-like Function-like TopSpace-yielding -> 1-sorted-yielding (set);
end;
:: WAYBEL18:exreg 1
registration
let a1 be set;
cluster Relation-like Function-like TopSpace-yielding ManySortedSet of a1;
end;
:: WAYBEL18:exreg 2
registration
let a1 be set;
cluster Relation-like Function-like non-Empty TopSpace-yielding ManySortedSet of a1;
end;
:: WAYBEL18:funcnot 1 => WAYBEL18:func 1
definition
let a1 be non empty set;
let a2 be TopSpace-yielding ManySortedSet of a1;
let a3 be Element of a1;
redefine func a2 . a3 -> TopStruct;
end;
:: WAYBEL18:funcnot 2 => WAYBEL18:func 2
definition
let a1 be set;
let a2 be TopSpace-yielding ManySortedSet of a1;
func product_prebasis A2 -> Element of bool bool product Carrier a2 means
for b1 being Element of bool product Carrier a2 holds
b1 in it
iff
ex b2 being set st
ex b3 being TopStruct st
ex b4 being Element of bool the carrier of b3 st
b2 in a1 & b4 is open(b3) & b3 = a2 . b2 & b1 = product ((Carrier a2) +*(b2,b4));
end;
:: WAYBEL18:def 2
theorem
for b1 being set
for b2 being TopSpace-yielding ManySortedSet of b1
for b3 being Element of bool bool product Carrier b2 holds
b3 = product_prebasis b2
iff
for b4 being Element of bool product Carrier b2 holds
b4 in b3
iff
ex b5 being set st
ex b6 being TopStruct st
ex b7 being Element of bool the carrier of b6 st
b5 in b1 & b7 is open(b6) & b6 = b2 . b5 & b4 = product ((Carrier b2) +*(b5,b7));
:: WAYBEL18:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
TopStruct(#b1,UniCl FinMeetCl b2#) is TopSpace-like;
:: WAYBEL18:funcnot 3 => WAYBEL18:func 3
definition
let a1 be set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
func product A2 -> strict TopSpace-like TopStruct means
the carrier of it = product Carrier a2 & product_prebasis a2 is prebasis of it;
end;
:: WAYBEL18:def 3
theorem
for b1 being set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being strict TopSpace-like TopStruct holds
b3 = product b2
iff
the carrier of b3 = product Carrier b2 & product_prebasis b2 is prebasis of b3;
:: WAYBEL18:funcreg 1
registration
let a1 be set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
cluster product a2 -> non empty strict TopSpace-like;
end;
:: WAYBEL18:funcnot 4 => WAYBEL18:func 4
definition
let a1 be non empty set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
let a3 be Element of a1;
redefine func a2 . a3 -> non empty TopStruct;
end;
:: WAYBEL18:funcreg 2
registration
let a1 be set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
cluster product a2 -> strict TopSpace-like constituted-Functions;
end;
:: WAYBEL18:funcnot 5 => WAYBEL18:func 5
definition
let a1 be non empty set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
let a3 be Element of the carrier of product a2;
let a4 be Element of a1;
redefine func a3 . a4 -> Element of the carrier of a2 . a4;
end;
:: WAYBEL18:funcnot 6 => WAYBEL18:func 6
definition
let a1 be non empty set;
let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
let a3 be Element of a1;
func proj(A2,A3) -> Function-like quasi_total Relation of the carrier of product a2,the carrier of a2 . a3 equals
proj(Carrier a2,a3);
end;
:: WAYBEL18:def 4
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds
proj(b2,b3) = proj(Carrier b2,b3);
:: WAYBEL18:th 5
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool the carrier of b2 . b3 holds
(proj(b2,b3)) " b4 = product ((Carrier b2) +*(b3,b4));
:: WAYBEL18:th 6
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds
proj(b2,b3) is continuous(product b2, b2 . b3);
:: WAYBEL18:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty set
for b3 being non-Empty TopSpace-yielding ManySortedSet of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of product b3 holds
b4 is continuous(b1, product b3)
iff
for b5 being Element of b2 holds
(proj(b3,b5)) * b4 is continuous(b1, b3 . b5);
:: WAYBEL18:attrnot 2 => WAYBEL18:attr 2
definition
let a1 be TopStruct;
attr a1 is injective means
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of a1
st b2 is continuous(b1, a1)
for b3 being non empty TopSpace-like TopStruct
st b1 is SubSpace of b3
holds ex b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of a1 st
b4 is continuous(b3, a1) & b4 | the carrier of b1 = b2;
end;
:: WAYBEL18:dfs 5
definiens
let a1 be TopStruct;
To prove
a1 is injective
it is sufficient to prove
thus for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of a1
st b2 is continuous(b1, a1)
for b3 being non empty TopSpace-like TopStruct
st b1 is SubSpace of b3
holds ex b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of a1 st
b4 is continuous(b3, a1) & b4 | the carrier of b1 = b2;
:: WAYBEL18:def 5
theorem
for b1 being TopStruct holds
b1 is injective
iff
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
st b3 is continuous(b2, b1)
for b4 being non empty TopSpace-like TopStruct
st b2 is SubSpace of b4
holds ex b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1 st
b5 is continuous(b4, b1) & b5 | the carrier of b2 = b3;
:: WAYBEL18:th 8
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is injective
holds product b2 is injective;
:: WAYBEL18:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is injective
for b2 being non empty SubSpace of b1
st b2 is_a_retract_of b1
holds b2 is injective;
:: WAYBEL18:funcnot 7 => WAYBEL18:func 7
definition
let a1 be 1-sorted;
let a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
func Image A3 -> SubSpace of a2 equals
a2 | rng a3;
end;
:: WAYBEL18:def 6
theorem
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
Image b3 = b2 | rng b3;
:: WAYBEL18:funcreg 3
registration
let a1 be non empty 1-sorted;
let a2 be non empty TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
cluster Image a3 -> non empty;
end;
:: WAYBEL18:th 10
theorem
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
the carrier of Image b3 = rng b3;
:: WAYBEL18:funcnot 8 => WAYBEL18:func 8
definition
let a1 be 1-sorted;
let a2 be non empty TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
func corestr A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of Image a3 equals
a3;
end;
:: WAYBEL18:def 7
theorem
for b1 being 1-sorted
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
corestr b3 = b3;
:: WAYBEL18:th 11
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is continuous(b1, b2)
holds corestr b3 is continuous(b1, Image b3);
:: WAYBEL18:funcreg 4
registration
let a1 be 1-sorted;
let a2 be non empty TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
cluster corestr a3 -> Function-like quasi_total onto;
end;
:: WAYBEL18:prednot 1 => WAYBEL18:pred 1
definition
let a1, a2 be TopStruct;
pred A1 is_Retract_of A2 means
ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a2 st
b1 is continuous(a2, a2) & b1 * b1 = b1 & Image b1,a1 are_homeomorphic;
end;
:: WAYBEL18:dfs 8
definiens
let a1, a2 be TopStruct;
To prove
a1 is_Retract_of a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a2 st
b1 is continuous(a2, a2) & b1 * b1 = b1 & Image b1,a1 are_homeomorphic;
:: WAYBEL18:def 8
theorem
for b1, b2 being TopStruct holds
b1 is_Retract_of b2
iff
ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 st
b3 is continuous(b2, b2) & b3 * b3 = b3 & Image b3,b1 are_homeomorphic;
:: WAYBEL18:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
st b1 is injective
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st corestr b3 is being_homeomorphism(b1, Image b3)
holds b1 is_Retract_of b2;
:: WAYBEL18:funcnot 9 => WAYBEL18:func 9
definition
func Sierpinski_Space -> strict TopStruct means
the carrier of it = {0,1} &
the topology of it = {{},{1},{0,1}};
end;
:: WAYBEL18:def 9
theorem
for b1 being strict TopStruct holds
b1 = Sierpinski_Space
iff
the carrier of b1 = {0,1} &
the topology of b1 = {{},{1},{0,1}};
:: WAYBEL18:funcreg 5
registration
cluster Sierpinski_Space -> non empty strict TopSpace-like;
end;
:: WAYBEL18:funcreg 6
registration
cluster Sierpinski_Space -> strict discerning;
end;
:: WAYBEL18:funcreg 7
registration
cluster Sierpinski_Space -> strict injective;
end;
:: WAYBEL18:funcreg 8
registration
let a1 be set;
let a2 be non empty 1-sorted;
cluster a1 --> a2 -> non-Empty;
end;
:: WAYBEL18:funcreg 9
registration
let a1 be set;
let a2 be TopStruct;
cluster a1 --> a2 -> TopSpace-yielding;
end;
:: WAYBEL18:funcreg 10
registration
let a1 be set;
let a2 be reflexive RelStr;
cluster a1 --> a2 -> reflexive-yielding;
end;
:: WAYBEL18:funcreg 11
registration
let a1 be non empty set;
let a2 be non empty antisymmetric RelStr;
cluster product (a1 --> a2) -> strict antisymmetric;
end;
:: WAYBEL18:funcreg 12
registration
let a1 be non empty set;
let a2 be non empty transitive RelStr;
cluster product (a1 --> a2) -> strict transitive;
end;
:: WAYBEL18:th 13
theorem
for b1 being Scott TopAugmentation of BoolePoset 1 holds
the topology of b1 = the topology of Sierpinski_Space;
:: WAYBEL18:th 14
theorem
for b1 being non empty set holds
{product ((Carrier (b1 --> Sierpinski_Space)) +*(b2,{1})) where b2 is Element of b1: TRUE} is prebasis of product (b1 --> Sierpinski_Space);
:: WAYBEL18:funcreg 13
registration
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
cluster product (a1 --> a2) -> strict with_suprema complete;
end;
:: WAYBEL18:funcreg 14
registration
let a1 be non empty set;
let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr;
cluster product (a1 --> a2) -> strict algebraic;
end;
:: WAYBEL18:th 15
theorem
for b1 being non empty set holds
ex b2 being Function-like quasi_total Relation of the carrier of BoolePoset b1,the carrier of product (b1 --> BoolePoset 1) st
b2 is isomorphic(BoolePoset b1, product (b1 --> BoolePoset 1)) &
(for b3 being Element of bool b1 holds
b2 . b3 = chi(b3,b1));
:: WAYBEL18:th 16
theorem
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> BoolePoset 1) holds
the topology of b2 = the topology of product (b1 --> Sierpinski_Space);
:: WAYBEL18:th 17
theorem
for b1, b2 being non empty TopSpace-like TopStruct
st the carrier of b1 = the carrier of b2 & the topology of b1 = the topology of b2 & b1 is injective
holds b2 is injective;
:: WAYBEL18:th 18
theorem
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> BoolePoset 1) holds
b2 is injective;
:: WAYBEL18:th 19
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
ex b2 being non empty set st
ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of product (b2 --> Sierpinski_Space) st
corestr b3 is being_homeomorphism(b1, Image b3);
:: WAYBEL18:th 20
theorem
for b1 being non empty TopSpace-like discerning TopStruct
st b1 is injective
holds ex b2 being non empty set st
b1 is_Retract_of product (b2 --> Sierpinski_Space);