Article TOPS_2, MML version 4.99.1005
:: TOPS_2:th 1
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
b2 c= bool [#] b1;
:: TOPS_2:th 3
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being set
st b3 c= b2
holds b3 is Element of bool bool the carrier of b1;
:: TOPS_2:th 5
theorem
for b1 being non empty 1-sorted
for b2 being Element of bool bool the carrier of b1
st b2 is_a_cover_of b1
holds b2 <> {};
:: TOPS_2:th 6
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool bool the carrier of b1 holds
(union b2) \ union b3 c= union (b2 \ b3);
:: TOPS_2:th 10
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 <> {}
iff
COMPLEMENT b2 <> {};
:: TOPS_2:th 11
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 <> {}
holds meet COMPLEMENT b2 = (union b2) `;
:: TOPS_2:th 12
theorem
for b1 being set
for b2 being Element of bool bool b1
st b2 <> {}
holds union COMPLEMENT b2 = (meet b2) `;
:: TOPS_2:th 13
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
COMPLEMENT b2 is finite
iff
b2 is finite;
:: TOPS_2:attrnot 1 => TOPS_2:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is open means
for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is open(a1);
end;
:: TOPS_2:dfs 1
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is open
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is open(a1);
:: TOPS_2:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is open(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is open(b1);
:: TOPS_2:attrnot 2 => TOPS_2:attr 2
definition
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
attr a2 is closed means
for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is closed(a1);
end;
:: TOPS_2:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool bool the carrier of a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is closed(a1);
:: TOPS_2:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is closed(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is closed(b1);
:: TOPS_2:th 16
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is closed(b1)
iff
COMPLEMENT b2 is open(b1);
:: TOPS_2:th 17
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
b2 is open(b1)
iff
COMPLEMENT b2 is closed(b1);
:: TOPS_2:th 18
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3 & b3 is open(b1)
holds b2 is open(b1);
:: TOPS_2:th 19
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 c= b3 & b3 is closed(b1)
holds b2 is closed(b1);
:: TOPS_2:th 20
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is open(b1) & b3 is open(b1)
holds b2 \/ b3 is open(b1);
:: TOPS_2:th 21
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is open(b1)
holds b2 /\ b3 is open(b1);
:: TOPS_2:th 22
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is open(b1)
holds b2 \ b3 is open(b1);
:: TOPS_2:th 23
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is closed(b1) & b3 is closed(b1)
holds b2 \/ b3 is closed(b1);
:: TOPS_2:th 24
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is closed(b1)
holds b2 /\ b3 is closed(b1);
:: TOPS_2:th 25
theorem
for b1 being TopStruct
for b2, b3 being Element of bool bool the carrier of b1
st b2 is closed(b1)
holds b2 \ b3 is closed(b1);
:: TOPS_2:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is open(b1)
holds union b2 is open(b1);
:: TOPS_2:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is open(b1) & b2 is finite
holds meet b2 is open(b1);
:: TOPS_2:th 28
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is closed(b1) & b2 is finite
holds union b2 is closed(b1);
:: TOPS_2:th 29
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
st b2 is closed(b1)
holds meet b2 is closed(b1);
:: TOPS_2:th 31
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool bool the carrier of b2 holds
b3 is Element of bool bool the carrier of b1;
:: TOPS_2:th 32
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b2 holds
b3 is open(b2)
iff
ex b4 being Element of bool the carrier of b1 st
b4 is open(b1) & b4 /\ [#] b2 = b3;
:: TOPS_2:th 33
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being SubSpace of b1
st b2 is open(b1)
for b4 being Element of bool the carrier of b3
st b4 = b2
holds b4 is open(b3);
:: TOPS_2:th 34
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being SubSpace of b1
st b2 is closed(b1)
for b4 being Element of bool the carrier of b3
st b4 = b2
holds b4 is closed(b3);
:: TOPS_2:th 35
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being SubSpace of b1
st b2 is open(b1)
for b4 being Element of bool bool the carrier of b3
st b4 = b2
holds b4 is open(b3);
:: TOPS_2:th 36
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1
for b3 being SubSpace of b1
st b2 is closed(b1)
for b4 being Element of bool bool the carrier of b3
st b4 = b2
holds b4 is closed(b3);
:: TOPS_2:th 38
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 /\ b3 is Element of bool the carrier of b1 | b3;
:: TOPS_2:funcnot 1 => TOPS_2:func 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
let a3 be Element of bool bool the carrier of a1;
func A3 | A2 -> Element of bool bool the carrier of a1 | a2 means
for b1 being Element of bool the carrier of a1 | a2 holds
b1 in it
iff
ex b2 being Element of bool the carrier of a1 st
b2 in a3 & b2 /\ a2 = b1;
end;
:: TOPS_2:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of b1 | b2 holds
b4 = b3 | b2
iff
for b5 being Element of bool the carrier of b1 | b2 holds
b5 in b4
iff
ex b6 being Element of bool the carrier of b1 st
b6 in b3 & b6 /\ b2 = b5;
:: TOPS_2:th 40
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of bool bool the carrier of b1
st b3 c= b4
holds b3 | b2 c= b4 | b2;
:: TOPS_2:th 41
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool bool the carrier of b1
st b2 in b4
holds b2 /\ b3 in b4 | b3;
:: TOPS_2:th 42
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool bool the carrier of b1
st b2 c= union b4
holds b2 /\ b3 c= union (b4 | b3);
:: TOPS_2:th 43
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b2 c= union b3
holds b2 = union (b3 | b2);
:: TOPS_2:th 44
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
union (b3 | b2) c= union b3;
:: TOPS_2:th 45
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b2 c= union (b3 | b2)
holds b2 c= union b3;
:: TOPS_2:th 46
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 is finite
holds b3 | b2 is finite;
:: TOPS_2:th 47
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 is open(b1)
holds b3 | b2 is open(b1 | b2);
:: TOPS_2:th 48
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
st b3 is closed(b1)
holds b3 | b2 is closed(b1 | b2);
:: TOPS_2:th 49
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool bool the carrier of b2
st b3 is open(b2)
holds ex b4 being Element of bool bool the carrier of b1 st
b4 is open(b1) &
(for b5 being Element of bool the carrier of b1
st b5 = [#] b2
holds b3 = b4 | b5);
:: TOPS_2:th 50
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
ex b4 being Relation-like Function-like set st
proj1 b4 = b3 &
proj2 b4 = b3 | b2 &
(for b5 being set
st b5 in b3
for b6 being Element of bool the carrier of b1
st b6 = b5
holds b4 . b5 = b6 /\ b2);
:: TOPS_2:th 52
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st ([#] b2 = {} implies [#] b1 = {})
holds b3 " [#] b2 = [#] b1;
:: TOPS_2:th 54
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2 holds
(" b3) .: b4 is Element of bool bool the carrier of b1;
:: TOPS_2:th 55
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st ([#] b2 = {} implies [#] b1 = {})
holds b3 is continuous(b1, b2)
iff
for b4 being Element of bool the carrier of b2
st b4 is open(b2)
holds b3 " b4 is open(b1);
:: TOPS_2:th 56
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is continuous(b1, b2)
iff
for b4 being Element of bool the carrier of b2 holds
Cl (b3 " b4) c= b3 " Cl b4;
:: TOPS_2:th 57
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is continuous(b1, b2)
iff
for b4 being Element of bool the carrier of b1 holds
b3 .: Cl b4 c= Cl (b3 .: b4);
:: TOPS_2:th 58
theorem
for b1, b2 being TopStruct
for b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st b4 is continuous(b1, b3) & b5 is continuous(b3, b2)
holds b5 * b4 is continuous(b1, b2);
:: TOPS_2:th 59
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2
st b3 is continuous(b1, b2) & b4 is open(b2)
for b5 being Element of bool bool the carrier of b1
st b5 = (" b3) .: b4
holds b5 is open(b1);
:: TOPS_2:th 60
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool bool the carrier of b2
st b3 is continuous(b1, b2) & b4 is closed(b2)
for b5 being Element of bool bool the carrier of b1
st b5 = (" b3) .: b4
holds b5 is closed(b1);
:: TOPS_2:funcnot 2 => TOPS_2:func 2
definition
let a1, a2 be set;
let a3 be Function-like quasi_total Relation of a1,a2;
assume rng a3 = a2 & a3 is one-to-one;
func A3 /" -> Function-like quasi_total Relation of a2,a1 equals
a3 ";
end;
:: TOPS_2:def 4
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b2
st rng b3 = b2 & b3 is one-to-one
holds b3 /" = b3 ";
:: TOPS_2:funcnot 3 => TOPS_2:func 2
notation
let a1, a2 be set;
let a3 be Function-like quasi_total Relation of a1,a2;
synonym a3 " for a3 /";
end;
:: TOPS_2:th 62
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st rng b3 = [#] b2 & b3 is one-to-one
holds dom (b3 /") = [#] b2 & rng (b3 /") = [#] b1;
:: TOPS_2:th 63
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st rng b3 = [#] b2 & b3 is one-to-one
holds b3 /" is one-to-one;
:: TOPS_2:th 64
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st rng b3 = [#] b2 & b3 is one-to-one
holds b3 /" /" = b3;
:: TOPS_2:th 65
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st rng b3 = [#] b2 & b3 is one-to-one
holds b3 /" * b3 = id dom b3 & b3 * (b3 /") = id rng b3;
:: TOPS_2:th 66
theorem
for b1 being 1-sorted
for b2, b3 being non empty 1-sorted
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st dom b4 = [#] b1 & rng b4 = [#] b2 & b4 is one-to-one & dom b5 = [#] b2 & rng b5 = [#] b3 & b5 is one-to-one
holds (b5 * b4) /" = b4 /" * (b5 /");
:: TOPS_2:th 67
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
st rng b3 = [#] b2 & b3 is one-to-one
holds b3 .: b4 = b3 /" " b4;
:: TOPS_2:th 68
theorem
for b1, b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2
st rng b3 = [#] b2 & b3 is one-to-one
holds b3 " b4 = b3 /" .: b4;
:: TOPS_2:attrnot 3 => TOPS_2:attr 3
definition
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is being_homeomorphism means
dom a3 = [#] a1 & rng a3 = [#] a2 & a3 is one-to-one & a3 is continuous(a1, a2) & a3 /" is continuous(a2, a1);
end;
:: TOPS_2:dfs 5
definiens
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is being_homeomorphism
it is sufficient to prove
thus dom a3 = [#] a1 & rng a3 = [#] a2 & a3 is one-to-one & a3 is continuous(a1, a2) & a3 /" is continuous(a2, a1);
:: TOPS_2:def 5
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_homeomorphism(b1, b2)
iff
dom b3 = [#] b1 & rng b3 = [#] b2 & b3 is one-to-one & b3 is continuous(b1, b2) & b3 /" is continuous(b2, b1);
:: TOPS_2:prednot 1 => TOPS_2:attr 3
notation
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
synonym a3 is_homeomorphism for being_homeomorphism;
end;
:: TOPS_2:th 70
theorem
for b1 being 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 b3 is being_homeomorphism(b1, b2)
holds b3 /" is being_homeomorphism(b2, b1);
:: TOPS_2:th 71
theorem
for b1, b2, b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
st b4 is being_homeomorphism(b1, b2) & b5 is being_homeomorphism(b2, b3)
holds b5 * b4 is being_homeomorphism(b1, b3);
:: TOPS_2:th 72
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_homeomorphism(b1, b2)
iff
dom b3 = [#] b1 &
rng b3 = [#] b2 &
b3 is one-to-one &
(for b4 being Element of bool the carrier of b1 holds
b4 is closed(b1)
iff
b3 .: b4 is closed(b2));
:: TOPS_2:th 73
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_homeomorphism(b1, b2)
iff
dom b3 = [#] b1 &
rng b3 = [#] b2 &
b3 is one-to-one &
(for b4 being Element of bool the carrier of b2 holds
b3 " Cl b4 = Cl (b3 " b4));
:: TOPS_2:th 74
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_homeomorphism(b1, b2)
iff
dom b3 = [#] b1 &
rng b3 = [#] b2 &
b3 is one-to-one &
(for b4 being Element of bool the carrier of b1 holds
b3 .: Cl b4 = Cl (b3 .: b4));
:: TOPS_2:th 75
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
for b4 being Element of bool the carrier of b1
st b3 is continuous(b1, b2) & b4 is connected(b1)
holds b3 .: b4 is connected(b2);
:: TOPS_2:th 76
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
for b4 being Element of bool the carrier of b2
st b3 is being_homeomorphism(b1, b2) & b4 is connected(b2)
holds b3 " b4 is connected(b1);