given a point (x) of that set, one may define a collection of sets "around" (that is, containing) x, used to approximate x. Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. For example, every point in X should approximate x to some degree of accuracy. Thus X should be in this family. Once we begin to define "smaller" sets containing x, we tend to approximate x to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about x is required to satisfy. Definitions Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one. Euclidean space A subset of the Euclidean -space is open if, for every point in , there exists a positive real number (depending on ) such that any point in whose Euclidean distance from is smaller than belongs to . Equivalently, a subset of is open if every point in is the center of an open ball contained in An example of a subset of that is not open is the closed interval , since neither nor belongs to for any , no matter how small. Metric space A subset U of a metric space is called open if, for any point x in U, there exists a real number ε > 0 such that any point satisfying belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space. Topological space A topology on a set is a set of subsets of with the properties below. Each member of is called an open set. and Any union of sets in belong to : if then Any finite intersection of sets in belong to : if then together with is called a topological space. Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form where is a positive integer, is the set which is not open in the real line. A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological