In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between them.
it seems that the definition does not force left-bounded must be smaller than right-bounded.
so, is the closed interval [1] legitimate technically?
generally [n], where $n \in \mathbb{N}$, the natural number set.
The set consisting of just the number $1$ is a closed interval, but if you want to display it as a closed interval, you have to write it as $[1,1]$. Writing $[1]$ is not a correct use of interval notation.