What would happen if all the sets weren't open? Earlier this year I started my course in multivariable calculus, and since the very beginning the importance of the open sets was very emphasized. Now that we have moved forward I can sense it is true, since they appear in every other definition or theorem pretty often. However, I don't really know the reason why that is true. For this reason, I've been wondering what would happen if we switched the open sets in the definitions, theorems... With other types of sets. Would it imply some kind of mathematical catastrophe, making everything fall apart and become nonsense? It wouldn't destroy everything but it would bring a loss of generality? Does the question even make sense in the first place?
I'm sorry if I haven't worded it very clearly, English isn't my first language (and math either...).
 A: The way your open sets in a space looks like give a lot of information about your space. One can define the sets of open sets (called the Topology of the space) even without a metric. The study of these sets (of open sets) is also called Topology.
To answer your question of importance of open sets in $\mathbb{R}^n$ in calculus more directly, one has to look at what do they give us. Mostly they keep things easy.
Lets say we want to make a definition of a property around a point $p$. Then we can say take a open set $U$ around $p$ and have already a good amount of properties. We know there exist a small open ball around $p$ $B_\epsilon(p)$ in which we have room to "wiggle" around, see definition of derivatives etc. Also we know that any point in $U$ has such an open ball so giving a definition for an open set we always can define and think in open balls. We don't have to worry about boundary points. Also if one wants to exclude a point $q$ one may always take the open neighborhood around a ball small enough such that it does not include the point $q$, even exclude an open ball around $q$.
Often one can replace the word open with more general statements but it highly depends on what exactly you want to define.
One may take these properties for granted in vector spaces but in different spaces without metric things can get more interesting.
A: As said in  the comments, this is a rather broad question, and would be better if you had a particular example. The main property of open sets is that all of their points are interior, which means that they are surrounded by open sets. That's contrasted with boundary points. A boundary point is one for which every open set that contains that point also contains points outside the set. There's a lot of properties that don't work well when you get to boundary points. For instance, suppose you're trying to optimize a function. A simplified explanation for how to do that is to simply follow the gradient. As long as you're going along the direction of the gradient, you're increasing the value of the function. Once you get to a point with a gradient of zero, you can't continue doing this anymore, so you're now at a local maximum. But if you reach a boundary point, then won't be able to follow the gradient if it points outside of the set.
