mod04lec24 - Closure and Limit points

A closed set can be associated with any set, which is called the closure of the set. It is the by definition the smallest closed set containing the given set. We give a description of the closure of a set in the subspace topology in terms of its closure in the larger topology, and an equivalent characterization of the closure in terms of neighbourhoods (open sets containing a point). The latter property is usually much easier to verify, as illustrated in a few examples. We then introduce the concept of limit points and show that the closure of a set is the union of the set with the set of limit points.