My Math Forum Is R open or closed?

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 April 1st, 2017, 05:56 AM #1 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,536 Thanks: 108 Is R open or closed? S is open if every point of S has a neighborhood entirely in S. -> R is open. S is closed if every limit point of S belongs to S. -> R is closed. S is a metric space (defined distance).
 April 1st, 2017, 12:44 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,121 Thanks: 1101 are $\pm \infty \in \mathbb{R}$ I'm pretty sure the answer is no. These are not numbers. There are no lack of sequences that are subsets of $\mathbb{R}$ that diverge to $\pm \infty$ Thus not every limit point of $\mathbb{R}$ belongs to $\mathbb{R}$ Thus $\mathbb{R}$ is not closed.
 April 1st, 2017, 05:23 PM #3 Member   Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 In any topological space X, X is both open and closed; i.e. clopen. See https://en.wikipedia.org/wiki/Clopen_set. So yes indeed R is both open and closed.
April 1st, 2017, 05:44 PM   #4
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Quote:
 Originally Posted by romsek There are no lack of sequences that are subsets of $\mathbb{R}$ that diverge to $\pm \infty$
Under the usual metric those sequences are not Cauchy and thus do not have a limit. At least, that's what I'd imagine the standard line to be.

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