:: BORSUK_3 semantic presentation

theorem Th1: :: BORSUK_3:1
for b1, b2 being TopSpace holds [#] [:b1,b2:] = [:([#] b1),([#] b2):]
proof end;

registration
let c1 be set ;
let c2 be empty set ;
cluster [:a1,a2:] -> empty ;
coherence
[:c1,c2:] is empty
by ZFMISC_1:113;
end;

registration
let c1 be empty set ;
let c2 be set ;
cluster [:a1,a2:] -> empty ;
coherence
[:c1,c2:] is empty
by ZFMISC_1:113;
end;

theorem Th2: :: BORSUK_3:2
for b1, b2 being non empty TopSpace
for b3 being Point of b1 holds
b2 --> b3 is continuous Function of b2,(b1 | {b3})
proof end;

registration
let c1 be non empty TopStruct ;
cluster id a1 -> being_homeomorphism ;
coherence
id c1 is being_homeomorphism
by TOPGRP_1:20;
end;

Lemma3: for b1 being non empty TopStruct holds b1,b1 are_homeomorphic
proof end;

Lemma4: for b1, b2 being non empty TopStruct holds
( b1,b2 are_homeomorphic implies b2,b1 are_homeomorphic )
proof end;

definition
let c1, c2 be non empty TopStruct ;
redefine pred are_homeomorphic as c1,c2 are_homeomorphic ;
reflexivity
for b1 being non empty TopStruct holds b1,b1 are_homeomorphic
by Lemma3;
symmetry
for b1, b2 being non empty TopStruct holds
( b1,b2 are_homeomorphic implies b2,b1 are_homeomorphic )
by Lemma4;
end;

theorem Th3: :: BORSUK_3:3
for b1, b2, b3 being non empty TopSpace holds
( b1,b2 are_homeomorphic & b2,b3 are_homeomorphic implies b1,b3 are_homeomorphic )
proof end;

registration
let c1 be TopStruct ;
let c2 be empty Subset of c1;
cluster a1 | a2 -> empty ;
coherence
c1 | c2 is empty
proof end;
end;

registration
cluster empty strict TopStruct ;
existence
ex b1 being TopSpace st
( b1 is strict & b1 is empty )
proof end;
end;

theorem Th4: :: BORSUK_3:4
for b1 being TopSpace
for b2 being empty TopSpace holds
( [:b1,b2:] is empty & [:b2,b1:] is empty )
proof end;

theorem Th5: :: BORSUK_3:5
for b1 being empty TopSpace holds b1 is compact
proof end;

registration
cluster empty -> compact TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is empty implies b1 is compact )
by Th5;
end;

registration
let c1 be TopSpace;
let c2 be empty TopSpace;
cluster [:a1,a2:] -> empty compact ;
coherence
[:c1,c2:] is empty
by Th4;
end;

theorem Th6: :: BORSUK_3:6
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:b2,(b1 | {b3}):],b2 holds
( b4 = pr1 the carrier of b2,{b3} implies b4 is one-to-one )
proof end;

theorem Th7: :: BORSUK_3:7
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:(b1 | {b3}),b2:],b2 holds
( b4 = pr2 {b3},the carrier of b2 implies b4 is one-to-one )
proof end;

theorem Th8: :: BORSUK_3:8
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:b2,(b1 | {b3}):],b2 holds
( b4 = pr1 the carrier of b2,{b3} implies b4 " = <:(id b2),(b2 --> b3):> )
proof end;

theorem Th9: :: BORSUK_3:9
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:(b1 | {b3}),b2:],b2 holds
( b4 = pr2 {b3},the carrier of b2 implies b4 " = <:(b2 --> b3),(id b2):> )
proof end;

theorem Th10: :: BORSUK_3:10
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:b2,(b1 | {b3}):],b2 holds
( b4 = pr1 the carrier of b2,{b3} implies b4 is_homeomorphism )
proof end;

theorem Th11: :: BORSUK_3:11
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Function of [:(b1 | {b3}),b2:],b2 holds
( b4 = pr2 {b3},the carrier of b2 implies b4 is_homeomorphism )
proof end;

theorem Th12: :: BORSUK_3:12
for b1 being non empty TopSpace
for b2 being non empty compact TopSpace
for b3 being open Subset of [:b1,b2:]
for b4 being set holds
not ( b4 in { b5 where B is Point of b1 : [:{b5},the carrier of b2:] c= b3 } & ( for b5 being ManySortedSet of the carrier of b2 holds
ex b6 being set st
( b6 in the carrier of b2 & ( for b7 being Subset of b1
for b8 being Subset of b2 holds
not ( b5 . b6 = [b7,b8] & [b4,b6] in [:b7,b8:] & b7 is open & b8 is open & [:b7,b8:] c= b3 ) ) ) ) )
proof end;

theorem Th13: :: BORSUK_3:13
for b1 being non empty TopSpace
for b2 being non empty compact TopSpace
for b3 being open Subset of [:b2,b1:]
for b4 being set holds
not ( b4 in { b5 where B is Point of b1 : [:([#] b2),{b5}:] c= b3 } & ( for b5 being open Subset of b1 holds
not ( b4 in b5 & b5 c= { b6 where B is Point of b1 : [:([#] b2),{b6}:] c= b3 } ) ) )
proof end;

theorem Th14: :: BORSUK_3:14
for b1 being non empty TopSpace
for b2 being non empty compact TopSpace
for b3 being open Subset of [:b2,b1:] holds { b4 where B is Point of b1 : [:([#] b2),{b4}:] c= b3 } in the topology of b1
proof end;

theorem Th15: :: BORSUK_3:15
for b1, b2 being non empty TopSpace
for b3 being Point of b1 holds [:(b1 | {b3}),b2:],b2 are_homeomorphic
proof end;

Lemma15: for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being non empty Subset of b1 holds
( b4 = {b3} implies [:b2,(b1 | b4):],b2 are_homeomorphic )
proof end;

theorem Th16: :: BORSUK_3:16
for b1, b2 being non empty TopSpace holds
( b1,b2 are_homeomorphic & b1 is compact implies b2 is compact )
proof end;

theorem Th17: :: BORSUK_3:17
for b1, b2 being TopSpace
for b3 being SubSpace of b1 holds
[:b2,b3:] is SubSpace of [:b2,b1:]
proof end;

Lemma18: for b1, b2 being TopSpace
for b3 being Subset of [:b2,b1:]
for b4 being Subset of b1 holds
( b3 = [:([#] b2),b4:] implies TopStruct(# the carrier of [:b2,(b1 | b4):],the topology of [:b2,(b1 | b4):] #) = TopStruct(# the carrier of ([:b2,b1:] | b3),the topology of ([:b2,b1:] | b3) #) )
proof end;

theorem Th18: :: BORSUK_3:18
for b1 being non empty TopSpace
for b2 being non empty compact TopSpace
for b3 being Point of b1
for b4 being Subset of [:b2,b1:] holds
( b4 = [:([#] b2),{b3}:] implies b4 is compact )
proof end;

theorem Th19: :: BORSUK_3:19
for b1 being non empty TopSpace
for b2 being non empty compact TopSpace
for b3 being Point of b1 holds [:b2,(b1 | {b3}):] is compact
proof end;

theorem Th20: :: BORSUK_3:20
for b1, b2 being non empty compact TopSpace
for b3 being Subset-Family of b1 holds
( b3 = { b4 where B is open Subset of b1 : [:([#] b2),b4:] c= union (Base-Appr ([#] [:b2,b1:])) } implies ( b3 is open & b3 is_a_cover_of [#] b1 ) )
proof end;

theorem Th21: :: BORSUK_3:21
for b1, b2 being non empty compact TopSpace
for b3 being Subset-Family of b1
for b4 being Subset-Family of [:b2,b1:] holds
( b4 is_a_cover_of [:b2,b1:] & b4 is open & b3 = { b5 where B is open Subset of b1 : ex b1 being Subset-Family of [:b2,b1:] st
( b6 c= b4 & b6 is finite & [:([#] b2),b5:] c= union b6 )
}
implies ( b3 is open & b3 is_a_cover_of b1 ) )
proof end;

theorem Th22: :: BORSUK_3:22
for b1, b2 being non empty compact TopSpace
for b3 being Subset-Family of b1
for b4 being Subset-Family of [:b2,b1:] holds
not ( b4 is_a_cover_of [:b2,b1:] & b4 is open & b3 = { b5 where B is open Subset of b1 : ex b1 being Subset-Family of [:b2,b1:] st
( b6 c= b4 & b6 is finite & [:([#] b2),b5:] c= union b6 )
}
& ( for b5 being Subset-Family of b1 holds
not ( b5 c= b3 & b5 is finite & b5 is_a_cover_of b1 ) ) )
proof end;

theorem Th23: :: BORSUK_3:23
for b1, b2 being non empty compact TopSpace
for b3 being Subset-Family of [:b2,b1:] holds
not ( b3 is_a_cover_of [:b2,b1:] & b3 is open & ( for b4 being Subset-Family of [:b2,b1:] holds
not ( b4 c= b3 & b4 is_a_cover_of [:b2,b1:] & b4 is finite ) ) )
proof end;

Lemma23: for b1, b2 being non empty compact TopSpace holds [:b1,b2:] is compact
proof end;

theorem Th24: :: BORSUK_3:24
for b1, b2 being TopSpace holds
( b1 is compact & b2 is compact implies [:b1,b2:] is compact )
proof end;

registration
let c1, c2 be compact TopSpace;
cluster [:a1,a2:] -> compact ;
coherence
[:c1,c2:] is compact
by Th24;
end;

Lemma25: for b1, b2 being TopSpace
for b3 being SubSpace of b1 holds
[:b3,b2:] is SubSpace of [:b1,b2:]
proof end;

theorem Th25: :: BORSUK_3:25
for b1, b2 being TopSpace
for b3 being SubSpace of b1
for b4 being SubSpace of b2 holds
[:b3,b4:] is SubSpace of [:b1,b2:]
proof end;

theorem Th26: :: BORSUK_3:26
for b1, b2 being TopSpace
for b3 being Subset of [:b2,b1:]
for b4 being Subset of b1
for b5 being Subset of b2 holds
( b3 = [:b5,b4:] implies 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) #) )
proof end;

registration
let c1 be TopStruct ;
cluster compact Element of bool the carrier of a1;
existence
ex b1 being Subset of c1 st b1 is compact
proof end;
end;

registration
let c1 be TopSpace;
let c2 be compact Subset of c1;
cluster a1 | a2 -> compact ;
coherence
c1 | c2 is compact
proof end;
end;

theorem Th27: :: BORSUK_3:27
for b1, b2 being TopSpace
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 is compact & b4 is compact implies [:b3,b4:] is compact Subset of [:b1,b2:] )
proof end;