In Topology (Greek is regarded as one of the first results on geometry that does not depend on any measurements, i.e., one of the first topological results. Georg Cantor, the inventor of set theory, had begun to study the theory of point sets in Euclidean space, in the later part... topology, a point x of a Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school. This article gives a brief and basic introduction... set S is called an isolated point, if there exists a neighbourhood of x not containing other points of S. In particular, in an In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclids concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the standard example of... Euclidean space (or in a In mathematics, a metric space is a set (or space) where a distance between points is defined. History Maurice Fréchet introduced metric spaces in his work is a set of points with an associated distance function (also called a metric) , to to on (((( to is some set and ) can... metric space), x is an isolated point of S, if one can find an Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. Examples With the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is the inside of... open ball around x which contains no other points of S.

A set which is made up only of isolated points is called a discrete set.

Examples

• For the set , the point 0 is an isolated point.

• For the set , each of the points 1/k is an isolated point, but 0 is not an isolated point because there are other points in S as close to 0 as desired.

• The set N={0, 1, 2, ...} of Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... natural numbers is a discrete set.

See also

• In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. Definitions Given a set is defined by letting the distance between any distinct points... discrete space