if I were to state the classification of intervals, I would provide the following list:
- the empty interval $\emptyset$,
- the degenerate intervals $\{a\}$ for any real number $a$,
- the non-degenerate, bounded intervals $(a,b)$ (open), $[a,b]$ (closed), $[a,b)$ (right half-open) and $(a,b]$ (left half-open) for any real numbers $a<b$
- the unbounded below, bounded above intervals $(-\infty,a)$ (open) and $(-\infty,a]$ (closed) for any real number $a$
- the bounded below, unbounded above intervals $(a,\infty)$ (open) and $[a,\infty)$ (closed) for any real number $a$