TITLE: Is $(-\infty,\infty)$ a closed **interval**?
QUESTION [8 upvotes]: Note that we are working in the reals, not the extended reals. Now consider two opposing claims:

$(-\infty,\infty)$ is a closed interval, because a closed
interval is an interval that is a closed set.
$(-\infty,\infty)$ is not a closed interval, because a closed
interval is an interval that includes both its endpoints.

Recently, I was very surprised to be informed by several mathematicians that $(-\infty,\infty)$ ought to be called a closed interval, not an open interval. Perhaps I could get some consensus from this community, if possible.
p.s. To be very clear, I am not asking if $(-\infty,\infty)$ is a closed set—it sure is.

Update
For what it's worth, the ISO 80000-2:2009 document explicitly calls $(a,b)$ an open interval and $[a,b]$ a closed interval.

REPLY [9 votes]: Amazing degree of consensus on the wrong answer. I won't bother giving my opinion (which I sort of hinted at in the previous sentence) because I am nobody and my opinion means nothing. Instead I will suggest a method you can use to get an answer, which may be slightly more valid than polling a handful of random internet addicts.
Go to the library and find the shelf with all the introductory analysis books. Look up the statements of theorems like "a continuous function on a finite closed interval attains its maximum" and "a continuous function on a finite closed interval is uniformly continuous". Count how many books include the word "finite" and how many omit it. I haven't done this myself, so I have no idea what result you will get.