So i've come across this question, with a follow up question of showing that the union of any two intervals need not be an interval.
I don't see how this could possibly be the case. The general structure of my proof would be to consider the case where:
- The intersection yields an empty set;
- The intersection yields a set with 1 element;
- The intersection yields a set with 2 or more elements;
and then consider each case and show that each is an interval.
But a union can only yield one of those three possibilities too. To elaborate: If any of the three possibilities were not an interval, then an intersection is not necessarily an interval, so each of them must be an interval. But each of these possibilities for the intersection are also the only possibilities for a union, meaning a union of intervals must be an interval too, which is not true.
I'm obviously wrong, but why?