It's a good question. A close reading of the Wikipedia page <https://en.wikipedia.org/wiki/Interval_%28mathematics%29#Ter... tells me that an open interval has no definite maximum or minimum.

This no doubt sounds strange, but it surely comes down to the slightly odd behavior of real numbers. In an open interval like (0, 1), there is no single real number which is less than one but is greater than all other real numbers less than one. (Proof by Cantor's diagonal: write out a list of all such numbers and you can always construct another which wasn't on the list, even when the list is infinite.) In contrast, the closed interval [0, 1] has a definite maximum element, the number 1.

This seems like a good example of poor UX in mathematics; the terms are too similar and can only be learned by memorization, the notations likewise. Good luck changing it though; Wikipedia informs me that there are no less than two ISO standards codifying it.

There is a perfectly clear notation in the Wikipedia page, however: (a,b) = {x∈ℝ|a<x<b}; [a,b] = {x∈ℝ|a≤x≤b}