Article CANTOR_1, MML version 4.99.1005
:: CANTOR_1:funcreg 1
registration
let a1 be set;
let a2 be non empty set;
cluster a1 --> a2 -> non-empty;
end;
:: CANTOR_1:funcnot 1 => CANTOR_1:func 1
definition
let a1 be set;
let a2 be Element of bool bool a1;
func UniCl A2 -> Element of bool bool a1 means
for b1 being Element of bool a1 holds
b1 in it
iff
ex b2 being Element of bool bool a1 st
b2 c= a2 & b1 = union b2;
end;
:: CANTOR_1:def 1
theorem
for b1 being set
for b2, b3 being Element of bool bool b1 holds
b3 = UniCl b2
iff
for b4 being Element of bool b1 holds
b4 in b3
iff
ex b5 being Element of bool bool b1 st
b5 c= b2 & b4 = union b5;
:: CANTOR_1:modenot 1 => CANTOR_1:mode 1
definition
let a1 be TopStruct;
mode Basis of A1 -> Element of bool bool the carrier of a1 means
it c= the topology of a1 & the topology of a1 c= UniCl it;
end;
:: CANTOR_1:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is Basis of a1
it is sufficient to prove
thus a2 c= the topology of a1 & the topology of a1 c= UniCl a2;
:: CANTOR_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is Basis of b1
iff
b2 c= the topology of b1 & the topology of b1 c= UniCl b2;
:: CANTOR_1:th 1
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 c= UniCl b2;
:: CANTOR_1:th 2
theorem
for b1 being TopStruct holds
the topology of b1 is Basis of b1;
:: CANTOR_1:th 3
theorem
for b1 being TopStruct holds
the topology of b1 is open(b1);
:: CANTOR_1:funcnot 2 => CANTOR_1:func 2
definition
let a1 be set;
let a2 be Element of bool bool a1;
func FinMeetCl A2 -> Element of bool bool a1 means
for b1 being Element of bool a1 holds
b1 in it
iff
ex b2 being Element of bool bool a1 st
b2 c= a2 & b2 is finite & b1 = Intersect b2;
end;
:: CANTOR_1:def 4
theorem
for b1 being set
for b2, b3 being Element of bool bool b1 holds
b3 = FinMeetCl b2
iff
for b4 being Element of bool b1 holds
b4 in b3
iff
ex b5 being Element of bool bool b1 st
b5 c= b2 & b5 is finite & b4 = Intersect b5;
:: CANTOR_1:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 c= FinMeetCl b2;
:: CANTOR_1:funcreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
cluster the topology of a1 -> non empty;
end;
:: CANTOR_1:th 5
theorem
for b1 being non empty TopSpace-like TopStruct holds
the topology of b1 = FinMeetCl the topology of b1;
:: CANTOR_1:th 6
theorem
for b1 being TopSpace-like TopStruct holds
the topology of b1 = UniCl the topology of b1;
:: CANTOR_1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct holds
the topology of b1 = UniCl FinMeetCl the topology of b1;
:: CANTOR_1:th 8
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b1 in FinMeetCl b2;
:: CANTOR_1:th 9
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
st b2 c= b3
holds UniCl b2 c= UniCl b3;
:: CANTOR_1:th 12
theorem
for b1 being set
for b2 being non empty Element of bool bool bool b1
for b3 being Element of bool bool b1
st b3 = {Intersect b4 where b4 is Element of b2: TRUE}
holds Intersect b3 = Intersect union b2;
:: CANTOR_1:funcnot 3 => CANTOR_1:func 3
definition
let a1, a2 be set;
let a3 be Element of bool bool a1;
let a4 be Function-like quasi_total Relation of a2,bool a3;
let a5 be set;
redefine func a4 . a5 -> Element of bool bool a1;
end;
:: CANTOR_1:th 13
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
FinMeetCl b2 = FinMeetCl FinMeetCl b2;
:: CANTOR_1:th 14
theorem
for b1 being set
for b2 being Element of bool bool b1
for b3, b4 being set
st b3 in FinMeetCl b2 & b4 in FinMeetCl b2
holds b3 /\ b4 in FinMeetCl b2;
:: CANTOR_1:th 15
theorem
for b1 being set
for b2 being Element of bool bool b1
for b3, b4 being set
st b3 c= FinMeetCl b2 & b4 c= FinMeetCl b2
holds INTERSECTION(b3,b4) c= FinMeetCl b2;
:: CANTOR_1:th 16
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
st b2 c= b3
holds FinMeetCl b2 c= FinMeetCl b3;
:: CANTOR_1:funcreg 3
registration
let a1 be set;
let a2 be Element of bool bool a1;
cluster FinMeetCl a2 -> non empty;
end;
:: CANTOR_1:th 17
theorem
for b1 being non empty set
for b2 being Element of bool bool b1 holds
TopStruct(#b1,UniCl FinMeetCl b2#) is TopSpace-like;
:: CANTOR_1:modenot 2 => CANTOR_1:mode 2
definition
let a1 be TopStruct;
mode prebasis of A1 -> Element of bool bool the carrier of a1 means
it c= the topology of a1 &
(ex b1 being Basis of a1 st
b1 c= FinMeetCl it);
end;
:: CANTOR_1:dfs 4
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is prebasis of a1
it is sufficient to prove
thus a2 c= the topology of a1 &
(ex b1 being Basis of a1 st
b1 c= FinMeetCl a2);
:: CANTOR_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is prebasis of b1
iff
b2 c= the topology of b1 &
(ex b3 being Basis of b1 st
b3 c= FinMeetCl b2);
:: CANTOR_1:th 18
theorem
for b1 being non empty set
for b2 being Element of bool bool b1 holds
b2 is Basis of TopStruct(#b1,UniCl b2#);
:: CANTOR_1:th 19
theorem
for b1, b2 being non empty strict TopSpace-like TopStruct
for b3 being prebasis of b1
st the carrier of b1 = the carrier of b2 & b3 is prebasis of b2
holds b1 = b2;
:: CANTOR_1:th 20
theorem
for b1 being non empty set
for b2 being Element of bool bool b1 holds
b2 is prebasis of TopStruct(#b1,UniCl FinMeetCl b2#);
:: CANTOR_1:funcnot 4 => CANTOR_1:func 4
definition
func the_Cantor_set -> non empty strict TopSpace-like TopStruct means
the carrier of it = product (omega --> {{},1}) &
(ex b1 being prebasis of it st
for b2 being Element of bool product (omega --> {{},1}) holds
b2 in b1
iff
ex b3, b4 being natural set st
for b5 being Element of product (omega --> {{},1}) holds
b5 in b2
iff
b5 . b3 = b4);
end;
:: CANTOR_1:def 6
theorem
for b1 being non empty strict TopSpace-like TopStruct holds
b1 = the_Cantor_set
iff
the carrier of b1 = product (omega --> {{},1}) &
(ex b2 being prebasis of b1 st
for b3 being Element of bool product (omega --> {{},1}) holds
b3 in b2
iff
ex b4, b5 being natural set st
for b6 being Element of product (omega --> {{},1}) holds
b6 in b3
iff
b6 . b4 = b5);