The first sentence sounds okay except I'm not sure what you are talking about. I am not familiar with the term "closed ball." The second sentence is not natural. You could fix the sentence to make it sound more natural by writing, "How could we have the nice properties come from compactness again?" That fixes it so it doesn't sound as strange, but it still is fairly meaningless, at least to me. What are "nice properties?"

After looking at an illustrated picture of closed balls in functional analysis class, we had a discussion regarding the absence of compactness. We were trying to figure out how to make the balls regain their properties of compactness again.

yes, it is about mathematical concepts

This seems technical/subject specific. What is this regarding?