The set of special orthogonal matrices is a closed set. What does that mean, and why do we care?
A question like this is usually discussed only in an upper-division set theory class, which is a class for seniors majoring in math on the theoretical side. Not math for engineering or science, but math for its own sake. By the time you get to a set theory class, you have passed all the difficult classes. Geometry, trigonometry, calculus and differential equations are behind you. As Terry Pratchett might say, you have gone through mathematics and come out the other side.
In an upper division set theory class, you will consider a math fact such as “a set contains its elements”. This fact will be given a fancy name, like “The Baire Category Theorem”, and you will be asked to prove it. Since you are in the habit of following along (or you wouldn’t have made it all the way through mathematics and out the other side), you know exactly what to do. You pull out a sharp pencil, and using the precise notation you were given earlier, you work out the proof in 4 or 5 lines. You are filled with a feeling of peace and confidence, as the rightness of the proof is crystal clear. Then you put the pencil away. You have finished your homework before your coffee has grown cold.
Meanwhile, your friends across the hall in the Comp Sci department are receiving their homework assignment: Write an operating system. From scratch. Due Tuesday. And those guys wondered why I majored in math.