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Let $X$ be a normal space, If we have a countable local base at the point m in the remainder of Stone -cech compactification $\beta X\setminus X$, can we say the point m has also a countable local base in $\beta X$?

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No. The Čech-Stone compactification of $\omega_1$ is $\omega_1+1$, its one-point compactification, so the remainder is a singleton, $\{\omega_1\}$. However, the character of the point $\omega_1$ in $\omega_1+1$ is $\omega_1$; it does not have a countable local base. The space $\omega_1$ has the topology induced by a linear order, so it is normal.

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thanks so much.. –  ege Nov 27 '12 at 9:55
    
@ege: You’re welcome. –  Brian M. Scott Nov 28 '12 at 3:16
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