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This is the second soft question I am asking today, so I apologise for that. This question, though probably a bit silly has been bugging me for a while and I have not come up with a satisfactory answer. Question is the reason as to why a open interval is used as the domain in the definition of a curve. One reason I can think of is the fact that this ensures a finite speed for the curves at every point which might not be the case at the endpoints of a closed interval. Is there something else to this??Again, my apologies if the question seems a bit too silly.

Edit: My bad, I thought this was common knowledge, so I didnt bother to mention the definition. A curve $ \alpha $ in $ \mathbb R^3 $ is defined as the differentiable function $ \alpha : I \rightarrow \mathbb R^3 $ where $ I $ is an open interval

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If you don't give your definition you are using, you have to at least link us to it. – rschwieb Oct 8 '12 at 16:01
As much as I know closed intervals are also used: for example circle parametrisation $(r\cos\theta,r\sin\theta)$ where $\theta$ belongs to $[0,2\pi]$. Notice that circle is closed curve. – Mykolas Oct 8 '12 at 16:17
What about a line? It has no endpoints (indeed, it is an open set), so you need to parametrize it with an open set. – Christopher A. Wong Oct 8 '12 at 18:32
So I guess this is a definition that can be manipulated on a case by case basis, although the standard definition in books is the one i stated. – Vishesh Oct 9 '12 at 3:09
@Mykolas. I was wondering the same thing,partly what made me question this definition. – Vishesh Oct 9 '12 at 3:10
up vote 1 down vote accepted

Open intervals are used for so colled open curves (line,parabola,hyperbola...). Closed intervals are used for closed curves (circles, elipse...). The reason for use of open intevals for open curvesand closed intervals for closed curves is that parametrisation is a homeomorphism between to "shapes". Circle is not homeomorphic to the line, for example. But it is to any closed loop.

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To avoid possibly confusing the OP: It is not claimed (and not true) that a closed loop is homeomorphic to the closed interval that many people would use to parametrize it. The two endpoints of the closed interval correspond to the same point in the loop, so the parametrization is not one-to-one. – Andreas Blass Aug 7 '13 at 4:11

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