# remainder of metrizable separable space

Hi,

A remainder of a space $X$ is a space $bX\setminus X$, where $bX$ is a compactification of $X$.

I want to ask does every separable metrizable space has a separable metrizable remainder?

thanks,

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Crossposted: math.stackexchange.com/questions/218446/… –  Qiaochu Yuan Oct 22 '12 at 2:35
The remainder as you stated it is not uniquely defined from $X$. Are you asking whether every compactification of $X$ has a metrizable separable remainder, or whether some compactification has? It’s certainly false for the Stone–Čech compactification: for example, $\beta\omega\setminus\omega$ is neither first-countable (or metrizable for that matter) nor separable. –  Emil Jeřábek Oct 22 '12 at 15:54
To go in the other direction from Emil's comment: any separable metrizable space has a metrizable compactification (for instance because any Polish space is homeomorphic to a $G_\delta$ subset of the Hilbert cube). In that case the remainder is obvisously separable and metrizable –  Julien Melleray Oct 22 '12 at 17:39