Prove that if $X$ is $T_4$ and $\Omega=\{\Omega_1\,\Omega_2,...,\Omega_n\}$ is an open cover of $X$, then there exists a refinement $\Omega'$ such that $\overline{\Omega'}=\{\overline{\Omega_1'},...,\overline{\Omega_n'}\}$ has the property that $\overline{\Omega_{i}'}\subseteq \Omega_i \forall i\in\{1,2,..,n\}$
My initial approach was to set $V_i=\Omega_i $\ $ \cup_{j=1}^{i-1}\Omega_j$ and try to use any of the characteristics of normality such as Tychonoff or Urysohn's Theorems. However, they all involve some properties of closed sets. I tried to use the complements of $\Omega_i$ but I don't know what to do with them.
How do I approach this?