Note that we are working in the reals, not the extended reals.
Do you understand a closed interval as "an interval that is a closed set" or as "an interval that includes both its endpoints"? If the former, then you would consider $(-\infty,\infty)$ to be a closed interval. If the latter, then you would not consider $(-\infty,\infty)$ to be a closed interval.
I belong to the latter camp, but I would like to get some consensus here, if possible. (I have been graded down for this, have asked around, and I am surprised that everyone I've consulted calls $(-\infty,\infty)$ a closed interval. Am I really crazy?)
p.s. To be very clear, I am not asking if $(-\infty,\infty)$ is a closed set -- it sure is.