(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We know that (a,b) is open in R. But is it open R^n?

2. Relevant equations

3. The attempt at a solution

I don't think (a,b) is open in R^n even if it is open in R. Let's take for example n=2, then

E = {(x,y) | x^2+y^2 < r^2} , where r^2 = |b|^2 + Y^2 , for all y belongs to R. However , we can see that even -b satisfies the above equation but -b is not part of (a,b). Hence, we have a point in R^2 which is not internal point of the segment (a,b). Hence, in R^2 (a,b) is not open.

It looks like (a,b) is not closed either for the neighborhood (-a-e, -a+e), for e >0 for any y belongs to R for any limit point (-a,y) in R^2 has no intersection with (a,b).

Any comments on the proof. Can this be extended to the n dimensional space?

Thanks.

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# Homework Help: Is (a,b) open in R^n

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