Article BORSUK_1, MML version 4.99.1005
:: BORSUK_1:th 5
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
st b1 c= b3 " b2
holds b3 .: b1 c= b2;
:: BORSUK_1:th 6
theorem
for b1, b2, b3 being set holds
(b1 --> b2) .: b3 c= {b2};
:: BORSUK_1:th 9
theorem
for b1, b2, b3 being set
st b1 c= [:b2,b3:]
holds (.: pr1(b2,b3)) . b1 = (pr1(b2,b3)) .: b1;
:: BORSUK_1:th 10
theorem
for b1, b2, b3 being set
st b1 c= [:b2,b3:]
holds (.: pr2(b2,b3)) . b1 = (pr2(b2,b3)) .: b1;
:: BORSUK_1:th 12
theorem
for b1, b2 being set
for b3 being Element of bool b1
for b4 being Element of bool b2
st [:b3,b4:] <> {}
holds (pr1(b1,b2)) .: [:b3,b4:] = b3 & (pr2(b1,b2)) .: [:b3,b4:] = b4;
:: BORSUK_1:th 13
theorem
for b1, b2 being set
for b3 being Element of bool b1
for b4 being Element of bool b2
st [:b3,b4:] <> {}
holds (.: pr1(b1,b2)) . [:b3,b4:] = b3 &
(.: pr2(b1,b2)) . [:b3,b4:] = b4;
:: BORSUK_1:th 14
theorem
for b1, b2 being set
for b3 being Element of bool [:b1,b2:]
for b4 being Element of bool bool [:b1,b2:]
st for b5 being set
st b5 in b4
holds b5 c= b3 &
(ex b6 being Element of bool b1 st
ex b7 being Element of bool b2 st
b5 = [:b6,b7:])
holds [:union ((.: pr1(b1,b2)) .: b4),meet ((.: pr2(b1,b2)) .: b4):] c= b3;
:: BORSUK_1:th 15
theorem
for b1, b2 being set
for b3 being Element of bool [:b1,b2:]
for b4 being Element of bool bool [:b1,b2:]
st for b5 being set
st b5 in b4
holds b5 c= b3 &
(ex b6 being Element of bool b1 st
ex b7 being Element of bool b2 st
b5 = [:b6,b7:])
holds [:meet ((.: pr1(b1,b2)) .: b4),union ((.: pr2(b1,b2)) .: b4):] c= b3;
:: BORSUK_1:th 16
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool bool b1 holds
union ((.: b3) .: b4) = b3 .: union b4;
:: BORSUK_1:th 17
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
union union b2 = union {union b3 where b3 is Element of bool b1: b3 in b2};
:: BORSUK_1:th 18
theorem
for b1 being set
for b2 being Element of bool bool b1
st union b2 = b1
for b3 being Element of bool b2
for b4 being Element of bool b1
st b4 = union b3
holds b4 ` c= union (b3 `);
:: BORSUK_1:th 19
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
st for b6, b7 being Element of b1
st b4 . b6 = b4 . b7
holds b5 . b6 = b5 . b7
holds ex b6 being Function-like quasi_total Relation of b2,b3 st
b6 * b4 = b5;
:: BORSUK_1:th 20
theorem
for b1, b2, b3 being non empty set
for b4 being Element of b2
for b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of b2,b3 holds
b5 " {b4} c= (b6 * b5) " {b6 . b4};
:: BORSUK_1:th 21
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of b1
for b6 being Element of b3 holds
[:b4,id b3:] .(b5,b6) = [b4 . b5,b6];
:: BORSUK_1:th 23
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of bool b1
for b6 being Element of bool b3 holds
[:b4,id b3:] .: [:b5,b6:] = [:b4 .: b5,b6:];
:: BORSUK_1:th 24
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Element of b2
for b6 being Element of b3 holds
[:b4,id b3:] " {[b5,b6]} = [:b4 " {b5},{b6}:];
:: BORSUK_1:th 25
theorem
for b1 being non empty set
for b2 being Element of bool bool b1
for b3 being Element of bool b2 holds
union b3 is Element of bool b1;
:: BORSUK_1:th 26
theorem
for b1 being set
for b2 being a_partition of b1
for b3, b4 being Element of bool b2 holds
union (b3 /\ b4) = (union b3) /\ union b4;
:: BORSUK_1:th 27
theorem
for b1 being non empty set
for b2 being a_partition of b1
for b3 being Element of bool b2
for b4 being Element of bool b1
st b4 = union b3
holds b4 ` = union (b3 `);
:: BORSUK_1:th 28
theorem
for b1 being non empty set
for b2 being symmetric transitive total Relation of b1,b1 holds
Class b2 is not empty;
:: BORSUK_1:exreg 1
registration
let a1 be non empty set;
cluster non empty with_non-empty_elements a_partition of a1;
end;
:: BORSUK_1:funcnot 1 => BORSUK_1:func 1
definition
let a1 be non empty set;
let a2 be non empty a_partition of a1;
func proj A2 -> Function-like quasi_total Relation of a1,a2 means
for b1 being Element of a1 holds
b1 in it . b1;
end;
:: BORSUK_1:def 1
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Function-like quasi_total Relation of b1,b2 holds
b3 = proj b2
iff
for b4 being Element of b1 holds
b4 in b3 . b4;
:: BORSUK_1:th 29
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b1
for b4 being Element of b2
st b3 in b4
holds b4 = (proj b2) . b3;
:: BORSUK_1:th 30
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b2 holds
b3 = (proj b2) " {b3};
:: BORSUK_1:th 31
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of bool b2 holds
(proj b2) " b3 = union b3;
:: BORSUK_1:th 32
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of b2 holds
ex b4 being Element of b1 st
(proj b2) . b4 = b3;
:: BORSUK_1:th 33
theorem
for b1 being non empty set
for b2 being non empty a_partition of b1
for b3 being Element of bool b1
st for b4 being Element of bool b1
st b4 in b2 & b4 meets b3
holds b4 c= b3
holds b3 = (proj b2) " ((proj b2) .: b3);
:: BORSUK_1:th 35
theorem
for b1 being TopStruct
for b2 being SubSpace of b1 holds
the carrier of b2 c= the carrier of b1;
:: BORSUK_1:funcnot 2 => BORSUK_1:func 2
definition
let a1 be 1-sorted;
let a2 be non empty 1-sorted;
let a3 be Element of the carrier of a2;
func A1 --> A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 equals
(the carrier of a1) --> a3;
end;
:: BORSUK_1:def 2
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Element of the carrier of b2 holds
b1 --> b3 = (the carrier of b1) --> b3;
:: BORSUK_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b2 holds
b1 --> b3 is continuous(b1, b2);
:: BORSUK_1:funcreg 1
registration
let a1 be TopSpace-like TopStruct;
let a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a2;
cluster a1 --> a3 -> Function-like quasi_total continuous;
end;
:: BORSUK_1:attrnot 1 => PRE_TOPC:attr 5
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 continuous means
for b1 being Element of the carrier of a1
for b2 being a_neighborhood of a3 . b1 holds
ex b3 being a_neighborhood of b1 st
a3 .: b3 c= b2;
end;
:: BORSUK_1:dfs 3
definiens
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is continuous
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being a_neighborhood of a3 . b1 holds
ex b3 being a_neighborhood of b1 st
a3 .: b3 c= b2;
:: BORSUK_1:def 3
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 holds
b3 is continuous(b1, b2)
iff
for b4 being Element of the carrier of b1
for b5 being a_neighborhood of b3 . b4 holds
ex b6 being a_neighborhood of b4 st
b3 .: b6 c= b5;
:: BORSUK_1:exreg 2
registration
let a1 be TopSpace-like TopStruct;
let a2 be non empty TopSpace-like TopStruct;
cluster Relation-like Function-like quasi_total total continuous Relation of the carrier of a1,the carrier of a2;
end;
:: BORSUK_1:funcreg 2
registration
let a1, a2, a3 be non empty TopSpace-like TopStruct;
let a4 be Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2;
let a5 be Function-like quasi_total continuous Relation of the carrier of a2,the carrier of a3;
cluster a4 * a5 -> Function-like quasi_total continuous;
end;
:: BORSUK_1:th 37
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2 holds
b3 " Int b4 c= Int (b3 " b4);
:: BORSUK_1:th 38
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1
for b5 being a_neighborhood of b3 holds
b4 " b5 is a_neighborhood of b4 " {b3};
:: BORSUK_1:funcnot 3 => BORSUK_1:func 3
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a2;
let a4 be Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2;
let a5 be a_neighborhood of a3;
redefine func a4 " a5 -> a_neighborhood of a4 " {a3};
end;
:: BORSUK_1:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being a_neighborhood of b3
st b2 c= b3
holds b4 is a_neighborhood of b2;
:: BORSUK_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
{b2} is compact(b1);
:: BORSUK_1:th 42
theorem
for b1 being TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 = b4
holds b3 is compact(b1)
iff
b4 is compact(b2);
:: BORSUK_1:funcnot 4 => BORSUK_1:func 4
definition
let a1, a2 be TopSpace-like TopStruct;
func [:A1,A2:] -> strict TopSpace-like TopStruct means
the carrier of it = [:the carrier of a1,the carrier of a2:] &
the topology of it = {union b1 where b1 is Element of bool bool the carrier of it: b1 c= {[:b2,b3:] where b2 is Element of bool the carrier of a1, b3 is Element of bool the carrier of a2: b2 in the topology of a1 & b3 in the topology of a2}};
end;
:: BORSUK_1:def 5
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being strict TopSpace-like TopStruct holds
b3 = [:b1,b2:]
iff
the carrier of b3 = [:the carrier of b1,the carrier of b2:] &
the topology of b3 = {union b4 where b4 is Element of bool bool the carrier of b3: b4 c= {[:b5,b6:] where b5 is Element of bool the carrier of b1, b6 is Element of bool the carrier of b2: b5 in the topology of b1 & b6 in the topology of b2}};
:: BORSUK_1:funcreg 3
registration
let a1, a2 be non empty TopSpace-like TopStruct;
cluster [:a1,a2:] -> non empty strict TopSpace-like;
end;
:: BORSUK_1:th 45
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
b3 is open([:b1,b2:])
iff
ex b4 being Element of bool bool the carrier of [:b1,b2:] st
b3 = union b4 &
(for b5 being set
st b5 in b4
holds ex b6 being Element of bool the carrier of b1 st
ex b7 being Element of bool the carrier of b2 st
b5 = [:b6,b7:] & b6 is open(b1) & b7 is open(b2));
:: BORSUK_1:funcnot 5 => BORSUK_1:func 5
definition
let a1, a2 be TopSpace-like TopStruct;
let a3 be Element of bool the carrier of a1;
let a4 be Element of bool the carrier of a2;
redefine func [:a3, a4:] -> Element of bool the carrier of [:a1,a2:];
end;
:: BORSUK_1:funcnot 6 => BORSUK_1:func 6
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
redefine func [a3, a4] -> Element of the carrier of [:a1,a2:];
end;
:: BORSUK_1:th 46
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st b3 is open(b1) & b4 is open(b2)
holds [:b3,b4:] is open([:b1,b2:]);
:: BORSUK_1:th 47
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2 holds
Int [:b3,b4:] = [:Int b3,Int b4:];
:: BORSUK_1:th 48
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being a_neighborhood of b3
for b6 being a_neighborhood of b4 holds
[:b5,b6:] is a_neighborhood of [b3,b4];
:: BORSUK_1:th 49
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being a_neighborhood of b3
for b6 being a_neighborhood of b4 holds
[:b5,b6:] is a_neighborhood of [:b3,b4:];
:: BORSUK_1:funcnot 7 => BORSUK_1:func 7
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
let a5 be a_neighborhood of a3;
let a6 be a_neighborhood of a4;
redefine func [:a5, a6:] -> a_neighborhood of [a3,a4];
end;
:: BORSUK_1:th 50
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of [:b1,b2:] holds
ex b4 being Element of the carrier of b1 st
ex b5 being Element of the carrier of b2 st
b3 = [b4,b5];
:: BORSUK_1:funcnot 8 => BORSUK_1:func 8
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of bool the carrier of a1;
let a4 be Element of the carrier of a2;
let a5 be a_neighborhood of a3;
let a6 be a_neighborhood of a4;
redefine func [:a5, a6:] -> a_neighborhood of [:a3,{a4}:];
end;
:: BORSUK_1:funcnot 9 => BORSUK_1:func 9
definition
let a1, a2 be TopSpace-like TopStruct;
let a3 be Element of bool the carrier of [:a1,a2:];
func Base-Appr A3 -> Element of bool bool the carrier of [:a1,a2:] equals
{[:b1,b2:] where b1 is Element of bool the carrier of a1, b2 is Element of bool the carrier of a2: [:b1,b2:] c= a3 & b1 is open(a1) & b2 is open(a2)};
end;
:: BORSUK_1:def 6
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
Base-Appr b3 = {[:b4,b5:] where b4 is Element of bool the carrier of b1, b5 is Element of bool the carrier of b2: [:b4,b5:] c= b3 & b4 is open(b1) & b5 is open(b2)};
:: BORSUK_1:th 51
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
Base-Appr b3 is open([:b1,b2:]);
:: BORSUK_1:th 52
theorem
for b1, b2 being TopSpace-like TopStruct
for b3, b4 being Element of bool the carrier of [:b1,b2:]
st b3 c= b4
holds Base-Appr b3 c= Base-Appr b4;
:: BORSUK_1:th 53
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
union Base-Appr b3 c= b3;
:: BORSUK_1:th 54
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
st b3 is open([:b1,b2:])
holds b3 = union Base-Appr b3;
:: BORSUK_1:th 55
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:] holds
Int b3 = union Base-Appr b3;
:: BORSUK_1:funcnot 10 => BORSUK_1:func 10
definition
let a1, a2 be non empty TopSpace-like TopStruct;
func Pr1(A1,A2) -> Function-like quasi_total Relation of bool the carrier of [:a1,a2:],bool the carrier of a1 equals
.: pr1(the carrier of a1,the carrier of a2);
end;
:: BORSUK_1:def 7
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
Pr1(b1,b2) = .: pr1(the carrier of b1,the carrier of b2);
:: BORSUK_1:funcnot 11 => BORSUK_1:func 11
definition
let a1, a2 be non empty TopSpace-like TopStruct;
func Pr2(A1,A2) -> Function-like quasi_total Relation of bool the carrier of [:a1,a2:],bool the carrier of a2 equals
.: pr2(the carrier of a1,the carrier of a2);
end;
:: BORSUK_1:def 8
theorem
for b1, b2 being non empty TopSpace-like TopStruct holds
Pr2(b1,b2) = .: pr2(the carrier of b1,the carrier of b2);
:: BORSUK_1:th 56
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
for b4 being Element of bool bool the carrier of [:b1,b2:]
st for b5 being set
st b5 in b4
holds b5 c= b3 &
(ex b6 being Element of bool the carrier of b1 st
ex b7 being Element of bool the carrier of b2 st
b5 = [:b6,b7:])
holds [:union ((Pr1(b1,b2)) .: b4),meet ((Pr2(b1,b2)) .: b4):] c= b3;
:: BORSUK_1:th 57
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being set
st b4 in (Pr1(b1,b2)) .: b3
holds ex b5 being Element of bool the carrier of [:b1,b2:] st
b5 in b3 &
b4 = (pr1(the carrier of b1,the carrier of b2)) .: b5;
:: BORSUK_1:th 58
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being set
st b4 in (Pr2(b1,b2)) .: b3
holds ex b5 being Element of bool the carrier of [:b1,b2:] st
b5 in b3 &
b4 = (pr2(the carrier of b1,the carrier of b2)) .: b5;
:: BORSUK_1:th 59
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
st b3 is open([:b1,b2:])
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2 holds
(b4 = (pr1(the carrier of b1,the carrier of b2)) .: b3 implies b4 is open(b1)) &
(b5 = (pr2(the carrier of b1,the carrier of b2)) .: b3 implies b5 is open(b2));
:: BORSUK_1:th 60
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
st b3 is open([:b1,b2:])
holds (Pr1(b1,b2)) .: b3 is open(b1) & (Pr2(b1,b2)) .: b3 is open(b2);
:: BORSUK_1:th 61
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
st ((Pr1(b1,b2)) .: b3 = {} or (Pr2(b1,b2)) .: b3 = {})
holds b3 = {};
:: BORSUK_1:th 62
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of [:b1,b2:]
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
st b3 is_a_cover_of [:b4,b5:]
holds (b5 = {} or (Pr1(b1,b2)) .: b3 is_a_cover_of b4) &
(b4 = {} or (Pr2(b1,b2)) .: b3 is_a_cover_of b5);
:: BORSUK_1:th 63
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool the carrier of b1
st b3 is_a_cover_of b4
holds ex b5 being Element of bool bool the carrier of b1 st
b5 c= b3 &
b5 is_a_cover_of b4 &
(for b6 being set
st b6 in b5
holds b6 meets b4);
:: BORSUK_1:th 64
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of [:b1,b2:]
st b3 is finite & b3 c= (Pr1(b1,b2)) .: b4
holds ex b5 being Element of bool bool the carrier of [:b1,b2:] st
b5 c= b4 & b5 is finite & b3 = (Pr1(b1,b2)) .: b5;
:: BORSUK_1:th 65
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
st [:b3,b4:] <> {}
holds (Pr1(b1,b2)) . [:b3,b4:] = b3 & (Pr2(b1,b2)) . [:b3,b4:] = b4;
:: BORSUK_1:th 67
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b2
st b4 is compact(b2)
for b5 being a_neighborhood of [:b4,{b3}:] holds
ex b6 being a_neighborhood of b4 st
ex b7 being a_neighborhood of b3 st
[:b6,b7:] c= b5;
:: BORSUK_1:funcnot 12 => BORSUK_1:func 12
definition
let a1 be 1-sorted;
func TrivDecomp A1 -> a_partition of the carrier of a1 equals
Class id the carrier of a1;
end;
:: BORSUK_1:def 9
theorem
for b1 being 1-sorted holds
TrivDecomp b1 = Class id the carrier of b1;
:: BORSUK_1:funcreg 4
registration
let a1 be non empty 1-sorted;
cluster TrivDecomp a1 -> non empty;
end;
:: BORSUK_1:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
st b2 in TrivDecomp b1
holds ex b3 being Element of the carrier of b1 st
b2 = {b3};
:: BORSUK_1:funcnot 13 => BORSUK_1:func 13
definition
let a1 be TopSpace-like TopStruct;
let a2 be a_partition of the carrier of a1;
func space A2 -> strict TopSpace-like TopStruct means
the carrier of it = a2 &
the topology of it = {b1 where b1 is Element of bool a2: union b1 in the topology of a1};
end;
:: BORSUK_1:def 10
theorem
for b1 being TopSpace-like TopStruct
for b2 being a_partition of the carrier of b1
for b3 being strict TopSpace-like TopStruct holds
b3 = space b2
iff
the carrier of b3 = b2 &
the topology of b3 = {b4 where b4 is Element of bool b2: union b4 in the topology of b1};
:: BORSUK_1:funcreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty a_partition of the carrier of a1;
cluster space a2 -> non empty strict TopSpace-like;
end;
:: BORSUK_1:th 69
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of bool b2 holds
union b3 in the topology of b1
iff
b3 in the topology of space b2;
:: BORSUK_1:funcnot 14 => BORSUK_1:func 14
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty a_partition of the carrier of a1;
func Proj A2 -> Function-like quasi_total continuous Relation of the carrier of a1,the carrier of space a2 equals
proj a2;
end;
:: BORSUK_1:def 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
Proj b2 = proj b2;
:: BORSUK_1:th 70
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in (Proj b2) . b3;
:: BORSUK_1:th 71
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1
for b3 being Element of the carrier of space b2 holds
ex b4 being Element of the carrier of b1 st
(Proj b2) . b4 = b3;
:: BORSUK_1:th 72
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
rng Proj b2 = the carrier of space b2;
:: BORSUK_1:funcnot 15 => BORSUK_1:func 15
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
let a3 be non empty a_partition of the carrier of a2;
func TrivExt A3 -> non empty a_partition of the carrier of a1 equals
a3 \/ {{b1} where b1 is Element of the carrier of a1: not b1 in the carrier of a2};
end;
:: BORSUK_1:def 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2 holds
TrivExt b3 = b3 \/ {{b4} where b4 is Element of the carrier of b1: not b4 in the carrier of b2};
:: BORSUK_1:th 74
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of bool the carrier of b1
st b4 in TrivExt b3 & not b4 in b3
holds ex b5 being Element of the carrier of b1 st
not b5 in [#] b2 & b4 = {b5};
:: BORSUK_1:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
st not b4 in the carrier of b2
holds {b4} in TrivExt b3;
:: BORSUK_1:th 76
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
st b4 in the carrier of b2
holds (Proj TrivExt b3) . b4 = (Proj b3) . b4;
:: BORSUK_1:th 77
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
st not b4 in the carrier of b2
holds (Proj TrivExt b3) . b4 = {b4};
:: BORSUK_1:th 78
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4, b5 being Element of the carrier of b1
st not b4 in the carrier of b2 &
(Proj TrivExt b3) . b4 = (Proj TrivExt b3) . b5
holds b4 = b5;
:: BORSUK_1:th 79
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being Element of the carrier of b1
st (Proj TrivExt b3) . b4 in the carrier of space b3
holds b4 in the carrier of b2;
:: BORSUK_1:th 80
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being non empty a_partition of the carrier of b2
for b4 being set
st b4 in the carrier of b2
holds (Proj TrivExt b3) . b4 in the carrier of space b3;
:: BORSUK_1:modenot 1 => BORSUK_1:mode 1
definition
let a1 be non empty TopSpace-like TopStruct;
mode u.s.c._decomposition of A1 -> non empty a_partition of the carrier of a1 means
for b1 being Element of bool the carrier of a1
st b1 in it
for b2 being a_neighborhood of b1 holds
ex b3 being Element of bool the carrier of a1 st
b3 is open(a1) &
b1 c= b3 &
b3 c= b2 &
(for b4 being Element of bool the carrier of a1
st b4 in it & b4 meets b3
holds b4 c= b3);
end;
:: BORSUK_1:dfs 12
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty a_partition of the carrier of a1;
To prove
a2 is u.s.c._decomposition of a1
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
for b2 being a_neighborhood of b1 holds
ex b3 being Element of bool the carrier of a1 st
b3 is open(a1) &
b1 c= b3 &
b3 c= b2 &
(for b4 being Element of bool the carrier of a1
st b4 in a2 & b4 meets b3
holds b4 c= b3);
:: BORSUK_1:def 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty a_partition of the carrier of b1 holds
b2 is u.s.c._decomposition of b1
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
for b4 being a_neighborhood of b3 holds
ex b5 being Element of bool the carrier of b1 st
b5 is open(b1) &
b3 c= b5 &
b5 c= b4 &
(for b6 being Element of bool the carrier of b1
st b6 in b2 & b6 meets b5
holds b6 c= b5);
:: BORSUK_1:th 81
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being u.s.c._decomposition of b1
for b3 being Element of the carrier of space b2
for b4 being a_neighborhood of (Proj b2) " {b3} holds
(Proj b2) .: b4 is a_neighborhood of b3;
:: BORSUK_1:th 82
theorem
for b1 being non empty TopSpace-like TopStruct holds
TrivDecomp b1 is u.s.c._decomposition of b1;
:: BORSUK_1:attrnot 2 => BORSUK_1:attr 1
definition
let a1 be TopSpace-like TopStruct;
let a2 be SubSpace of a1;
attr a2 is closed means
for b1 being Element of bool the carrier of a1
st b1 = the carrier of a2
holds b1 is closed(a1);
end;
:: BORSUK_1:dfs 13
definiens
let a1 be TopSpace-like TopStruct;
let a2 be SubSpace 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 = the carrier of a2
holds b1 is closed(a1);
:: BORSUK_1:def 14
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1 holds
b2 is closed(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 = the carrier of b2
holds b3 is closed(b1);
:: BORSUK_1:exreg 3
registration
let a1 be TopSpace-like TopStruct;
cluster strict TopSpace-like closed SubSpace of a1;
end;
:: BORSUK_1:exreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty strict TopSpace-like closed SubSpace of a1;
end;
:: BORSUK_1:funcnot 16 => BORSUK_1:func 16
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty closed SubSpace of a1;
let a3 be u.s.c._decomposition of a2;
redefine func TrivExt a3 -> u.s.c._decomposition of a1;
end;
:: BORSUK_1:attrnot 3 => BORSUK_1:attr 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be u.s.c._decomposition of a1;
attr a2 is DECOMPOSITION-like means
for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is compact(a1);
end;
:: BORSUK_1:dfs 14
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be u.s.c._decomposition of a1;
To prove
a2 is DECOMPOSITION-like
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 in a2
holds b1 is compact(a1);
:: BORSUK_1:def 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being u.s.c._decomposition of b1 holds
b2 is DECOMPOSITION-like(b1)
iff
for b3 being Element of bool the carrier of b1
st b3 in b2
holds b3 is compact(b1);
:: BORSUK_1:exreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty with_non-empty_elements DECOMPOSITION-like u.s.c._decomposition of a1;
end;
:: BORSUK_1:modenot 2
definition
let a1 be non empty TopSpace-like TopStruct;
mode DECOMPOSITION of a1 is DECOMPOSITION-like u.s.c._decomposition of a1;
end;
:: BORSUK_1:funcnot 17 => BORSUK_1:func 17
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty closed SubSpace of a1;
let a3 be DECOMPOSITION-like u.s.c._decomposition of a2;
redefine func TrivExt a3 -> DECOMPOSITION-like u.s.c._decomposition of a1;
end;
:: BORSUK_1:funcnot 18 => BORSUK_1:func 18
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty closed SubSpace of a1;
let a3 be DECOMPOSITION-like u.s.c._decomposition of a2;
redefine func space a3 -> strict closed SubSpace of space TrivExt a3;
end;
:: BORSUK_1:funcnot 19 => BORSUK_1:func 19
definition
func I[01] -> TopStruct means
for b1 being Element of bool the carrier of TopSpaceMetr RealSpace
st b1 = [.0,1.]
holds it = (TopSpaceMetr RealSpace) | b1;
end;
:: BORSUK_1:def 16
theorem
for b1 being TopStruct holds
b1 = I[01]
iff
for b2 being Element of bool the carrier of TopSpaceMetr RealSpace
st b2 = [.0,1.]
holds b1 = (TopSpaceMetr RealSpace) | b2;
:: BORSUK_1:funcreg 6
registration
cluster I[01] -> non empty strict TopSpace-like;
end;
:: BORSUK_1:th 83
theorem
the carrier of I[01] = [.0,1.];
:: BORSUK_1:funcnot 20 => BORSUK_1:func 20
definition
func 0[01] -> Element of the carrier of I[01] equals
0;
end;
:: BORSUK_1:def 17
theorem
0[01] = 0;
:: BORSUK_1:funcnot 21 => BORSUK_1:func 21
definition
func 1[01] -> Element of the carrier of I[01] equals
1;
end;
:: BORSUK_1:def 18
theorem
1[01] = 1;
:: BORSUK_1:attrnot 4 => BORSUK_1:attr 3
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is being_a_retraction means
for b1 being Element of the carrier of a1
st b1 in the carrier of a2
holds a3 . b1 = b1;
end;
:: BORSUK_1:dfs 18
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is being_a_retraction
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 in the carrier of a2
holds a3 . b1 = b1;
:: BORSUK_1:def 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is being_a_retraction(b1, b2)
iff
for b4 being Element of the carrier of b1
st b4 in the carrier of b2
holds b3 . b4 = b4;
:: BORSUK_1:prednot 1 => BORSUK_1:attr 3
notation
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
synonym a3 is_a_retraction for being_a_retraction;
end;
:: BORSUK_1:prednot 2 => BORSUK_1:pred 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
pred A2 is_a_retract_of A1 means
ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
b1 is being_a_retraction(a1, a2);
end;
:: BORSUK_1:dfs 19
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
To prove
a2 is_a_retract_of a1
it is sufficient to prove
thus ex b1 being Function-like quasi_total continuous Relation of the carrier of a1,the carrier of a2 st
b1 is being_a_retraction(a1, a2);
:: BORSUK_1:def 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is_a_retract_of b1
iff
ex b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2 st
b3 is being_a_retraction(b1, b2);
:: BORSUK_1:prednot 3 => BORSUK_1:pred 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
pred A2 is_an_SDR_of A1 means
ex b1 being Function-like quasi_total continuous Relation of the carrier of [:a1,I[01]:],the carrier of a1 st
for b2 being Element of the carrier of a1 holds
b1 . [b2,0[01]] = b2 &
b1 . [b2,1[01]] in the carrier of a2 &
(b2 in the carrier of a2 implies for b3 being Element of the carrier of I[01] holds
b1 . [b2,b3] = b2);
end;
:: BORSUK_1:dfs 20
definiens
let a1 be non empty TopSpace-like TopStruct;
let a2 be non empty SubSpace of a1;
To prove
a2 is_an_SDR_of a1
it is sufficient to prove
thus ex b1 being Function-like quasi_total continuous Relation of the carrier of [:a1,I[01]:],the carrier of a1 st
for b2 being Element of the carrier of a1 holds
b1 . [b2,0[01]] = b2 &
b1 . [b2,1[01]] in the carrier of a2 &
(b2 in the carrier of a2 implies for b3 being Element of the carrier of I[01] holds
b1 . [b2,b3] = b2);
:: BORSUK_1:def 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
b2 is_an_SDR_of b1
iff
ex b3 being Function-like quasi_total continuous Relation of the carrier of [:b1,I[01]:],the carrier of b1 st
for b4 being Element of the carrier of b1 holds
b3 . [b4,0[01]] = b4 &
b3 . [b4,1[01]] in the carrier of b2 &
(b4 in the carrier of b2 implies for b5 being Element of the carrier of I[01] holds
b3 . [b4,b5] = b4);
:: BORSUK_1:th 84
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being DECOMPOSITION-like u.s.c._decomposition of b2
st b2 is_a_retract_of b1
holds space b3 is_a_retract_of space TrivExt b3;
:: BORSUK_1:th 85
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being DECOMPOSITION-like u.s.c._decomposition of b2
st b2 is_an_SDR_of b1
holds space b3 is_an_SDR_of space TrivExt b3;
:: BORSUK_1:th 86
theorem
for b1 being real set holds
0 <= b1 & b1 <= 1
iff
b1 in the carrier of I[01];