I have a question, which is in the title. And the answer in the book is $\mathbf{false}$.
I'm new in topology, but by studying the questions on this site, I understand that for Hausdorff space (which is $\mathbb{R}$) the intersection of an arbitrary number (possibly uncountable) of compact sets is the compact (bounded and closed).
Any closed line segment in $\mathbb{R}$ is a compact. So, if we need to represent a compact subset of $\mathbb{R}$ which is a line segment as the intersection of closed line segments we can take this segment and the statement will be true, if the compact set, which we need to represent as an intersection is the point we can use Cantor's intersection theorem according to which the intersection of closed nested line segments, with lengths go to zero, gives one point.
Thus, any compact subset of $\mathbb{R}$ we can represent as the intersection of closed line segments. Am I wrong, or there is the mistake in the book?
Would be very grateful for help.
You're wrong, because an intersection of line segments is either empty or a line segment. This would imply the only non-empty compact sets are line segments.
Unfortunately a finite set in $\mathbf R$ is compact, as it is closed and bounded.