I am new to both here and real analysis. It would be quite silly question to most of you guys. But I spent all day, seriously, to figure out. But I couldn't. My question is from the proposition of Royden. [p.17, Royden, 4th]
Proposition. Every nonempty open set is the disjoint union of a countable collection of open intervals.
My question is when we pick one interval with one rational number, what are endpoints of the interval. In my thinking, both endpoints would be included in original open set. But Royden tells me that both endpoints are not included in that. What am I missing? What are the both endpoints?