In textbook I found the following statement: "Any open set on a line is a disjoint union of finite or countable number of intervals."
I undertand the proof (but cannot understand, why the disjoint union of interval with itself -- this interval. I think that this union must be to consist of set of pairs $(a,i)$, where $i\in I$ -- element of index-set. But the interval is the set of points, not the set of the pairs).
And why this statement cannot be generalize to dimensions greater than $1$?
It can be generalised: every open set in $\mathbb{R}^n$ is an at most countable union (can be finite too) of disjoint connected open sets.
It just so happens that intervals only make sense in a linear order and the connected subsets of $\mathbb{R}$ are precisely the order-intervals (possibly of infinite length, like $(1,\infty)$).