TITLE: Motivation for the concept of "open set" in topology
QUESTION [12 upvotes]: I am looking at the section "Motivation" for the Wikipedia entry on "open sets": https://en.wikipedia.org/wiki/Open_set#Motivation and I am not sure it is doing such a good object of motivating open sets, as opposed to closed sets. 
I quote: "Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces."
It seems to me (I am sure this is too naive) that I could equally well distinguish two points by using closed sets instead of opens. What am I missing? What is so crucial about opens that closed sets don't have/do?
Update: I get the feeling that maybe opens are "less precise" than closed sets, in the sense that the boundary of a set, that the closed sets have, is an object with a rather precise "position". Opens seem to avoid having to be that precise about anything - I know this is vague. It might even be wrong, but it seems like it could be a useful intuition to differentiate opens and closeds? Or not?

REPLY [4 votes]: Your intuition about the "fuzziness" of open sets is correct.
Inuitively, if $U$ is an open set in the topological space $X$ and $x \in U$, then $U$ necessarily contains any other point in $X$ that is "near enough" to $x$.  
Now this statement has a precise meaning in a metric space: in that setting, "near enough" means "within distance $\epsilon$" for some $\epsilon > 0$", and then the preceding statement becomes the definition of an open subset of a metric space.
In a general topological space, open sets are what provide the sense of "nearness", so the preceding statement is circular at best, but it is still what you should keep in mind.  You should contrast it with closed sets: if $x$ is a point on the boundary of a closed set $F$, then there will be points in $X$ that come as close as you like to $x$ but are not in $F$ (the points on the "other side of the boundary" to $F$, intuitively speaking: think of the end-points of a closed interval, or a point on the boundary of a closed disk).  
The preceding discussion is purely intuitive: to really see how the definitions of closed and open sets are used you have to get a little further in your study of topology and see the technical way in which these notions are used.  If you haven't already done so, it can be good to start with the metric space case: then the open and closed subsets are defined in terms of the metric, as is the concept "sufficiently close to $x$", and the above intuitions become precise.
Then, in your study, you will see that lots of concepts can be defined, and that lots of theorems can be proved, purely in terms of open and closed subsets, without having to explicitly refer to the metric.  (E.g. the definitions and basic properties of continuity, convergence, compactness, connectedness, etc.) At this point it will seem natural to abstract out the properties of open and closed sets, without using the crutch of a metric to define them.