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Consider the circle with center in $(0,0)$ and radius 1 and the circle in $(2,0)$ and radius 1.5 in the plane. Are they homotopic (a) if we remove the origin, (b) if no point is removed? I think that the answers are NO for (a) and YES for (b). Am I right?

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Yes, you are right. Do you know why? –  Hagen von Eitzen Oct 30 '12 at 22:39
    
@HagenvonEitzen: I believe the first case is quite trivial, for the second I believe that it would not be too difficult to consider a continuous transformation that translates the big circle near the little one while shrinking it too. Thanks however. –  Flast9 Oct 30 '12 at 22:49

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For the second case, you can use the family of homotheties: $$f_t(x,y)=(-4,0)+(1+\tfrac{t}{2})(x+4,y)$$ restricted to $x^2+y^2=1$, $0\leq t\leq 1$. For $t=0$, it is the identity and, for $t=1$, you obtain the second sphere.

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