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Suppose that $X$ is a Hausdorff completely regular space. $X$ embeds homeomorphically into its Stone-Cech compactifaction $\beta{X}$ and is dense in $\beta{X}$; we can identify $X \subset \beta{X}$. Suppose we have a cover $\mathcal{V}$ of $X$ by $\beta{X}$-open sets. Must $\mathcal{V}$ also cover all of $\beta{X}$ (and therefore have a finite subcover)?

Thanks.

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No: $\beta\omega_1$ is (homeomorphic to) $\omega_1+1$, and $\omega_1$ is open in $\omega_1+1$.

Added: More generally, if $X$ is locally compact, then $X$ is open in $\beta X$, so any cover of $X$ by $X$-open sets is a cover of $X$ by $\beta X$-open sets. In particular, $\beta\omega$ (or $\beta\Bbb N$, if you prefer) is an example: $\omega$ is locally compact and therefore open in $\beta\omega$.

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