Neighborhood Space (Topology)

Give a set X and, for each element x in X, a complete system of neighborhoods of x, let “fancy N” be the collection of all of these complete systems. We call (X, “fancy N”) a neighborhood space. It’s a useful framework for telling someone about a point x in X along with “useful” subsets related to that point. Concepts like continuity can be translated into this language of neighborhoods, and that’s interesting because we no longer need to directly appeal to the concept of a metric/distance function.