:: TOPALG_5 semantic presentation

set o = |[0,0]|;

set I = the carrier of I[01] ;

set R = the carrier of R^1 ;

Lm1: 0 in INT
by INT_1:def 1;

Lm2: 0 in the carrier of I[01]
by JORDAN5A:2;

then Lm3: {0} c= the carrier of I[01]
by ZFMISC_1:37;

Lm4: 0 in {0}
by TARSKI:def 1;

Lm5: the carrier of [:I[01] ,I[01] :] = [:the carrier of I[01] ,the carrier of I[01] :]
by BORSUK_1:def 5;

reconsider j0 = 0, j1 = 1 as Point of I[01] by BORSUK_1:def 17, BORSUK_1:def 18;

Lm6: [#] I[01] = the carrier of I[01]
;

Lm7: I[01] | ([#] I[01] ) = I[01]
by TSEP_1:3;

Lm8: 1 - 0 <= 1
;

Lm9: (3 / 2) - (1 / 2) <= 1
;

registration

cluster -> integer Element of the carrier of INT.Group ;
coherence by INT_1:def 2, GR_CY_1:def 4;

end;

registration

cluster INT.Group -> infinite ;
coherence

proof end;

end;

registration

let S be infinite 1-sorted ;
cluster the carrier of S -> infinite ;
coherence by GROUP_1:def 14;

end;

theorem Th1: :: TOPALG_5:1

for r, s, a being real number st r <= s & 0 < a holds
for p being Point of (Closed-Interval-MSpace r,s) holds
( Ball p,a = [.r,s.] or Ball p,a = [.r,(p + a).[ or Ball p,a = ].(p - a),s.] or Ball p,a = ].(p - a),(p + a).[ )

proof end;

theorem Th2: :: TOPALG_5:2

for r, s being real number st r <= s holds
ex B being Basis of Closed-Interval-TSpace r,s st
( ex f being ManySortedSet of (Closed-Interval-TSpace r,s) st
for y being Point of (Closed-Interval-MSpace r,s) holds
( f . y = { (Ball y,(1 / n)) where n is Nat : n <> 0 } & B = Union f ) & ( for X being Subset of (Closed-Interval-TSpace r,s) st X in B holds
X is connected ) )

proof end;

theorem Th3: :: TOPALG_5:3

for T being TopStruct
for A being Subset of T
for t being Point of T st t in A holds
skl t,A c= A

proof end;

registration

let T be TopSpace;
let A be open Subset of T;
cluster T | A -> open ;
coherence

proof end;

end;

theorem Th4: :: TOPALG_5:4

for T being TopSpace
for S being SubSpace of T
for A being Subset of T
for B being Subset of S st A = B holds
T | A = S | B

proof end;

theorem Th5: :: TOPALG_5:5

for S, T being TopSpace
for A, B being Subset of T
for C, D being Subset of S st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & A = C & B = D & A,B are_separated holds
C,D are_separated

proof end;

theorem :: TOPALG_5:6

for S, T being TopSpace st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & S is connected holds
T is connected

proof end;

theorem Th7: :: TOPALG_5:7

for S, T being TopSpace
for A being Subset of S
for B being Subset of T st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & A = B & A is connected holds
B is connected

proof end;

theorem Th8: :: TOPALG_5:8

for S, T being non empty TopSpace
for s being Point of S
for t being Point of T
for A being a_neighborhood of s st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & s = t holds
A is a_neighborhood of t

proof end;

theorem :: TOPALG_5:9

for S, T being non empty TopSpace
for A being Subset of S
for B being Subset of T
for N being a_neighborhood of A st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & A = B holds
N is a_neighborhood of B

proof end;

theorem Th10: :: TOPALG_5:10

for S, T being non empty TopSpace
for A, B being Subset of T
for f being Function of S,T st f is_homeomorphism & A is_a_component_of B holds
f " A is_a_component_of f " B

proof end;

theorem Th11: :: TOPALG_5:11

for T being non empty TopSpace
for S being non empty SubSpace of T
for A being non empty Subset of T
for B being non empty Subset of S st A = B & A is locally_connected holds
B is locally_connected

proof end;

theorem Th12: :: TOPALG_5:12

for S, T being non empty TopSpace st TopStruct(# the carrier of S,the topology of S #) = TopStruct(# the carrier of T,the topology of T #) & S is locally_connected holds
T is locally_connected

proof end;

theorem Th13: :: TOPALG_5:13

for T being non empty TopSpace holds
( T is locally_connected iff [#] T is locally_connected )

proof end;

Lm10: for T being non empty TopSpace
for S being non empty open SubSpace of T st T is locally_connected holds
TopStruct(# the carrier of S,the topology of S #) is locally_connected

proof end;

theorem Th14: :: TOPALG_5:14

for T being non empty TopSpace
for S being non empty open SubSpace of T st T is locally_connected holds
S is locally_connected

proof end;

theorem :: TOPALG_5:15

for S, T being non empty TopSpace st S,T are_homeomorphic & S is locally_connected holds
T is locally_connected

proof end;

theorem Th16: :: TOPALG_5:16

for T being non empty TopSpace st ex B being Basis of T st
for X being Subset of T st X in B holds
X is connected holds
T is locally_connected

proof end;

theorem Th17: :: TOPALG_5:17

for r, s being real number st r <= s holds
Closed-Interval-TSpace r,s is locally_connected

proof end;

registration

cluster I[01] -> locally_connected ;
coherence by Th17, TOPMETR:27;

end;

registration

let A be non empty open Subset of I[01] ;
cluster I[01] | A -> locally_connected open ;
coherence by Th14;

end;

definition

let r be real number ;
func ExtendInt r -> Function of I[01] ,R^1 means :Def1: :: TOPALG_5:def 1
for x being Point of I[01] holds it . x = r * x;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines ExtendInt TOPALG_5:def 1 :

registration

let r be real number ;
cluster ExtendInt r -> continuous ;
coherence

proof end;

end;

definition

let r be real number ;
:: original: ExtendInt
redefine func ExtendInt r -> Path of R^1 0, R^1 r;
coherence

proof end;

end;

definition

let S, T, Y be non empty TopSpace;
let H be Function of [:S,T:],Y;
let t be Point of T;
func Prj1 t,H -> Function of S,Y means :Def2: :: TOPALG_5:def 2
for s being Point of S holds it . s = H . s,t;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines Prj1 TOPALG_5:def 2 :

definition

let S, T, Y be non empty TopSpace;
let H be Function of [:S,T:],Y;
let s be Point of S;
func Prj2 s,H -> Function of T,Y means :Def3: :: TOPALG_5:def 3
for t being Point of T holds it . t = H . s,t;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines Prj2 TOPALG_5:def 3 :

registration

let S, T, Y be non empty TopSpace;
let H be continuous Function of [:S,T:],Y;
let t be Point of T;
cluster Prj1 t,H -> continuous ;
coherence

proof end;

end;

registration

let S, T, Y be non empty TopSpace;
let H be continuous Function of [:S,T:],Y;
let s be Point of S;
cluster Prj2 s,H -> continuous ;
coherence

proof end;

end;

theorem :: TOPALG_5:18

for T being non empty TopSpace
for a, b being Point of T
for P, Q being Path of a,b
for H being Homotopy of P,Q
for t being Point of I[01] st H is continuous holds
Prj1 t,H is continuous

proof end;

theorem :: TOPALG_5:19

for T being non empty TopSpace
for a, b being Point of T
for P, Q being Path of a,b
for H being Homotopy of P,Q
for s being Point of I[01] st H is continuous holds
Prj2 s,H is continuous

proof end;

set TUC = Tunit_circle 2;

set cS1 = the carrier of (Tunit_circle 2);

Lm11: now

Tunit_circle 2 = Tcircle (0.REAL 2),1 by TOPREALB:def 7;
hence the carrier of (Tunit_circle 2) = Sphere |[0,0]|,1 by TOPREALB:9, EUCLID:58; :: thesis:

end;

Lm12: dom CircleMap = REAL
by FUNCT_2:def 1, TOPMETR:24;

definition

let r be real number ;
func cLoop r -> Function of I[01] ,(Tunit_circle 2) means :Def4: :: TOPALG_5:def 4
for x being Point of I[01] holds it . x = |[(cos (((2 * PI ) * r) * x)),(sin (((2 * PI ) * r) * x))]|;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines cLoop TOPALG_5:def 4 :

theorem Th20: :: TOPALG_5:20

for r being real number holds cLoop r = CircleMap * (ExtendInt r)

proof end;

definition

let n be Integer;
:: original: cLoop
redefine func cLoop n -> Loop of c[10] ;
coherence

proof end;

end;

Lm13: ex F being Subset-Family of (Tunit_circle 2) st
( F is_a_cover_of Tunit_circle 2 & F is open & ( for U being Subset of (Tunit_circle 2) st U in F holds
ex D being mutually-disjoint open Subset-Family of R^1 st
( union D = CircleMap " U & ( for d being Subset of R^1 st d in D holds
for f being Function of (R^1 | d),((Tunit_circle 2) | U) st f = CircleMap | d holds
f is_homeomorphism ) ) ) )

proof end;

Lm14: the carrier of (Sspace 0[01] ) = {0}
by BORSUK_1:def 17, TEX_2:def 4;

then Lm15: [#] (Sspace 0[01] ) = {0}
;

Lm16: for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace r,s)
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is_a_cover_of Closed-Interval-TSpace r,s & F is open & F is connected & r <= s holds
not G is empty

proof end;

theorem Th21: :: TOPALG_5:21

for UL being Subset-Family of (Tunit_circle 2) st UL is_a_cover_of Tunit_circle 2 & UL is open holds
for Y being non empty TopSpace
for F being continuous Function of [:Y,I[01] :],(Tunit_circle 2)
for y being Point of Y ex T being non empty FinSequence of REAL st
( T . 1 = 0 & T . (len T) = 1 & T is increasing & ex N being open Subset of Y st
( y in N & ( for i being natural number st i in dom T & i + 1 in dom T holds
ex Ui being non empty Subset of (Tunit_circle 2) st
( Ui in UL & F .: [:N,[.(T . i),(T . (i + 1)).]:] c= Ui ) ) ) )

proof end;

theorem Th22: :: TOPALG_5:22

for Y being non empty TopSpace
for F being Function of [:Y,I[01] :],(Tunit_circle 2)
for Ft being Function of [:Y,(Sspace 0[01] ):],R^1 st F is continuous & Ft is continuous & F | [:the carrier of Y,{0}:] = CircleMap * Ft holds
ex G being Function of [:Y,I[01] :],R^1 st
( G is continuous & F = CircleMap * G & G | [:the carrier of Y,{0}:] = Ft & ( for H being Function of [:Y,I[01] :],R^1 st H is continuous & F = CircleMap * H & H | [:the carrier of Y,{0}:] = Ft holds
G = H ) )

proof end;

theorem Th23: :: TOPALG_5:23

for x0, y0 being Point of (Tunit_circle 2)
for xt being Point of R^1
for f being Path of x0,y0 st xt in CircleMap " {x0} holds
ex ft being Function of I[01] ,R^1 st
( ft . 0 = xt & f = CircleMap * ft & ft is continuous & ( for f1 being Function of I[01] ,R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds
ft = f1 ) )

proof end;

theorem Th24: :: TOPALG_5:24

for x0, y0 being Point of (Tunit_circle 2)
for P, Q being Path of x0,y0
for F being Homotopy of P,Q
for xt being Point of R^1 st P,Q are_homotopic & xt in CircleMap " {x0} holds
ex yt being Point of R^1 ex Pt, Qt being Path of xt,yt ex Ft being Homotopy of Pt,Qt st
( Pt,Qt are_homotopic & F = CircleMap * Ft & yt in CircleMap " {y0} & ( for F1 being Homotopy of Pt,Qt st F = CircleMap * F1 holds
Ft = F1 ) )

proof end;

definition

func Ciso -> Function of INT.Group ,(pi_1 (Tunit_circle 2),c[10] ) means :Def5: :: TOPALG_5:def 5
for n being Integer holds it . n = Class (EqRel (Tunit_circle 2),c[10] ),(cLoop n);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines Ciso TOPALG_5:def 5 :

theorem Th25: :: TOPALG_5:25

for i being Integer
for f being Path of R^1 0, R^1 i holds Ciso . i = Class (EqRel (Tunit_circle 2),c[10] ),(CircleMap * f)

proof end;

definition

:: original: Ciso
redefine func Ciso -> Homomorphism of INT.Group ,(pi_1 (Tunit_circle 2),c[10] );
coherence

proof end;

end;

registration

cluster Ciso -> one-to-one onto ;
coherence

proof end;

end;

theorem Th26: :: TOPALG_5:26

Ciso is being_isomorphism

proof end;

Lm17: for r being real positive number
for o being Point of (TOP-REAL 2)
for x being Point of (Tcircle o,r) holds INT.Group , pi_1 (Tcircle o,r),x are_isomorphic

proof end;

theorem Th27: :: TOPALG_5:27

for S being being_simple_closed_curve SubSpace of TOP-REAL 2
for x being Point of S holds INT.Group , pi_1 S,x are_isomorphic

proof end;

registration

let S be being_simple_closed_curve SubSpace of TOP-REAL 2;
let x be Point of S;
cluster FundamentalGroup S,x -> infinite ;
coherence

proof end;

end;