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$X$ is a topological space that is contractible and compact. Show that if $U$ is an open set in $X$ containing $x_0$ then there exists $t_0<1$ so that $H(x,t)∈U$, for all $x∈X$, and all $t_0≤t≤1$. Here $H(x,t)$ is the homotopy from the identity map to the constant map.

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The set $H^{-1}[U]$ is an open neighbourhood of $X\times\{1\}$. What can you conclude? – Miha Habič Nov 26 '12 at 22:17

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