:: BORSUK_2 semantic presentation
Lemma30:
for r being real number holds
( ( 0 <= r & r <= 1 ) iff r 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 :
Lemma81:
( 0 in [.0,1.] & 1 in [.0,1.] )
theorem Th5: :: BORSUK_2:5
Lemma85:
for G being non empty TopSpace
for w1, w2, w3 being Point of G
for h1, h2 being Function of I[01] ,G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) )
definition
let T be non
empty TopSpace;
let a be
Point of
T,
b be
Point of
T,
c be
Point of
T;
let P be
Path of
a,
b;
let Q be
Path of
b,
c;
assume that E32:
a,
b are_connected
and E33:
b,
c are_connected
;
func c5 + c6 -> Path of
a2,
a4 means :
Def5:
:: BORSUK_2:def 5
for
t being
Point of
I[01] holds
( (
t <= 1
/ 2 implies
it . t = P . (2 * t) ) & ( 1
/ 2
<= t implies
it . t = Q . ((2 * t) - 1) ) );
existence
ex b1 being Path of a,c st
for t being Point of I[01] holds
( ( t <= 1 / 2 implies b1 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b1 . t = Q . ((2 * t) - 1) ) )
uniqueness
for b1, b2 being Path of a,c st ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b1 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b1 . t = Q . ((2 * t) - 1) ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b2 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b2 . t = Q . ((2 * t) - 1) ) ) ) holds
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 :
Lemma145:
for r being Real st 0 <= r & r <= 1 holds
( 0 <= 1 - r & 1 - r <= 1 )
Lemma146:
for r being Real st r in the carrier of I[01] holds
1 - r in the carrier of I[01]
theorem Th8: :: BORSUK_2:8
theorem Th9: :: BORSUK_2:9
definition
let S1 be non
empty TopSpace,
S2 be non
empty TopSpace,
T1 be non
empty TopSpace,
T2 be non
empty TopSpace;
let f be
Function of
S1,
S2;
let g be
Function of
T1,
T2;
redefine func [: as
[:c5,c6:] -> Function of
[:a1,a3:],
[:a2,a4:];
coherence
[:f,g:] is Function of [:S1,T1:],[:S2,T2:]
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
Lemma171:
for T1, T2 being non empty TopSpace st T1 is_T2 & T2 is_T2 holds
[:T1,T2:] is_T2
definition
let T be non
empty TopStruct ;
let a be
Point of
T,
b be
Point of
T;
let P be
Path of
a,
b,
Q be
Path of
a,
b;
pred c4,
c5 are_homotopic means :: BORSUK_2:def 7
ex
f being
Function of
[:I[01] ,I[01] :],
T st
(
f is
continuous & ( for
s being
Point of
I[01] holds
(
f . s,0
= P . s &
f . s,1
= Q . s & ( for
t being
Point of
I[01] holds
(
f . 0,
t = a &
f . 1,
t = b ) ) ) ) );
symmetry
for P, Q being Path of a,b st ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = P . s & f . s,1 = Q . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) ) holds
ex f being Function of [:I[01] ,I[01] :],T st
( f is continuous & ( for s being Point of I[01] holds
( f . s,0 = Q . s & f . s,1 = P . s & ( for t being Point of I[01] holds
( f . 0,t = a & f . 1,t = b ) ) ) ) )
end;
:: deftheorem Def7 defines are_homotopic BORSUK_2:def 7 :
for
T being non
empty TopStruct for
a,
b being
Point of
T for
P,
Q being
Path of
a,
b holds
(
P,
Q are_homotopic iff ex
f being
Function of
[:I[01] ,I[01] :],
T st
(
f is
continuous & ( for
s being
Point of
I[01] holds
(
f . s,0
= P . s &
f . s,1
= Q . s & ( for
t being
Point of
I[01] holds
(
f . 0,
t = a &
f . 1,
t = b ) ) ) ) ) );
theorem Th15: :: BORSUK_2:15
theorem Th16: :: BORSUK_2:16
theorem Th17: :: BORSUK_2:17
Lemma191:
for p1, p2 being Point of I[01] st p1 <= p2 holds
[.p1,p2.] is non empty Subset of I[01]
by BORSUK_1:83, RCOMP_1:15, RCOMP_1:16;
theorem Th18: :: BORSUK_2:18