|
|
| Line 9: |
Line 9: |
|
Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math> |
|
Suppose <math>x=(x_1,x_2,...)\in S^{\infty}</math> then <math>\sum x_n^2=1</math> |
|
|
|
|
|
Define <math>F:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F(x,t)=((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)/||((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)||</math> |
|
Define <math>F_1:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_1(x,t)=((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)/||((1-t)x_1+t\sqrt{x_1^2+x_2^2},(1-t)x_2,x_3,x_4,...)||</math> |
|
|
|
|
|
<math>F_2:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_2(x,t)=((1-t)x_1+t\sqrt{x_1^2+x_3^2},0,(1-t)x_3,x_4,x_5,...)/||((1-t)x_1+t\sqrt{x_1^2+x_3^2},0,(1-t)x_3,x_4,x_5,...)||</math> |
|
|
|
|
|
<math>F_3:S^{\infty}\times I\rightarrow S^{\infty}</math> by <math>F_3(x,t)=((1-t)x_1+t\sqrt{x_1^2+x_4^2},0,0,(1-t)x_4,x_5,x_6,...)/||((1-t)x_1+t\sqrt{x_1^2+x_4^2},0,0,(1-t)x_4,x_5,x_6,...)||</math> |
|
|
|
|
|
and so on ... |
|
|
|
|
|
applying the homotopy <math>F_1</math> in the time interval <math>[0,1/2]</math>, <math>F_2</math> in the interval <math>[1/2,3/4]</math>, <math>F_3</math> in <math>[3/4,5/6]</math> etc... |
|
|
|
|
|
we get the desired contraction to the point <math>(1,0,0,...)</math>. |
Let 
and define 
Claim
is contractible
Proof
Suppose 
then 
Define 
by 
by 
by 
and so on ...
applying the homotopy 
in the time interval ![{\displaystyle [0,1/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf40737bcb67d2b44e2827874fcbbc9e9d133ee1)
, 
in the interval ![{\displaystyle [1/2,3/4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1cdfbb0bf7b7883367f10f7d0cf0c6b897c41a)
, 
in ![{\displaystyle [3/4,5/6]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94cb6091fd78124b0afcf1073464aa6528f2f850)
etc...
we get the desired contraction to the point 
.
