The Čech-Stone compactification of the real line (with discrete topology) $\beta{\mathbb{R}}$ is not countable. I want to investigate to some properties of $\beta{\mathbb{R}}$ and the remainder $\beta{\mathbb{R}}\setminus\mathbb{R}$. I think the remainder is not Lindelöf. How can I proof this? And what about the metrizability of $\beta{\mathbb{R}}$?


$\newcommand{\cl}{\operatorname{cl}}\beta\Bbb R$ is not even first countable, let alone metrizable: $\cl_{\beta\Bbb R}\Bbb Z$ is homeomorphic to $\beta\Bbb N$, and $\beta\Bbb N$ contains no convergent sequences. Corollary 3.7 in this paper by Henriksen and Isbell shows that $\beta\Bbb R\setminus\Bbb R$ is Lindelöf. Both of these statements hold both for the usual topology on $\Bbb R$ and for the discrete topology on $\Bbb R$.

  • $\begingroup$ Interesting. Uncountable set with the discrete topology is Lindelöf... $\endgroup$ – Asaf Karagila Dec 11 '12 at 7:51
  • $\begingroup$ @Asaf: No, the Čech-Stone remainder of an uncountable discrete space is Lindelöf, because every discrete space is metrizable. $\endgroup$ – Brian M. Scott Dec 11 '12 at 7:54
  • $\begingroup$ I see. That makes much more sense! $\endgroup$ – Asaf Karagila Dec 11 '12 at 7:56
  • $\begingroup$ Thanks so much Prof. Scott. It was very useful for me. $\endgroup$ – ege Dec 11 '12 at 9:14
  • $\begingroup$ @ege: Glad to hear it; you’re very welcome. $\endgroup$ – Brian M. Scott Dec 11 '12 at 9:15

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