Article BORSUK_2, MML version 4.99.1005
:: BORSUK_2:sch 1
scheme BORSUK_2:sch 1
{F1 -> non empty set,
F2 -> set,
F3 -> set}:
Card {F3(b1) where b1 is Element of F1(): b1 in F2() & P1[b1]} c= Card F2()
:: BORSUK_2:th 1
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
st b1 is SubSpace of b4 &
b3 is SubSpace of b4 &
([#] b1) \/ [#] b3 = [#] b4 &
b1 is compact &
b3 is compact &
b4 is being_T2 &
b5 is continuous(b1, b2) &
b6 is continuous(b3, b2) &
(for b7 being set
st b7 in ([#] b1) /\ [#] b3
holds b5 . b7 = b6 . b7)
holds ex b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2 st
b7 = b5 +* b6 & b7 is continuous(b4, b2);
:: BORSUK_2:funcreg 1
registration
let a1 be TopStruct;
cluster id a1 -> Function-like quasi_total continuous open;
end;
:: BORSUK_2:exreg 1
registration
let a1 be TopStruct;
cluster Relation-like Function-like one-to-one quasi_total continuous Relation of the carrier of a1,the carrier of a1;
end;
:: BORSUK_2:th 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
st b3 is being_homeomorphism(b1, b2)
holds b3 /" is open(b2, b1);
:: BORSUK_2:condreg 1
registration
cluster -> real (Element of the carrier of I[01]);
end;
:: BORSUK_2:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
ex b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
b3 is continuous(I[01], b1) & b3 . 0 = b2 & b3 . 1 = b2;
:: BORSUK_2:prednot 1 => BORSUK_2:pred 1
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_connected means
ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 st
b1 is continuous(I[01], a1) & b1 . 0 = a2 & b1 . 1 = a3;
end;
:: BORSUK_2:dfs 1
definiens
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_connected
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 st
b1 is continuous(I[01], a1) & b1 . 0 = a2 & b1 . 1 = a3;
:: BORSUK_2:def 1
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_connected
iff
ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
b4 is continuous(I[01], b1) & b4 . 0 = b2 & b4 . 1 = b3;
:: BORSUK_2:prednot 2 => BORSUK_2:pred 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of the carrier of a1;
redefine pred a2,a3 are_connected;
reflexivity;
:: for a1 being non empty TopSpace-like TopStruct
:: for a2 being Element of the carrier of a1 holds
:: a2,a2 are_connected;
end;
:: BORSUK_2:modenot 1 => BORSUK_2:mode 1
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
assume a2,a3 are_connected;
mode Path of A2,A3 -> Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 means
it is continuous(I[01], a1) & it . 0 = a2 & it . 1 = a3;
end;
:: BORSUK_2:dfs 2
definiens
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Function-like quasi_total Relation of the carrier of I[01],the carrier of a1;
To prove
a4 is Path of a2,a3
it is sufficient to prove
thus a2,a3 are_connected;
thus a4 is continuous(I[01], a1) & a4 . 0 = a2 & a4 . 1 = a3;
:: BORSUK_2:def 2
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1
st b2,b3 are_connected
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 holds
b4 is Path of b2,b3
iff
b4 is continuous(I[01], b1) & b4 . 0 = b2 & b4 . 1 = b3;
:: BORSUK_2:exreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster non empty Relation-like Function-like quasi_total continuous total Path of a2,a2;
end;
:: BORSUK_2:attrnot 1 => BORSUK_2:attr 1
definition
let a1 be TopStruct;
attr a1 is arcwise_connected means
for b1, b2 being Element of the carrier of a1 holds
b1,b2 are_connected;
end;
:: BORSUK_2:dfs 3
definiens
let a1 be TopStruct;
To prove
a1 is arcwise_connected
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
b1,b2 are_connected;
:: BORSUK_2:def 3
theorem
for b1 being TopStruct holds
b1 is arcwise_connected
iff
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_connected;
:: BORSUK_2:exreg 3
registration
cluster non empty TopSpace-like arcwise_connected TopStruct;
end;
:: BORSUK_2:modenot 2 => BORSUK_2:mode 1
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
mode Path of A2,A3 -> Function-like quasi_total Relation of the carrier of I[01],the carrier of a1 means
it is continuous(I[01], a1) & it . 0 = a2 & it . 1 = a3;
end;
:: BORSUK_2:dfs 4
definiens
let a1 be arcwise_connected TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Function-like quasi_total Relation of the carrier of I[01],the carrier of a1;
To prove
a4 is Path of a2,a3
it is sufficient to prove
thus a4 is continuous(I[01], a1) & a4 . 0 = a2 & a4 . 1 = a3;
:: BORSUK_2:def 4
theorem
for b1 being arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 holds
b4 is Path of b2,b3
iff
b4 is continuous(I[01], b1) & b4 . 0 = b2 & b4 . 1 = b3;
:: BORSUK_2:condreg 2
registration
let a1 be arcwise_connected TopStruct;
let a2, a3 be Element of the carrier of a1;
cluster -> continuous (Path of a2,a3);
end;
:: BORSUK_2:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is arcwise_connected
holds b1 is connected;
:: BORSUK_2:condreg 3
registration
cluster non empty TopSpace-like arcwise_connected -> connected (TopStruct);
end;
:: BORSUK_2:funcnot 1 => BORSUK_2:func 1
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3, a4 be Element of the carrier of a1;
let a5 be Path of a2,a3;
let a6 be Path of a3,a4;
assume a2,a3 are_connected & a3,a4 are_connected;
func A5 + A6 -> Path of a2,a4 means
for b1 being Element of the carrier of I[01] holds
(b1 <= 1 / 2 implies it . b1 = a5 . (2 * b1)) &
(1 / 2 <= b1 implies it . b1 = a6 . ((2 * b1) - 1));
end;
:: BORSUK_2:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4
st b2,b3 are_connected & b3,b4 are_connected
for b7 being Path of b2,b4 holds
b7 = b5 + b6
iff
for b8 being Element of the carrier of I[01] holds
(b8 <= 1 / 2 implies b7 . b8 = b5 . (2 * b8)) &
(1 / 2 <= b8 implies b7 . b8 = b6 . ((2 * b8) - 1));
:: BORSUK_2:exreg 4
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
cluster non empty Relation-like Function-like constant quasi_total total Path of a2,a2;
end;
:: BORSUK_2:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being constant Path of b2,b2 holds
b3 = I[01] --> b2;
:: BORSUK_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being constant Path of b2,b2 holds
b3 + b3 = b3;
:: BORSUK_2:funcreg 2
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
let a3 be constant Path of a2,a2;
cluster a3 + a3 -> constant;
end;
:: BORSUK_2:funcnot 2 => BORSUK_2:func 2
definition
let a1 be non empty TopSpace-like TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4 be Path of a2,a3;
assume a2,a3 are_connected;
func - A4 -> Path of a3,a2 means
for b1 being Element of the carrier of I[01] holds
it . b1 = a4 . (1 - b1);
end;
:: BORSUK_2:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
for b5 being Path of b3,b2 holds
b5 = - b4
iff
for b6 being Element of the carrier of I[01] holds
b5 . b6 = b4 . (1 - b6);
:: BORSUK_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being constant Path of b2,b2 holds
- b3 = b3;
:: BORSUK_2:funcreg 3
registration
let a1 be non empty TopSpace-like TopStruct;
let a2 be Element of the carrier of a1;
let a3 be constant Path of a2,a2;
cluster - a3 -> constant;
end;
:: BORSUK_2:th 9
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool bool the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b4 " union b3 = union (b4 " b3);
:: BORSUK_2:funcnot 3 => BORSUK_2:func 3
definition
let a1, a2, a3, a4 be non empty TopSpace-like TopStruct;
let a5 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
let a6 be Function-like quasi_total Relation of the carrier of a3,the carrier of a4;
redefine func [:a5, a6:] -> Function-like quasi_total Relation of the carrier of [:a1,a3:],the carrier of [:a2,a4:];
end;
:: BORSUK_2:th 10
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4
for b7, b8 being Element of bool the carrier of [:b3,b4:]
st b8 in Base-Appr b7
holds [:b5,b6:] " b8 is open([:b1,b2:]);
:: BORSUK_2:th 11
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4
for b7 being Element of bool the carrier of [:b3,b4:]
st b7 is open([:b3,b4:])
holds [:b5,b6:] " b7 is open([:b1,b2:]);
:: BORSUK_2:th 12
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b3
for b6 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b4 holds
[:b5,b6:] is continuous([:b1,b2:], [:b3,b4:]);
:: BORSUK_2:condreg 4
registration
cluster empty -> discerning (TopStruct);
end;
:: BORSUK_2:funcreg 4
registration
let a1, a2 be non empty TopSpace-like discerning TopStruct;
cluster [:a1,a2:] -> strict TopSpace-like discerning;
end;
:: BORSUK_2:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
st b1 is being_T1 & b2 is being_T1
holds [:b1,b2:] is being_T1;
:: BORSUK_2:funcreg 5
registration
let a1, a2 be non empty TopSpace-like being_T1 TopStruct;
cluster [:a1,a2:] -> strict TopSpace-like being_T1;
end;
:: BORSUK_2:funcreg 6
registration
let a1, a2 be non empty TopSpace-like being_T2 TopStruct;
cluster [:a1,a2:] -> strict TopSpace-like being_T2;
end;
:: BORSUK_2:funcreg 7
registration
cluster I[01] -> compact being_T2;
end;
:: BORSUK_2:prednot 3 => BORSUK_2:pred 3
definition
let a1 be non empty TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4, a5 be Path of a2,a3;
pred A4,A5 are_homotopic means
ex b1 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of a1 st
b1 is continuous([:I[01],I[01]:], a1) &
(for b2 being Element of the carrier of I[01] holds
b1 .(b2,0) = a4 . b2 & b1 .(b2,1) = a5 . b2 & b1 .(0,b2) = a2 & b1 .(1,b2) = a3);
symmetry;
:: for a1 being non empty TopStruct
:: for a2, a3 being Element of the carrier of a1
:: for a4, a5 being Path of a2,a3
:: st a4,a5 are_homotopic
:: holds a5,a4 are_homotopic;
end;
:: BORSUK_2:dfs 7
definiens
let a1 be non empty TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4, a5 be Path of a2,a3;
To prove
a4,a5 are_homotopic
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of a1 st
b1 is continuous([:I[01],I[01]:], a1) &
(for b2 being Element of the carrier of I[01] holds
b1 .(b2,0) = a4 . b2 & b1 .(b2,1) = a5 . b2 & b1 .(0,b2) = a2 & b1 .(1,b2) = a3);
:: BORSUK_2:def 7
theorem
for b1 being non empty TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Path of b2,b3 holds
b4,b5 are_homotopic
iff
ex b6 being Function-like quasi_total Relation of the carrier of [:I[01],I[01]:],the carrier of b1 st
b6 is continuous([:I[01],I[01]:], b1) &
(for b7 being Element of the carrier of I[01] holds
b6 .(b7,0) = b4 . b7 & b6 .(b7,1) = b5 . b7 & b6 .(0,b7) = b2 & b6 .(1,b7) = b3);
:: BORSUK_2:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds b4,b4 are_homotopic;
:: BORSUK_2:prednot 4 => BORSUK_2:pred 4
definition
let a1 be non empty TopSpace-like arcwise_connected TopStruct;
let a2, a3 be Element of the carrier of a1;
let a4, a5 be Path of a2,a3;
redefine pred a4,a5 are_homotopic;
symmetry;
:: for a1 being non empty TopSpace-like arcwise_connected TopStruct
:: for a2, a3 being Element of the carrier of a1
:: for a4, a5 being Path of a2,a3
:: st a4,a5 are_homotopic
:: holds a5,a4 are_homotopic;
reflexivity;
:: for a1 being non empty TopSpace-like arcwise_connected TopStruct
:: for a2, a3 being Element of the carrier of a1
:: for a4 being Path of a2,a3 holds
:: a4,a4 are_homotopic;
end;
:: BORSUK_2:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1
st b5 is continuous(I[01], b1) & b2 = b5 . 0 & b3 = b5 . 1 & b6 is continuous(I[01], b1) & b3 = b6 . 0 & b4 = b6 . 1
holds ex b7 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
b7 is continuous(I[01], b1) & b2 = b7 . 0 & b4 = b7 . 1 & rng b7 c= (rng b5) \/ rng b6;
:: BORSUK_2:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4
st b5 is continuous(I[01], b1) & b6 is continuous(I[01], b1) & b5 . 0 = b2 & b5 . 1 = b3 & b6 . 0 = b3 & b6 . 1 = b4
holds b5 + b6 is continuous(I[01], b1) & (b5 + b6) . 0 = b2 & (b5 + b6) . 1 = b4;
:: BORSUK_2:exreg 5
registration
let a1 be non empty TopSpace-like TopStruct;
cluster non empty connected compact Element of bool the carrier of a1;
end;
:: BORSUK_2:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
st ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
b4 is continuous(I[01], b1) & b4 . 0 = b2 & b4 . 1 = b3
holds ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 st
b4 is continuous(I[01], b1) & b4 . 0 = b3 & b4 . 1 = b2;
:: BORSUK_2:funcreg 8
registration
cluster I[01] -> arcwise_connected;
end;