:: TOPALG_5 semantic presentation
set c1 = |[0,0]|;
set c2 = the carrier of I[01] ;
set c3 = the carrier of R^1 ;
Lemma1:
0 in INT
by INT_1:def 1;
Lemma2:
0 in the carrier of I[01]
by BORSUK_1:86;
then Lemma3:
{0} c= the carrier of I[01]
by ZFMISC_1:37;
Lemma4:
0 in {0}
by TARSKI:def 1;
Lemma5:
the carrier of [:I[01] ,I[01] :] = [:the carrier of I[01] ,the carrier of I[01] :]
by BORSUK_1:def 5;
reconsider c4 = 0, c5 = 1 as Point of I[01] by BORSUK_1:def 17, BORSUK_1:def 18;
Lemma6:
[#] I[01] = the carrier of I[01]
;
Lemma7:
I[01] | ([#] I[01] ) = I[01]
by TSEP_1:3;
Lemma8:
1 - 0 <= 1
;
Lemma9:
(3 / 2) - (1 / 2) <= 1
;
theorem Th1: :: TOPALG_5:1
for b
1, b
2, b
3 being
real number holds
( b
1 <= b
2 & 0
< b
3 implies for b
4 being
Point of
(Closed-Interval-MSpace b1,b2) holds
not ( not
Ball b
4,b
3 = [.b1,b2.] & not
Ball b
4,b
3 = [.b1,(b4 + b3).[ & not
Ball b
4,b
3 = ].(b4 - b3),b2.] & not
Ball b
4,b
3 = ].(b4 - b3),(b4 + b3).[ ) )
theorem Th2: :: TOPALG_5:2
theorem Th3: :: TOPALG_5:3
theorem Th4: :: TOPALG_5:4
theorem Th5: :: TOPALG_5:5
theorem Th6: :: TOPALG_5:6
theorem Th7: :: TOPALG_5:7
theorem Th8: :: TOPALG_5:8
theorem Th9: :: TOPALG_5:9
theorem Th10: :: TOPALG_5:10
theorem Th11: :: TOPALG_5:11
theorem Th12: :: TOPALG_5:12
theorem Th13: :: TOPALG_5:13
Lemma21:
for b1 being non empty TopSpace
for b2 being non empty open SubSpace of b1 holds
( b1 is locally_connected implies TopStruct(# the carrier of b2,the topology of b2 #) is locally_connected )
theorem Th14: :: TOPALG_5:14
theorem Th15: :: TOPALG_5:15
theorem Th16: :: TOPALG_5:16
theorem Th17: :: TOPALG_5:17
:: deftheorem Def1 defines ExtendInt TOPALG_5:def 1 :
definition
let c
6, c
7, c
8 be non
empty TopSpace;
let c
9 be
Function of
[:c6,c7:],c
8;
let c
10 be
Point of c
7;
func Prj1 c
5,c
4 -> Function of a
1,a
3 means :
Def2:
:: TOPALG_5:def 2
for b
1 being
Point of a
1 holds a
6 . b
1 = a
4 . b
1,a
5;
existence
ex b1 being Function of c6,c8 st
for b2 being Point of c6 holds b1 . b2 = c9 . b2,c10
uniqueness
for b1, b2 being Function of c6,c8 holds
( ( for b3 being Point of c6 holds b1 . b3 = c9 . b3,c10 ) & ( for b3 being Point of c6 holds b2 . b3 = c9 . b3,c10 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Prj1 TOPALG_5:def 2 :
definition
let c
6, c
7, c
8 be non
empty TopSpace;
let c
9 be
Function of
[:c6,c7:],c
8;
let c
10 be
Point of c
6;
func Prj2 c
5,c
4 -> Function of a
2,a
3 means :
Def3:
:: TOPALG_5:def 3
for b
1 being
Point of a
2 holds a
6 . b
1 = a
4 . a
5,b
1;
existence
ex b1 being Function of c7,c8 st
for b2 being Point of c7 holds b1 . b2 = c9 . c10,b2
uniqueness
for b1, b2 being Function of c7,c8 holds
( ( for b3 being Point of c7 holds b1 . b3 = c9 . c10,b3 ) & ( for b3 being Point of c7 holds b2 . b3 = c9 . c10,b3 ) implies b1 = b2 )
end;
:: deftheorem Def3 defines Prj2 TOPALG_5:def 3 :
theorem Th18: :: TOPALG_5:18
theorem Th19: :: TOPALG_5:19
set c6 = Tunit_circle 2;
set c7 = the carrier of (Tunit_circle 2);
Lemma29:
dom CircleMap = REAL
by FUNCT_2:def 1, TOPMETR:24;
:: deftheorem Def4 defines cLoop TOPALG_5:def 4 :
theorem Th20: :: TOPALG_5:20
Lemma32:
ex b1 being Subset-Family of (Tunit_circle 2) st
( b1 is_a_cover_of Tunit_circle 2 & b1 is open & ( for b2 being Subset of (Tunit_circle 2) holds
not ( b2 in b1 & ( for b3 being mutually-disjoint open Subset-Family of R^1 holds
not ( union b3 = CircleMap " b2 & ( for b4 being Subset of R^1 holds
( b4 in b3 implies for b5 being Function of (R^1 | b4),((Tunit_circle 2) | b2) holds
( b5 = CircleMap | b4 implies b5 is_homeomorphism ) ) ) ) ) ) ) )
Lemma33:
the carrier of (Sspace 0[01] ) = {0}
by BORSUK_1:def 17, TEX_2:def 4;
then Lemma34:
[#] (Sspace 0[01] ) = {0}
;
Lemma35:
for b1, b2 being real number
for b3 being Subset-Family of (Closed-Interval-TSpace b1,b2)
for b4 being IntervalCover of b3
for b5 being IntervalCoverPts of b4 holds
not ( b3 is_a_cover_of Closed-Interval-TSpace b1,b2 & b3 is open & b3 is connected & b1 <= b2 & b5 is empty )
theorem Th21: :: TOPALG_5:21
theorem Th22: :: TOPALG_5:22
for b
1 being non
empty TopSpacefor b
2 being
Function of
[:b1,I[01] :],
(Tunit_circle 2)for b
3 being
Function of
[:b1,(Sspace 0[01] ):],
R^1 holds
not ( b
2 is
continuous & b
3 is
continuous & b
2 | [:the carrier of b1,{0}:] = CircleMap * b
3 & ( for b
4 being
Function of
[:b1,I[01] :],
R^1 holds
not ( b
4 is
continuous & b
2 = CircleMap * b
4 & b
4 | [:the carrier of b1,{0}:] = b
3 & ( for b
5 being
Function of
[:b1,I[01] :],
R^1 holds
( b
5 is
continuous & b
2 = CircleMap * b
5 & b
5 | [:the carrier of b1,{0}:] = b
3 implies b
4 = b
5 ) ) ) ) )
theorem Th23: :: TOPALG_5:23
theorem Th24: :: TOPALG_5:24
for b
1, b
2 being
Point of
(Tunit_circle 2)for b
3, b
4 being
Path of b
1,b
2for b
5 being
Homotopy of b
3,b
4for b
6 being
Point of
R^1 holds
not ( b
3,b
4 are_homotopic & b
6 in CircleMap " {b1} & ( for b
7 being
Point of
R^1 for b
8, b
9 being
Path of b
6,b
7for b
10 being
Homotopy of b
8,b
9 holds
not ( b
8,b
9 are_homotopic & b
5 = CircleMap * b
10 & b
7 in CircleMap " {b2} & ( for b
11 being
Homotopy of b
8,b
9 holds
( b
5 = CircleMap * b
11 implies b
10 = b
11 ) ) ) ) )
definition
func Ciso -> Function of
INT.Group ,
(pi_1 (Tunit_circle 2),c[10] ) means :
Def5:
:: TOPALG_5:def 5
for b
1 being
Integer holds a
1 . b
1 = Class (EqRel (Tunit_circle 2),c[10] ),
(cLoop b1);
existence
ex b1 being Function of INT.Group ,(pi_1 (Tunit_circle 2),c[10] ) st
for b2 being Integer holds b1 . b2 = Class (EqRel (Tunit_circle 2),c[10] ),(cLoop b2)
uniqueness
for b1, b2 being Function of INT.Group ,(pi_1 (Tunit_circle 2),c[10] ) holds
( ( for b3 being Integer holds b1 . b3 = Class (EqRel (Tunit_circle 2),c[10] ),(cLoop b3) ) & ( for b3 being Integer holds b2 . b3 = Class (EqRel (Tunit_circle 2),c[10] ),(cLoop b3) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Ciso TOPALG_5:def 5 :
theorem Th25: :: TOPALG_5:25
theorem Th26: :: TOPALG_5:26
Lemma43:
for b1 being positive real number
for b2 being Point of (TOP-REAL 2)
for b3 being Point of (Tcircle b2,b1) holds INT.Group , pi_1 (Tcircle b2,b1),b3 are_isomorphic
theorem Th27: :: TOPALG_5:27