My Math Forum Is R open or closed?

 Real Analysis Real Analysis Math Forum

 April 1st, 2017, 05:56 AM #1 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,060 Thanks: 86 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: CA Posts: 1,238 Thanks: 638 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: 44 Thanks: 26 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: 6,783
Thanks: 2197

Math Focus: Mainly analysis and algebra
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.

 Tags closed, open

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post yuceelly Topology 3 March 8th, 2016 10:45 AM mizunoami Real Analysis 3 November 30th, 2011 07:06 PM rqeeb Real Analysis 3 August 29th, 2011 01:59 AM blbl Real Analysis 1 May 27th, 2010 05:23 PM JC Real Analysis 2 June 6th, 2009 12:13 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top