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?
Best Answer
You could just as well define your topology by defining the closed sets, demanding that your space and the empty set be closed, and that arbitrary intersections and finite unions of closed sets are closed. Since open sets are the complements of closed sets, this would then give us the open sets, and define the exact same topology.
For a more pedagogic or philosophical answer, see this mathoverflow thread for some good discussion on the motivation/interpretation of open sets: https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets