a question on the $\beta X$

Suppose that $X$ is a topological space and $\beta X$ is Stone–Čech compactification of $X$; and let $U$ is an open set of $X$ and $U'=\operatorname{Int}_{\beta X} \operatorname{cl}_{\beta X} U$.

The questions are these:

1. What's the relation between $U$ and $U'$?
2. If $\{U_n: n\in N\}$ is the cover of $X$, then the $\{U'_n: n\in N\}$ is the cover of $\beta X$?

Thanks for any help:)

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Just to be clear, is this $\beta X$ the Stone–Čech compactification? – Dylan Moreland Jun 30 '12 at 6:57
yes, it is same. – Paul Jun 30 '12 at 7:01

Let $X=\omega$; then $\{n\}$ is clopen in $\beta\omega$ for each $n\in\omega$, so $\{n\}'=\{n\}$. On the other hand, $\omega\,'=\beta\omega$. There’s really not much that you can say about the relationship between $U$ and $U'$.
For the second question, $\big\{\{n\}:n\in\omega\big\}$ is an open cover of $\omega$ such that $\big\{\{n\}':n\in\omega\big\}$ is not an open cover of $\beta\omega$.
A more useful set related to $U$ is $\operatorname{Ex}(U)=\beta X\setminus\operatorname{cl}_{\beta X}(X\setminus U)$: $\{\operatorname{Ex}(U):U\subseteq X\text{ is open}\}$ is a base for the topology of $\beta X$, and $\operatorname{Ex}(U)\cap X=U$.