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Not in general. Let $\overline X=(\omega_1+1)\times(\omega+1)$ where the ordinals are given their usual order topology, $X=\omega_1\times\omega$, and $U=\omega_1\times\{\omega\}$. If $W\subseteq\overline X$ is any open set that includes $U$, then for every $\alpha<\omega_1$, there is $n_\alpha<\omega$ such that $\{\alpha\}\times[n_\alpha,\omega]\subseteq W$. Since $\mathrm{cf}(\omega_1)>\omega$, there is $n<\omega$ such that $\{\alpha:n=n_\alpha\}$ is cofinal in $\omega_1$. Thus, $\overline W\supseteq\{\omega_1\}\times[n,\omega]$. On the other hand, $\overline U=(\omega_1+1)\times\{\omega\}$, so $\langle\omega_1,n\rangle$ is an element of $\overline W\cap\nu X$ outside $\overline U$.