Contractible and Compact space can be contained in an open set after time $t_0$?

Question

$X$ 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.