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 October 31st, 2012, 05:51 AM #1 Newbie   Joined: Oct 2012 From: Pakistan Gujranwala Posts: 9 Thanks: 0 Topology Give two examples of a complete metric space having two disjoint dense subsets
 October 31st, 2012, 08:58 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Topology Hint: $\mathbb{Q}$ is dense in $\mathbb{R}.$ Can you think of a dense subset of $\mathbb{R}$ that is disjoint from $\mathbb{Q}?$
 October 31st, 2012, 09:03 AM #3 Member   Joined: Nov 2010 Posts: 48 Thanks: 0 Re: Topology One example is $\mathbb{R}$ with the usual topology and rational numbers as a countable dense subset and irrational numbers also is dense subset, which are two disjoint dense subsets.
 October 31st, 2012, 09:35 AM #4 Member   Joined: Nov 2010 Posts: 48 Thanks: 0 Re: Topology Also, the space $C[0,1]$ is complete and the set of polynomial functions by Weierstrass theorem is dense. I'm not sure, but I think the set with trigonometric functions is also dense.
 November 3rd, 2012, 08:41 PM #5 Newbie   Joined: Oct 2012 From: Pakistan Gujranwala Posts: 9 Thanks: 0 Re: Topology thank you guys.........

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