Answer
False.
Work Step by Step
A half-closed (or half-open) interval has one of the endpoints included, and the other excluded.
For example,
$(2,5]$ is an interval, $2 \lt x \leq 5$
that contains 2.1,2.0001,2.000000001... but not 2 itself.
It is open from the left,
and closed from the right, as it contains the number 5.
So, you don't need $\pm\infty$ to be endpoints of half-closed intervals.
