The interval $(-\infty,\infty)$ is open/closed

Working in the reals, not the extended reals, I recently encountered several mathematicians, including my lecturer whose area of interest is topology, independently refer to $(-\infty,\infty),$ presumably since it is a closed set, as a closed interval. I was surprised, because, say, a tall preadolescent isn't necessarily a tall person. Perhaps I could get some consensus from this community, if possible, regarding these opposing claims about the closed set$(-\infty,\infty):$

• $(-\infty,\infty)$is an open interval, because a closedinterval means an interval that includes both its endpoints (i.e., that has a minimum and maximum);
• $(-\infty,\infty)$is a closed interval, because a closedinterval means an interval that is a closed set (i.e., that is topologically closed).

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.