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April 27th, 2013, 03:49 PM   #1
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Help! Question regarding open sets

I'm stuck on the following question: "Let D be a non-empty 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
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April 27th, 2013, 10:52 PM   #2
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Re: Help! Question regarding open sets

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Originally Posted by chappyform
I'm stuck on the following question: "Let D be a non-empty 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
What's the intuition? E is just a stretch or shrinkage of D. Can you do the problem if, say, D = the open unit interval and a = 1/2? a = 2? The general case should be pretty obvious once you look at some simple cases.
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April 28th, 2013, 05:19 PM   #3
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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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