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:
- What's the relation between $U$ and $U'$?
- 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:)