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I was watching a video stream a little bit ago and noticed on an equation without context that had the interval $\left[{-\infty, \infty}\right]$. This was preculiar to me as I've never seen the interval for $\mathbb{R}$ expressed this way before, however, I do vaguely remember hearing something about this a long time ago and its use in the construction of certain uncountably infinite sets in axiomatic set theory, but I'm not sure. Is there any use in writing ${-\infty, \infty}$ in a closed interval? If so, what would its use be?

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  • $\begingroup$ The extended real number line will be of interest. $\endgroup$ – Irregular User May 10 '16 at 18:52
  • $\begingroup$ @IrregularUser Thanks, that's just what I was looking for! $\endgroup$ – joshumax May 10 '16 at 18:53
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    $\begingroup$ Hopefully someone can answer the part of your question "If so, what would its use be?"! $\endgroup$ – Irregular User May 10 '16 at 18:54
  • $\begingroup$ The Wikipedia article goes over its uses in measure theory pretty well though $\endgroup$ – joshumax May 10 '16 at 19:04
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$[-\infty,+\infty]$ refers to the extended reals. In general, though, $\mathbb{R}=(-\infty,+\infty)$.

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