Can you have an open interval over the set of natural numbers? Can you have an open interval for instance, over the set of natural numbers? Wouldn't every open interval on the natural numbers simply be another way of writing a closed interval over the naturals?
Do the concepts of "open" and "closed" intervals only make sense for "continuous stuff"?
How do "open" and "closed"  intervals generalize for non-'numerical' stuff?
What is professional mathematical terminology for "stuff"? 

Short answer: yes.
Long answer:
Let $\mathbb R$ be endowed with the usual topology of open sets generated by open intervals (a,b). The subset topology on $\mathbb N\subset\mathbb R$ has as open sets all open sets in $\mathbb R$ intersected with $\mathbb N$. Same for closed sets. This is means any subset of $\mathbb N$ is both closed and open (sometimes known as clopen https://www.youtube.com/watch?v=SyD4p8_y8Kw) in this topology. This is also known as the discrete topology. So a range, or interval, in the natural numbers such as $\{i,i+1,i+2,\dots,i+k\}=[i,i+k]\cap\mathbb N=(i-1/2,i+k+1/2)\cap\mathbb N$ is closed (and open) in the usual topology associated with the natural numbers.
