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April 1st, 2017, 05:56 AM   #1
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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).
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April 1st, 2017, 12:44 PM   #2
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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.
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April 1st, 2017, 05:23 PM   #3
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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.
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April 1st, 2017, 05:44 PM   #4
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Quote:
Originally Posted by romsek View Post
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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