Let $\beta \mathbb{N}$ be the Stone Cech compactification of the natural numbers and let $ x\in \beta \mathbb{N}$. Is it true that there exists a sequence of open sets $\{U_n\}_{n=1}^\infty$ in $\beta \mathbb{N}$ such that $\bigcap_{n=1}^\infty U_n = \{x\}$? What about if $ x$ is in the corona?
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4$\begingroup$ Any $G_\delta$ point of a compact Hausdorff space has a countable local base, and this certainly doesn't happen at any point of the remainder in $\beta\mathbb{N}$. In fact, any nonempty closed $G_\delta$ subset contained in the remainder has cardinality $\geq2^\mathfrak{c}$. $\endgroup$– TyroneFeb 18 at 7:12
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Yes if the point is from $\mathbb{N}$ (it is isolated). No if the point is in $\beta\mathbb{N}\setminus\mathbb{N}$ because in that subspace every nonempty $G_\delta$set has nonempty interior, see this paper for more elementary properties of $\beta\mathbb{N}$.