We form our topologies around the “basis” of open sets. However, we must also take a look at the closed sets that make the “holes” in our topology. So what is a closed set? A closed set is a set such that given an open set of a topology , we can say that is open. Let us now apply this to the real numbers . Our closed sets in this case are closed intervals . This is valid because the complement which is an open set.
With this notion of closed sets in topologies, we may develop some conditions involve in any topological space :
- and are closed,
- arbitrary intersections of closed sets are closed, and
- finite unions of closed sets are closed.
We may prove this with DeMorgan’s law, however we will leave that to the reader as an exercise. However now we move on to what we call the interior and closure of a given subset of a topological space . Essentially, the interior is the union of all open sets contained in and the closure is the intersection of all closed sets in . We denote the interior of as and the closure of as . Automatically you should be thinking that is open and is closed because the finite union of open sets are open and the arbitrary intersection of closed sets are closed. Therefore relative to , we get
Since we have been defining topologies through closed sets, we should define how bases would work. So given a topological space and a a subset , we get the following properties pertaining to the basis of :
- iff every open set containing intersects , and
- iff every basis element containing intersects .
And this brings back connections to our definition of a basis with respect to the open sets of the topological space .