What is the mathematical distinction between closed and open sets? If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would give you an example with a set of numbers $(1, 2)$ vs $[1,2]$ and tell you which bracket signifies open or closed.
But in many theorems the author is dead set about using either closed or open sets. What is the strict mathematical difference that distinguish between the two sets and signifies the importance for such distinction?
Can someone demonstrate with an example where using closed set for a theorem associated with open set would cause some sort of a problem?

(I am assuming everything inside a metric space)
consider any cauchy-sequence inside a closed set...then this will always converge where this is not true for any open set..
same time for any point in the open set there always exist a neighbour hood which is properly contained in the open set..which is in general not true for any closed set...
theses two things are very important for proving a lots of theorem.
