Homotopy Extension Property

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  • #1
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Can someone elaborate?

Let I = [0,1], A = {0, 1, 1/2, 1/3, 1/4, \cdots}. Show that the homotopy extension property does not hold on the pair (I, A).

Thanks in advance,

A
 

Answers and Replies

  • #2
you have to specify the space you are mapping into...I will assume it is R. I'm having a little bit of trouble with this one. But, if you let f:I --> R and g:A-->R be any continuous functions such that f(1/n)-g(1/n) converges to 0 as n goes to infinity, then you can construct a homotopy H:AxI --> R and it naturally extends to a homotopy E:XxI -->R. So, you could look at what happens when f(1/n) -g(1/n) does converge to zero. I think this may be the way to solve the problem because there is a natural way of defining the homotopy H whenever this happens. For instance if f(1/n) and g(1/n) both converge to 1 as n goes to infinity. Then we can define a homotopy H = f(a)*g(t).

To me, it seems there must be two functions on A which are homotopic but do not converge to the same point. Once you have that the solution might be easy.
 

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