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This might be a trivial question, but I'll ask it anyway.

Lectures on Advanced Mathematical Methods for Physicists by Sunil Mukhi and N. Mukunda said:[tex]X=\{x\in\mathbb{R}\,|\,x > 0\}[/tex]

is an open set.This is not, however, an open interval.

Why is this not an open interval? Let a, b be two elements in X, such that a < b. Then, every real number c satisfying the order relation a < c < b is necessarily in (a, b) and hence in X. Hence X is an interval. Since X is equivalent to the interval (0, [itex]\infty[/itex]), I assumed it is also an open interval. Does the definition of an open interval only apply to intervals with finite end points? I thought otherwise: http://en.wikipedia.org/wiki/Interval_(mathematics)#Infinite_endpoints.

Thanks in advance.