Understanding Interval Schedulling problem requirement
Let us check some simple examples.

Suppose there are three intervals [1,3], [2,5], [4, 6]. Then the wanted number is 1. Here is how. Let $T$ be any two of the intervals. Then every interval will has a nonempty intersection with one of the intervals in $T$. For example, you can let $T$ be {[1,3], [2,5]}. You can also let $T$ be {[1,3], [4,6]}. On the other hand, Let $M$ be the set of [2,5], i.e., $M$={[2,5]}. Then every interval will has a nonempty intersection with the only one interval in $M$, [2,5]. Since the wanted number can not be smaller than 1, it is 1.

For a slightly more interesting example, suppose there are 5 intervals [1, 3], [2,5], [4, 7], [6,9], [8,10], where each interval intersects its two neighboring intervals. 


The wanted number is 2. Why? Firstly, any interval will not intersect either [1,3] or [8,10]. So one interval is not enough. Secondly, consider intervals [2,5] and [8,10]. We can see that every interval will intersect one of [2,5] and [8,10]. So two intervals are enough.
If you name those five intervals A, B, C, D, E respectively and connect a pair of letters if the corresponding two intervals intersects, you get a graph with vertices {A,B,C,D,E} with edges AB, BC, CD, DE. (Graphs obtained this way are called interval graphs.) Now it is easy to translate our problem about intersecting intervals into a problem about graphs. The new problem is called the minimum dominating set of a graph.




Here is an equivalent version of the problem that is much simpler.


  Given $n$ fixed intervals all of whose endpoints are distinct. Select some intervals such that the intersection of every original interval and the union of selected intervals is nonempty. How to minimize the number of intervals selected?