Article BORSUK_3, MML version 4.99.1005
:: BORSUK_3:th 1
theorem
for b1, b2 being TopSpace-like TopStruct holds
[#] [:b1,b2:] = [:[#] b1,[#] b2:];
:: BORSUK_3:funcreg 1
registration
let a1 be set;
let a2 be empty set;
cluster [:a1,a2:] -> empty;
end;
:: BORSUK_3:funcreg 2
registration
let a1 be empty set;
let a2 be set;
cluster [:a1,a2:] -> empty;
end;
:: BORSUK_3:th 2
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1 holds
b2 --> b3 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1 | {b3};
:: BORSUK_3:funcreg 3
registration
let a1 be TopStruct;
cluster id a1 -> Function-like quasi_total being_homeomorphism;
end;
:: BORSUK_3:prednot 1 => BORSUK_3:pred 1
definition
let a1, a2 be TopStruct;
redefine pred a1,a2 are_homeomorphic;
reflexivity;
:: for a1 being TopStruct holds
:: a1,a1 are_homeomorphic;
end;
:: BORSUK_3:prednot 2 => BORSUK_3:pred 2
definition
let a1, a2 be non empty TopStruct;
redefine pred a1,a2 are_homeomorphic;
symmetry;
:: for a1, a2 being non empty TopStruct
:: st a1,a2 are_homeomorphic
:: holds a2,a1 are_homeomorphic;
reflexivity;
:: for a1 being non empty TopStruct holds
:: a1,a1 are_homeomorphic;
end;
:: BORSUK_3:th 3
theorem
for b1, b2, b3 being non empty TopSpace-like TopStruct
st b1,b2 are_homeomorphic & b2,b3 are_homeomorphic
holds b1,b3 are_homeomorphic;
:: BORSUK_3:funcreg 4
registration
let a1 be TopStruct;
let a2 be empty Element of bool the carrier of a1;
cluster a1 | a2 -> empty strict;
end;
:: BORSUK_3:exreg 1
registration
cluster empty strict TopSpace-like TopStruct;
end;
:: BORSUK_3:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being empty TopSpace-like TopStruct holds
[:b1,b2:] is empty & [:b2,b1:] is empty;
:: BORSUK_3:th 5
theorem
for b1 being empty TopSpace-like TopStruct holds
b1 is compact;
:: BORSUK_3:condreg 1
registration
cluster empty TopSpace-like -> compact (TopStruct);
end;
:: BORSUK_3:funcreg 5
registration
let a1 be TopSpace-like TopStruct;
let a2 be empty TopSpace-like TopStruct;
cluster [:a1,a2:] -> empty strict TopSpace-like;
end;
:: BORSUK_3:th 6
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 Relation of the carrier of [:b2,b1 | {b3}:],the carrier of b2
st b4 = pr1(the carrier of b2,{b3})
holds b4 is one-to-one;
:: BORSUK_3:th 7
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 Relation of the carrier of [:b1 | {b3},b2:],the carrier of b2
st b4 = pr2({b3},the carrier of b2)
holds b4 is one-to-one;
:: BORSUK_3:th 8
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 Relation of the carrier of [:b2,b1 | {b3}:],the carrier of b2
st b4 = pr1(the carrier of b2,{b3})
holds b4 /" = <:id b2,b2 --> b3:>;
:: BORSUK_3:th 9
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 Relation of the carrier of [:b1 | {b3},b2:],the carrier of b2
st b4 = pr2({b3},the carrier of b2)
holds b4 /" = <:b2 --> b3,id b2:>;
:: BORSUK_3:th 10
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 Relation of the carrier of [:b2,b1 | {b3}:],the carrier of b2
st b4 = pr1(the carrier of b2,{b3})
holds b4 is being_homeomorphism([:b2,b1 | {b3}:], b2);
:: BORSUK_3:th 11
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 Relation of the carrier of [:b1 | {b3},b2:],the carrier of b2
st b4 = pr2({b3},the carrier of b2)
holds b4 is being_homeomorphism([:b1 | {b3},b2:], b2);
:: BORSUK_3:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like compact TopStruct
for b3 being open Element of bool the carrier of [:b1,b2:]
for b4 being set
st [:{b4},the carrier of b2:] c= b3
holds ex b5 being ManySortedSet of the carrier of b2 st
for b6 being set
st b6 in the carrier of b2
holds ex b7 being Element of bool the carrier of b1 st
ex b8 being Element of bool the carrier of b2 st
b5 . b6 = [b7,b8] & [b4,b6] in [:b7,b8:] & b7 is open(b1) & b8 is open(b2) & [:b7,b8:] c= b3;
:: BORSUK_3:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like compact TopStruct
for b3 being open Element of bool the carrier of [:b2,b1:]
for b4 being set
st [:[#] b2,{b4}:] c= b3
holds ex b5 being open Element of bool the carrier of b1 st
b4 in b5 &
b5 c= {b6 where b6 is Element of the carrier of b1: [:[#] b2,{b6}:] c= b3};
:: BORSUK_3:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like compact TopStruct
for b3 being open Element of bool the carrier of [:b2,b1:] holds
{b4 where b4 is Element of the carrier of b1: [:[#] b2,{b4}:] c= b3} in the topology of b1;
:: BORSUK_3:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1 holds
[:b1 | {b3},b2:],b2 are_homeomorphic;
:: BORSUK_3:th 16
theorem
for b1, b2 being non empty TopSpace-like TopStruct
st b1,b2 are_homeomorphic & b1 is compact
holds b2 is compact;
:: BORSUK_3:th 17
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being SubSpace of b1 holds
[:b2,b3:] is SubSpace of [:b2,b1:];
:: BORSUK_3:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of [:b2,b1:]
st b4 = [:[#] b2,{b3}:]
holds b4 is compact([:b2,b1:]);
:: BORSUK_3:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of the carrier of b1 holds
[:b2,b1 | {b3}:] is compact;
:: BORSUK_3:th 20
theorem
for b1, b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of bool bool the carrier of b1
st b3 = {b4 where b4 is open Element of bool the carrier of b1: [:[#] b2,b4:] c= union Base-Appr [#] [:b2,b1:]}
holds b3 is open(b1) & b3 is_a_cover_of [#] b1;
:: BORSUK_3:th 21
theorem
for b1, b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of [:b2,b1:]
st b4 is_a_cover_of [:b2,b1:] &
b4 is open([:b2,b1:]) &
b3 = {b5 where b5 is open Element of bool the carrier of b1: ex b6 being Element of bool bool the carrier of [:b2,b1:] st
b6 c= b4 & b6 is finite & [:[#] b2,b5:] c= union b6}
holds b3 is open(b1) & b3 is_a_cover_of b1;
:: BORSUK_3:th 22
theorem
for b1, b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of bool bool the carrier of b1
for b4 being Element of bool bool the carrier of [:b2,b1:]
st b4 is_a_cover_of [:b2,b1:] &
b4 is open([:b2,b1:]) &
b3 = {b5 where b5 is open Element of bool the carrier of b1: ex b6 being Element of bool bool the carrier of [:b2,b1:] st
b6 c= b4 & b6 is finite & [:[#] b2,b5:] c= union b6}
holds ex b5 being Element of bool bool the carrier of b1 st
b5 c= b3 & b5 is finite & b5 is_a_cover_of b1;
:: BORSUK_3:th 23
theorem
for b1, b2 being non empty TopSpace-like compact TopStruct
for b3 being Element of bool bool the carrier of [:b2,b1:]
st b3 is_a_cover_of [:b2,b1:] & b3 is open([:b2,b1:])
holds ex b4 being Element of bool bool the carrier of [:b2,b1:] st
b4 c= b3 & b4 is_a_cover_of [:b2,b1:] & b4 is finite;
:: BORSUK_3:th 24
theorem
for b1, b2 being TopSpace-like TopStruct
st b1 is compact & b2 is compact
holds [:b1,b2:] is compact;
:: BORSUK_3:funcreg 6
registration
let a1, a2 be TopSpace-like compact TopStruct;
cluster [:a1,a2:] -> strict TopSpace-like compact;
end;
:: BORSUK_3:th 25
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being SubSpace of b1
for b4 being SubSpace of b2 holds
[:b3,b4:] is SubSpace of [:b1,b2:];
:: BORSUK_3:th 26
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b2,b1:]
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
st b3 = [:b5,b4:]
holds TopStruct(#the carrier of [:b2 | b5,b1 | b4:],the topology of [:b2 | b5,b1 | b4:]#) = TopStruct(#the carrier of [:b2,b1:] | b3,the topology of [:b2,b1:] | b3#);
:: BORSUK_3:exreg 2
registration
let a1 be TopStruct;
cluster compact Element of bool the carrier of a1;
end;
:: BORSUK_3:funcreg 7
registration
let a1 be TopSpace-like TopStruct;
let a2 be compact Element of bool the carrier of a1;
cluster a1 | a2 -> strict compact;
end;
:: BORSUK_3:th 27
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 compact(b1) & b4 is compact(b2)
holds [:b3,b4:] is compact Element of bool the carrier of [:b1,b2:];