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April 27th, 2013, 03:49 PM  #1 
Newbie Joined: Feb 2013 Posts: 3 Thanks: 0  Help! Question regarding open sets
I'm stuck on the following question: "Let D be a nonempty subset of the real numbers, and E={ax, x in D} where a>0. Prove that E is open if and only if D is open." I think I must be missing something really obvious, but can't figure out what. Any help greatly appreciated I 
April 27th, 2013, 10:52 PM  #2  
Senior Member Joined: Aug 2012 Posts: 2,007 Thanks: 574  Re: Help! Question regarding open sets Quote:
 
April 28th, 2013, 05:19 PM  #3 
Newbie Joined: Apr 2013 Posts: 10 Thanks: 0  Re: Help! Question regarding open sets
Also, note that it suffices to prove just one direction of the "if and only if" once you observe that: [E is a "stretch" of D by a factor of a] if and only if [D is a "stretch" of E by a factor of 1/a]. So after proving [if D is open, then any stretch E of D is also open], you automatically also know that if E is open, then so is D, since D is a just a "stretch" of E (we're interchanging the roles of D and E here). 

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