:: TOPALG_3 semantic presentation
set c1 = the carrier of I[01] ;
set c2 = the carrier of R^1 ;
Lemma1:
the carrier of [:I[01] ,I[01] :] = [:the carrier of I[01] ,the carrier of I[01] :]
by BORSUK_1:def 5;
reconsider c3 = 0, c4 = 1 as Point of I[01] by BORSUK_1:def 17, BORSUK_1:def 18;
theorem Th1: :: TOPALG_3:1
theorem Th2: :: TOPALG_3:2
theorem Th3: :: TOPALG_3:3
canceled;
theorem Th4: :: TOPALG_3:4
theorem Th5: :: TOPALG_3:5
theorem Th6: :: TOPALG_3:6
theorem Th7: :: TOPALG_3:7
theorem Th8: :: TOPALG_3:8
theorem Th9: :: TOPALG_3:9
theorem Th10: :: TOPALG_3:10
theorem Th11: :: TOPALG_3:11
theorem Th12: :: TOPALG_3:12
theorem Th13: :: TOPALG_3:13
theorem Th14: :: TOPALG_3:14
theorem Th15: :: TOPALG_3:15
theorem Th16: :: TOPALG_3:16
theorem Th17: :: TOPALG_3:17
theorem Th18: :: TOPALG_3:18
theorem Th19: :: TOPALG_3:19
theorem Th20: :: TOPALG_3:20
for b
1, b
2, b
3, b
4, b
5 being non
empty TopSpacefor b
6 being
Function of
[:b5,b3:],b
1for b
7 being
Function of
[:b5,b4:],b
1for b
8, b
9 being
closed Subset of b
2 holds
not ( b
3 is
SubSpace of b
2 & b
4 is
SubSpace of b
2 & b
8 = [#] b
3 & b
9 = [#] b
4 &
([#] b3) \/ ([#] b4) = [#] b
2 & b
6 is
continuous & b
7 is
continuous & ( for b
10 being
set holds
( b
10 in ([#] [:b5,b3:]) /\ ([#] [:b5,b4:]) implies b
6 . b
10 = b
7 . b
10 ) ) & ( for b
10 being
Function of
[:b5,b2:],b
1 holds
not ( b
10 = b
6 +* b
7 & b
10 is
continuous ) ) )
theorem Th21: :: TOPALG_3:21
for b
1, b
2, b
3, b
4, b
5 being non
empty TopSpacefor b
6 being
Function of
[:b3,b5:],b
1for b
7 being
Function of
[:b4,b5:],b
1for b
8, b
9 being
closed Subset of b
2 holds
not ( b
3 is
SubSpace of b
2 & b
4 is
SubSpace of b
2 & b
8 = [#] b
3 & b
9 = [#] b
4 &
([#] b3) \/ ([#] b4) = [#] b
2 & b
6 is
continuous & b
7 is
continuous & ( for b
10 being
set holds
( b
10 in ([#] [:b3,b5:]) /\ ([#] [:b4,b5:]) implies b
6 . b
10 = b
7 . b
10 ) ) & ( for b
10 being
Function of
[:b2,b5:],b
1 holds
not ( b
10 = b
6 +* b
7 & b
10 is
continuous ) ) )
theorem Th22: :: TOPALG_3:22
theorem Th23: :: TOPALG_3:23
theorem Th24: :: TOPALG_3:24
theorem Th25: :: TOPALG_3:25
theorem Th26: :: TOPALG_3:26
theorem Th27: :: TOPALG_3:27
theorem Th28: :: TOPALG_3:28
theorem Th29: :: TOPALG_3:29
for b
1, b
2 being non
empty TopSpacefor b
3 being
continuous Function of b
1,b
2for b
4, b
5 being
Point of b
1for b
6, b
7 being
Path of b
4,b
5for b
8, b
9 being
Path of b
3 . b
4,b
3 . b
5for b
10 being
Homotopy of b
6,b
7 holds
( b
6,b
7 are_homotopic & b
8 = b
3 * b
6 & b
9 = b
3 * b
7 implies b
3 * b
10 is
Homotopy of b
8,b
9 )
theorem Th30: :: TOPALG_3:30
for b
1, b
2 being non
empty TopSpacefor b
3 being
continuous Function of b
1,b
2for b
4, b
5, b
6 being
Point of b
1for b
7 being
Path of b
4,b
5for b
8 being
Path of b
5,b
6for b
9 being
Path of b
3 . b
4,b
3 . b
5for b
10 being
Path of b
3 . b
5,b
3 . b
6 holds
( b
4,b
5 are_connected & b
5,b
6 are_connected & b
9 = b
3 * b
7 & b
10 = b
3 * b
8 implies b
9 + b
10 = b
3 * (b7 + b8) )
theorem Th31: :: TOPALG_3:31
definition
let c
5, c
6 be non
empty TopSpace;
let c
7 be
Point of c
5;
let c
8 be
Function of c
5,c
6;
assume E15:
c
8 is
continuous
;
set c
9 =
pi_1 c
5,c
7;
set c
10 =
pi_1 c
6,
(c8 . c7);
func FundGrIso c
4,c
3 -> Function of
(pi_1 a1,a3),
(pi_1 a2,(a4 . a3)) means :
Def1:
:: TOPALG_3:def 1
for b
1 being
Element of
(pi_1 a1,a3) holds
ex b
2 being
Loop of a
3ex b
3 being
Loop of a
4 . a
3 st
( b
1 = Class (EqRel a1,a3),b
2 & b
3 = a
4 * b
2 & a
5 . b
1 = Class (EqRel a2,(a4 . a3)),b
3 );
existence
ex b1 being Function of (pi_1 c5,c7),(pi_1 c6,(c8 . c7)) st
for b2 being Element of (pi_1 c5,c7) holds
ex b3 being Loop of c7ex b4 being Loop of c8 . c7 st
( b2 = Class (EqRel c5,c7),b3 & b4 = c8 * b3 & b1 . b2 = Class (EqRel c6,(c8 . c7)),b4 )
uniqueness
for b1, b2 being Function of (pi_1 c5,c7),(pi_1 c6,(c8 . c7)) holds
( ( for b3 being Element of (pi_1 c5,c7) holds
ex b4 being Loop of c7ex b5 being Loop of c8 . c7 st
( b3 = Class (EqRel c5,c7),b4 & b5 = c8 * b4 & b1 . b3 = Class (EqRel c6,(c8 . c7)),b5 ) ) & ( for b3 being Element of (pi_1 c5,c7) holds
ex b4 being Loop of c7ex b5 being Loop of c8 . c7 st
( b3 = Class (EqRel c5,c7),b4 & b5 = c8 * b4 & b2 . b3 = Class (EqRel c6,(c8 . c7)),b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines FundGrIso TOPALG_3:def 1 :
for b
1, b
2 being non
empty TopSpacefor b
3 being
Point of b
1for b
4 being
Function of b
1,b
2 holds
( b
4 is
continuous implies for b
5 being
Function of
(pi_1 b1,b3),
(pi_1 b2,(b4 . b3)) holds
( b
5 = FundGrIso b
4,b
3 iff for b
6 being
Element of
(pi_1 b1,b3) holds
ex b
7 being
Loop of b
3ex b
8 being
Loop of b
4 . b
3 st
( b
6 = Class (EqRel b1,b3),b
7 & b
8 = b
4 * b
7 & b
5 . b
6 = Class (EqRel b2,(b4 . b3)),b
8 ) ) );
theorem Th32: :: TOPALG_3:32
definition
let c
5, c
6 be non
empty TopSpace;
let c
7 be
Point of c
5;
let c
8 be
continuous Function of c
5,c
6;
redefine func FundGrIso as
FundGrIso c
4,c
3 -> Homomorphism of
(pi_1 a1,a3),
(pi_1 a2,(a4 . a3));
coherence
FundGrIso c8,c7 is Homomorphism of (pi_1 c5,c7),(pi_1 c6,(c8 . c7))
end;
theorem Th33: :: TOPALG_3:33
theorem Th34: :: TOPALG_3:34
theorem Th35: :: TOPALG_3:35