If a set is closed and open, it may be bounded (e.g., the empty set), or it may be unbounded (e.g., the set of real numbers).
But what about a closed set that is not open? Such as:
- a singleton
- a set of finite points
- a closed interval
- the union of a finite collection of closed intervals
- the nonempty intersection of an arbitrary (possibly infinite) collection of closed intervals
These are all examples of closed, non-open sets, and they are all bounded.
Prove that if a set is closed and non-open, then it is bounded; or disprove by providing an example of a closed, non-open set that is unbounded.