April 1st, 2017, 05:56 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,158 Thanks: 90  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: Southern California, USA Posts: 1,511 Thanks: 761 
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: 51 Thanks: 27 
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 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,973 Thanks: 2296 Math Focus: Mainly analysis and algebra  

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