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How does one show that any open interval in n-dimensional Euclidean space is connected?

This is homework, and I have been stuck on it for a few hours now. Seems like it should be easy, but I can't get a proof that I am comfortable with.

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@kurt: How do you define an interval in n dimensional euclidean space –  anonymous Nov 17 '10 at 23:05
    
@Kurt: Give an example of interval in $\mathbb{R}^{2}$ –  anonymous Nov 17 '10 at 23:15
    
@Chandru1 A subset of R^n that is the Cartesian product of n open intervals on R. So an open interval of R^1 is (0, 1), and open interval of R^2 is an open rectangle ({(x, y) | 0 < x < 1, 0 < y < 2 for x,y in R}) –  Kurt Kaylor Nov 17 '10 at 23:18
    
@Kurt: So this: $(0,1) \times (0,1) \times, \cdots \times (0,1)$ is an interval in $\mathbb{R}^{n}$ –  anonymous Nov 17 '10 at 23:19
    
@Chandru1 Yes, exactly. (and much better said) ;-) –  Kurt Kaylor Nov 17 '10 at 23:21
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2 Answers

up vote 3 down vote accepted

OK. This is a basic result which states that Cartesian product of connected sets are connected. You may look here for the proof:

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Can you show that an interval is connected in $\mathbb{R}$? Then can you say (a,b) is connected to (c,b) is connected to (c,d)?

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