We noted above that many important results in metric spaces can be proved using only the basic properties of open sets that the empty set and underlying set X are both open, the intersection of any two open sets is open, and unions of arbitrarily many open sets are open. We will call any collection of sets on X satisfying these properties a topology. In the following section, we also seek to give alternative ways of describing this important collection of sets.
T is closed under finite intersection.
In particular, it is sometimes suffices to use sets which contain open sets but are not necessarily open. We call such sets neighborhoods. Note here that every nhd of a point in X is open. It often happens that the open sets of a space can be very complicated and yet they can all be described using a selection of fairly simple special ones. In addition, it is fortunate that many topological concepts can be characterized in terms of these simpler base or subbase elements.