Divide line into 3 segments of infinite length Is it possible to divide line (I mean line of infinite length, $[-\infty, +\infty]$) into a three segments of infinite length? My common sense says it's not possible, because every 2 points define segment of finite length, but am I right?

Your intuition is correct.  It is not possible to partition the real line into three non-overlapping infinite intervals.
Suppose we have three such intervals.
The intervals do not overlap, so only one can extend to $-\infty$.  Thus, two of the intervals must have finite left endpoints.  Similarly, only one can extend to $\infty$, so two of the intervals must have finite right endpoints.
By the pigeonhole principle, at least one interval must have both endpoints finite.  That makes it a finite interval, contradicting our assumption.
Therefore, no such intervals can exist.
