What does it mean that a production set is closed and non-empty? And why is it important?
It is just mathematical assumptions in calculus that makes the mathematical analysis has real economic meaning. Not of great importance unless you want to look deeply into the derivation of results.
You may read this paper: https://web.stanford.edu/~jdlevin/Econ%20202/Producer%20Theory.pdf
"Throughout our analysis, we will make the innocent technical assumptions that Y is non-empty (so as to have something to study!) and closed (to make it more likely that optimal production plans exist)."
Hope this helps.