So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t \in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?
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Can you highlight what exactly the question is.. – Jan 05 '19 at 21:00
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Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C \times [0,1] \rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t \in [0,1]$ – J.M.W Turner Jan 05 '19 at 21:10
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Can you add that I the question.. your $Y$ here is singleton.. Right? – Jan 06 '19 at 03:26