:: TOPALG_1 semantic presentation
theorem Th1: :: TOPALG_1:1
theorem Th2: :: TOPALG_1:2
theorem Th3: :: TOPALG_1:3
theorem Th4: :: TOPALG_1:4
theorem Th5: :: TOPALG_1:5
theorem Th6: :: TOPALG_1:6
theorem Th7: :: TOPALG_1:7
theorem Th8: :: TOPALG_1:8
theorem Th9: :: TOPALG_1:9
theorem Th10: :: TOPALG_1:10
theorem Th11: :: TOPALG_1:11
theorem Th12: :: TOPALG_1:12
for b
1, b
2 being
real number for b
3 being
Natfor b
4, b
5, b
6, b
7, b
8, b
9 being
Point of
(Euclid b3)for b
10, b
11, b
12, b
13 being
Point of
(TOP-REAL b3) holds
not ( b
4 = b
10 & b
5 = b
11 & b
6 = b
12 & b
7 = b
13 & b
8 = b
10 + b
12 & b
9 = b
11 + b
13 &
dist b
4,b
5 < b
1 &
dist b
6,b
7 < b
2 & not
dist b
8,b
9 < b
1 + b
2 )
theorem Th13: :: TOPALG_1:13
theorem Th14: :: TOPALG_1:14
for b
1, b
2, b
3, b
4 being
real number for b
5 being
Natfor b
6, b
7, b
8, b
9, b
10, b
11 being
Point of
(Euclid b5)for b
12, b
13, b
14, b
15 being
Point of
(TOP-REAL b5) holds
not ( b
6 = b
12 & b
7 = b
13 & b
8 = b
14 & b
9 = b
15 & b
10 = (b1 * b12) + (b2 * b14) & b
11 = (b1 * b13) + (b2 * b15) &
dist b
6,b
7 < b
3 &
dist b
8,b
9 < b
4 & b
1 <> 0 & b
2 <> 0 & not
dist b
10,b
11 < ((abs b1) * b3) + ((abs b2) * b4) )
Lemma12:
for b1 being Nat
for b2 being non empty TopSpace
for b3, b4, b5 being Function of b2,(TOP-REAL b1) holds
( b3 is continuous & b4 is continuous & ( for b6 being Point of b2 holds b5 . b6 = (b3 . b6) + (b4 . b6) ) implies b5 is continuous )
theorem Th15: :: TOPALG_1:15
canceled;
theorem Th16: :: TOPALG_1:16
theorem Th17: :: TOPALG_1:17
theorem Th18: :: TOPALG_1:18
theorem Th19: :: TOPALG_1:19
theorem Th20: :: TOPALG_1:20
theorem Th21: :: TOPALG_1:21
theorem Th22: :: TOPALG_1:22
theorem Th23: :: TOPALG_1:23
theorem Th24: :: TOPALG_1:24
theorem Th25: :: TOPALG_1:25
theorem Th26: :: TOPALG_1:26
theorem Th27: :: TOPALG_1:27
theorem Th28: :: TOPALG_1:28
theorem Th29: :: TOPALG_1:29
theorem Th30: :: TOPALG_1:30
theorem Th31: :: TOPALG_1:31
theorem Th32: :: TOPALG_1:32
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected & b
5,b
6 are_connected implies for b
7 being
Path of b
2,b
3for b
8 being
Path of b
3,b
4for b
9 being
Path of b
4,b
5for b
10 being
Path of b
5,b
6 holds
((b7 + b8) + b9) + b
10,
(b7 + (b8 + b9)) + b
10 are_homotopic )
theorem Th33: :: TOPALG_1:33
theorem Th34: :: TOPALG_1:34
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected & b
5,b
6 are_connected implies for b
7 being
Path of b
2,b
3for b
8 being
Path of b
3,b
4for b
9 being
Path of b
4,b
5for b
10 being
Path of b
5,b
6 holds
((b7 + b8) + b9) + b
10,b
7 + ((b8 + b9) + b10) are_homotopic )
theorem Th35: :: TOPALG_1:35
theorem Th36: :: TOPALG_1:36
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected & b
5,b
6 are_connected implies for b
7 being
Path of b
2,b
3for b
8 being
Path of b
3,b
4for b
9 being
Path of b
4,b
5for b
10 being
Path of b
5,b
6 holds
(b7 + (b8 + b9)) + b
10,
(b7 + b8) + (b9 + b10) are_homotopic )
theorem Th37: :: TOPALG_1:37
theorem Th38: :: TOPALG_1:38
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5 being
Point of b
1 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
3,b
5 are_connected implies for b
6 being
Path of b
2,b
3for b
7 being
Path of b
5,b
3for b
8 being
Path of b
3,b
4 holds
((b6 + (- b7)) + b7) + b
8,b
6 + b
8 are_homotopic )
theorem Th39: :: TOPALG_1:39
theorem Th40: :: TOPALG_1:40
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5 being
Point of b
1 holds
( b
2,b
3 are_connected & b
2,b
4 are_connected & b
4,b
5 are_connected implies for b
6 being
Path of b
2,b
3for b
7 being
Path of b
4,b
5for b
8 being
Path of b
2,b
4 holds
(((b6 + (- b6)) + b8) + b7) + (- b7),b
8 are_homotopic )
theorem Th41: :: TOPALG_1:41
theorem Th42: :: TOPALG_1:42
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5 being
Point of b
1 holds
( b
2,b
3 are_connected & b
2,b
4 are_connected & b
5,b
4 are_connected implies for b
6 being
Path of b
2,b
3for b
7 being
Path of b
4,b
5for b
8 being
Path of b
2,b
4 holds
(b6 + (((- b6) + b8) + b7)) + (- b7),b
8 are_homotopic )
theorem Th43: :: TOPALG_1:43
theorem Th44: :: TOPALG_1:44
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Point of b
1 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected & b
5,b
6 are_connected & b
2,b
7 are_connected implies for b
8 being
Path of b
2,b
3for b
9 being
Path of b
3,b
4for b
10 being
Path of b
4,b
5for b
11 being
Path of b
5,b
6for b
12 being
Path of b
7,b
4 holds
(b8 + (b9 + b10)) + b
11,
((b8 + b9) + (- b12)) + ((b12 + b10) + b11) are_homotopic )
theorem Th45: :: TOPALG_1:45
for b
1 being non
empty arcwise_connected TopSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Point of b
1for b
8 being
Path of b
2,b
3for b
9 being
Path of b
3,b
4for b
10 being
Path of b
4,b
5for b
11 being
Path of b
5,b
6for b
12 being
Path of b
7,b
4 holds
(b8 + (b9 + b10)) + b
11,
((b8 + b9) + (- b12)) + ((b12 + b10) + b11) are_homotopic
:: deftheorem Def1 defines Loops TOPALG_1:def 1 :
Lemma30:
for b1 being non empty TopSpace
for b2 being Point of b1 holds
ex b3 being Equivalence_Relation of Loops b2 st
for b4, b5 being set holds
( [b4,b5] in b3 iff ( b4 in Loops b2 & b5 in Loops b2 & ex b6, b7 being Loop of b2 st
( b6 = b4 & b7 = b5 & b6,b7 are_homotopic ) ) )
definition
let c
1 be non
empty TopSpace;
let c
2 be
Point of c
1;
func EqRel c
1,c
2 -> Relation of
Loops a
2 means :
Def2:
:: TOPALG_1:def 2
for b
1, b
2 being
Loop of a
2 holds
(
[b1,b2] in a
3 iff b
1,b
2 are_homotopic );
existence
ex b1 being Relation of Loops c2 st
for b2, b3 being Loop of c2 holds
( [b2,b3] in b1 iff b2,b3 are_homotopic )
uniqueness
for b1, b2 being Relation of Loops c2 holds
( ( for b3, b4 being Loop of c2 holds
( [b3,b4] in b1 iff b3,b4 are_homotopic ) ) & ( for b3, b4 being Loop of c2 holds
( [b3,b4] in b2 iff b3,b4 are_homotopic ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines EqRel TOPALG_1:def 2 :
theorem Th46: :: TOPALG_1:46
theorem Th47: :: TOPALG_1:47
definition
let c
1 be non
empty TopSpace;
let c
2 be
Point of c
1;
func FundamentalGroup c
1,c
2 -> strict HGrStr means :
Def3:
:: TOPALG_1:def 3
( the
carrier of a
3 = Class (EqRel a1,a2) & ( for b
1, b
2 being
Element of a
3 holds
ex b
3, b
4 being
Loop of a
2 st
( b
1 = Class (EqRel a1,a2),b
3 & b
2 = Class (EqRel a1,a2),b
4 & the
mult of a
3 . b
1,b
2 = Class (EqRel a1,a2),
(b3 + b4) ) ) );
existence
ex b1 being strict HGrStr st
( the carrier of b1 = Class (EqRel c1,c2) & ( for b2, b3 being Element of b1 holds
ex b4, b5 being Loop of c2 st
( b2 = Class (EqRel c1,c2),b4 & b3 = Class (EqRel c1,c2),b5 & the mult of b1 . b2,b3 = Class (EqRel c1,c2),(b4 + b5) ) ) )
uniqueness
for b1, b2 being strict HGrStr holds
( the carrier of b1 = Class (EqRel c1,c2) & ( for b3, b4 being Element of b1 holds
ex b5, b6 being Loop of c2 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the mult of b1 . b3,b4 = Class (EqRel c1,c2),(b5 + b6) ) ) & the carrier of b2 = Class (EqRel c1,c2) & ( for b3, b4 being Element of b2 holds
ex b5, b6 being Loop of c2 st
( b3 = Class (EqRel c1,c2),b5 & b4 = Class (EqRel c1,c2),b6 & the mult of b2 . b3,b4 = Class (EqRel c1,c2),(b5 + b6) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines FundamentalGroup TOPALG_1:def 3 :
for b
1 being non
empty TopSpacefor b
2 being
Point of b
1for b
3 being
strict HGrStr holds
( b
3 = FundamentalGroup b
1,b
2 iff ( the
carrier of b
3 = Class (EqRel b1,b2) & ( for b
4, b
5 being
Element of b
3 holds
ex b
6, b
7 being
Loop of b
2 st
( b
4 = Class (EqRel b1,b2),b
6 & b
5 = Class (EqRel b1,b2),b
7 & the
mult of b
3 . b
4,b
5 = Class (EqRel b1,b2),
(b6 + b7) ) ) ) );
theorem Th48: :: TOPALG_1:48
Lemma36:
for b1 being non empty TopSpace
for b2 being Point of b1
for b3, b4 being Element of (pi_1 b1,b2)
for b5, b6 being Loop of b2 holds
( b3 = Class (EqRel b1,b2),b5 & b4 = Class (EqRel b1,b2),b6 implies b3 * b4 = Class (EqRel b1,b2),(b5 + b6) )
definition
let c
1 be non
empty TopSpace;
let c
2, c
3 be
Point of c
1;
let c
4 be
Path of c
2,c
3;
assume E37:
c
2,c
3 are_connected
;
func pi_1-iso c
4 -> Function of
(pi_1 a1,a3),
(pi_1 a1,a2) means :
Def4:
:: TOPALG_1:def 4
for b
1 being
Loop of a
3 holds a
5 . (Class (EqRel a1,a3),b1) = Class (EqRel a1,a2),
((a4 + b1) + (- a4));
existence
ex b1 being Function of (pi_1 c1,c3),(pi_1 c1,c2) st
for b2 being Loop of c3 holds b1 . (Class (EqRel c1,c3),b2) = Class (EqRel c1,c2),((c4 + b2) + (- c4))
uniqueness
for b1, b2 being Function of (pi_1 c1,c3),(pi_1 c1,c2) holds
( ( for b3 being Loop of c3 holds b1 . (Class (EqRel c1,c3),b3) = Class (EqRel c1,c2),((c4 + b3) + (- c4)) ) & ( for b3 being Loop of c3 holds b2 . (Class (EqRel c1,c3),b3) = Class (EqRel c1,c2),((c4 + b3) + (- c4)) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines pi_1-iso TOPALG_1:def 4 :
for b
1 being non
empty TopSpacefor b
2, b
3 being
Point of b
1for b
4 being
Path of b
2,b
3 holds
( b
2,b
3 are_connected implies for b
5 being
Function of
(pi_1 b1,b3),
(pi_1 b1,b2) holds
( b
5 = pi_1-iso b
4 iff for b
6 being
Loop of b
3 holds b
5 . (Class (EqRel b1,b3),b6) = Class (EqRel b1,b2),
((b4 + b6) + (- b4)) ) );
theorem Th49: :: TOPALG_1:49
theorem Th50: :: TOPALG_1:50
theorem Th51: :: TOPALG_1:51
theorem Th52: :: TOPALG_1:52
theorem Th53: :: TOPALG_1:53
theorem Th54: :: TOPALG_1:54
theorem Th55: :: TOPALG_1:55
theorem Th56: :: TOPALG_1:56
theorem Th57: :: TOPALG_1:57
theorem Th58: :: TOPALG_1:58
theorem Th59: :: TOPALG_1:59
definition
let c
1 be
Nat;
let c
2, c
3 be
Point of
(TOP-REAL c1);
let c
4, c
5 be
Path of c
2,c
3;
func RealHomotopy c
4,c
5 -> Function of
[:I[01] ,I[01] :],
(TOP-REAL a1) means :
Def5:
:: TOPALG_1:def 5
for b
1, b
2 being
Element of
I[01] holds a
6 . b
1,b
2 = ((1 - b2) * (a4 . b1)) + (b2 * (a5 . b1));
existence
ex b1 being Function of [:I[01] ,I[01] :],(TOP-REAL c1) st
for b2, b3 being Element of I[01] holds b1 . b2,b3 = ((1 - b3) * (c4 . b2)) + (b3 * (c5 . b2))
uniqueness
for b1, b2 being Function of [:I[01] ,I[01] :],(TOP-REAL c1) holds
( ( for b3, b4 being Element of I[01] holds b1 . b3,b4 = ((1 - b4) * (c4 . b3)) + (b4 * (c5 . b3)) ) & ( for b3, b4 being Element of I[01] holds b2 . b3,b4 = ((1 - b4) * (c4 . b3)) + (b4 * (c5 . b3)) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines RealHomotopy TOPALG_1:def 5 :
Lemma46:
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1)
for b4, b5 being Path of b2,b3 holds RealHomotopy b4,b5 is continuous
Lemma47:
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1)
for b4, b5 being Path of b2,b3
for b6 being Point of I[01] holds
( (RealHomotopy b4,b5) . b6,0 = b4 . b6 & (RealHomotopy b4,b5) . b6,1 = b5 . b6 & ( for b7 being Point of I[01] holds
( (RealHomotopy b4,b5) . 0,b7 = b2 & (RealHomotopy b4,b5) . 1,b7 = b3 ) ) )
theorem Th60: :: TOPALG_1:60
theorem Th61: :: TOPALG_1:61
theorem Th62: :: TOPALG_1:62