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

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.

I don't know the reference and thus the context, but are you sure the authors are not referring to a 'bounded interval' ?

Or are the authors distinguishing between the reals (which do not contain [tex] \pm \infty [/tex] ) and the extended reals which do?
This distinction is often ignored.

Look in that book for the definition of "open interval". Your set is an open set and an interval, but perhaps there is a technical definition of "open interval" used in that book, requiring an "open interval" to have the form (a,b) with b a real number.

If it turns out out that a,b are supposed to be members of R, then it makes sense that (5, infinity) is not an open interval (because open sets are the ones of the form (a,b) where both a AND b are in R. (infinity is not in R).