:: BORSUK_2 semantic presentation
Lemma1:
for b1 being real number holds
( ( 0 <= b1 & b1 <= 1 ) iff b1 in the carrier of I[01] )
theorem Th1: :: BORSUK_2:1
theorem Th2: :: BORSUK_2:2
canceled;
theorem Th3: :: BORSUK_2:3
theorem Th4: :: BORSUK_2:4
:: deftheorem Def1 defines are_connected BORSUK_2:def 1 :
:: deftheorem Def2 defines Path BORSUK_2:def 2 :
:: deftheorem Def3 defines arcwise_connected BORSUK_2:def 3 :
:: deftheorem Def4 defines Path BORSUK_2:def 4 :
Lemma7:
( 0 in [.0,1.] & 1 in [.0,1.] )
theorem Th5: :: BORSUK_2:5
Lemma9:
for b1 being non empty TopSpace
for b2, b3, b4 being Point of b1
for b5, b6 being Function of I[01] ,b1 holds
not ( b5 is continuous & b2 = b5 . 0 & b3 = b5 . 1 & b6 is continuous & b3 = b6 . 0 & b4 = b6 . 1 & ( for b7 being Function of I[01] ,b1 holds
not ( b7 is continuous & b2 = b7 . 0 & b4 = b7 . 1 & rng b7 c= (rng b5) \/ (rng b6) ) ) )
definition
let c
1 be non
empty TopSpace;
let c
2, c
3, c
4 be
Point of c
1;
let c
5 be
Path of c
2,c
3;
let c
6 be
Path of c
3,c
4;
assume that E10:
c
2,c
3 are_connected
and E11:
c
3,c
4 are_connected
;
func c
5 + c
6 -> Path of a
2,a
4 means :
Def5:
:: BORSUK_2:def 5
for b
1 being
Point of
I[01] holds
( ( b
1 <= 1
/ 2 implies a
7 . b
1 = a
5 . (2 * b1) ) & ( 1
/ 2
<= b
1 implies a
7 . b
1 = a
6 . ((2 * b1) - 1) ) );
existence
ex b1 being Path of c2,c4 st
for b2 being Point of I[01] holds
( ( b2 <= 1 / 2 implies b1 . b2 = c5 . (2 * b2) ) & ( 1 / 2 <= b2 implies b1 . b2 = c6 . ((2 * b2) - 1) ) )
uniqueness
for b1, b2 being Path of c2,c4 holds
( ( for b3 being Point of I[01] holds
( ( b3 <= 1 / 2 implies b1 . b3 = c5 . (2 * b3) ) & ( 1 / 2 <= b3 implies b1 . b3 = c6 . ((2 * b3) - 1) ) ) ) & ( for b3 being Point of I[01] holds
( ( b3 <= 1 / 2 implies b2 . b3 = c5 . (2 * b3) ) & ( 1 / 2 <= b3 implies b2 . b3 = c6 . ((2 * b3) - 1) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines + BORSUK_2:def 5 :
theorem Th6: :: BORSUK_2:6
theorem Th7: :: BORSUK_2:7
:: deftheorem Def6 defines - BORSUK_2:def 6 :
Lemma13:
for b1 being Real holds
( 0 <= b1 & b1 <= 1 implies ( 0 <= 1 - b1 & 1 - b1 <= 1 ) )
Lemma14:
for b1 being Real holds
( b1 in the carrier of I[01] implies 1 - b1 in the carrier of I[01] )
theorem Th8: :: BORSUK_2:8
theorem Th9: :: BORSUK_2:9
definition
let c
1, c
2, c
3, c
4 be non
empty TopSpace;
let c
5 be
Function of c
1,c
2;
let c
6 be
Function of c
3,c
4;
redefine func [: as
[:c5,c6:] -> Function of
[:a1,a3:],
[:a2,a4:];
coherence
[:c5,c6:] is Function of [:c1,c3:],[:c2,c4:]
end;
theorem Th10: :: BORSUK_2:10
theorem Th11: :: BORSUK_2:11
theorem Th12: :: BORSUK_2:12
theorem Th13: :: BORSUK_2:13
canceled;
theorem Th14: :: BORSUK_2:14
Lemma21:
for b1, b2 being non empty TopSpace holds
( b1 is_T2 & b2 is_T2 implies [:b1,b2:] is_T2 )
definition
let c
1 be non
empty TopStruct ;
let c
2, c
3 be
Point of c
1;
let c
4, c
5 be
Path of c
2,c
3;
pred c
4,c
5 are_homotopic means :: BORSUK_2:def 7
ex b
1 being
Function of
[:I[01] ,I[01] :],a
1 st
( b
1 is
continuous & ( for b
2 being
Point of
I[01] holds
( b
1 . b
2,0
= a
4 . b
2 & b
1 . b
2,1
= a
5 . b
2 & ( for b
3 being
Point of
I[01] holds
( b
1 . 0,b
3 = a
2 & b
1 . 1,b
3 = a
3 ) ) ) ) );
symmetry
for b1, b2 being Path of c2,c3 holds
not ( ex b3 being Function of [:I[01] ,I[01] :],c1 st
( b3 is continuous & ( for b4 being Point of I[01] holds
( b3 . b4,0 = b1 . b4 & b3 . b4,1 = b2 . b4 & ( for b5 being Point of I[01] holds
( b3 . 0,b5 = c2 & b3 . 1,b5 = c3 ) ) ) ) ) & ( for b3 being Function of [:I[01] ,I[01] :],c1 holds
not ( b3 is continuous & ( for b4 being Point of I[01] holds
( b3 . b4,0 = b2 . b4 & b3 . b4,1 = b1 . b4 & ( for b5 being Point of I[01] holds
( b3 . 0,b5 = c2 & b3 . 1,b5 = c3 ) ) ) ) ) ) )
end;
:: deftheorem Def7 defines are_homotopic BORSUK_2:def 7 :
for b
1 being non
empty TopStruct for b
2, b
3 being
Point of b
1for b
4, b
5 being
Path of b
2,b
3 holds
( b
4,b
5 are_homotopic iff ex b
6 being
Function of
[:I[01] ,I[01] :],b
1 st
( b
6 is
continuous & ( for b
7 being
Point of
I[01] holds
( b
6 . b
7,0
= b
4 . b
7 & b
6 . b
7,1
= b
5 . b
7 & ( for b
8 being
Point of
I[01] holds
( b
6 . 0,b
8 = b
2 & b
6 . 1,b
8 = b
3 ) ) ) ) ) );
theorem Th15: :: BORSUK_2:15
theorem Th16: :: BORSUK_2:16
theorem Th17: :: BORSUK_2:17
Lemma24:
for b1, b2 being Point of I[01] holds
( b1 <= b2 implies [.b1,b2.] is non empty Subset of I[01] )
by BORSUK_1:83, RCOMP_1:15, RCOMP_1:16;
theorem Th18: :: BORSUK_2:18