TITLE: Is The Union of Intervals an Interval or not?
QUESTION [8 upvotes]: I have a question on union of intervals.
My teacher says the union of intervals is not an interval.
Is this always true? 
I mean
$[0,2] \cup [4,5]$ is not an interval, because $[0,2] \cap [4,5]= \varnothing ,$
but what about
$ (0,8) \cup (7,9).$
I think this is an interval because $(0,8) \cup (7,9)=(0,9)$.

REPLY [6 votes]: The problem is that you and your teacher have different interpretations of an ambiguous English statement as a precise mathematical statement.
Your teacher's intended meaning.
"The union of [any two] intervals is not [necessarily] an interval". This statement is easily proved true by providing two disjoint intervals whose union is not an interval, such as [0,2] and [4,5]. 
In other words, there is counterexample to the false statement "For all intervals $X$ and $Y$, $X \cup Y$ is also an interval."
Your interpretation
"The union of [two particular] intervals is not [ever] an interval". This statement is easily proved false providing an example of two overlapping intervals whose union is an interval, such as (0,8) and (7,9).