In topology, a branch of mathematics, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x which contains no other points of S. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric Equivalently, a point x is not isolated if and only if x is an accumulation point. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"

A set which is made up only of isolated points is called a discrete set. A discrete subset of Euclidean space is countable; however, a set can be countable but not discrete, e. g. the rational numbers. See also discrete space. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "

A closed set with no isolated point is called a perfect set. In Mathematics, more specifically in Point-set topology, the derived set of a subset S of a Topological space is the set of all Limit points

The number of isolated points is a topological invariant, i. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is e. if two topological spaces X and Y are homeomorphic, the number of isolated points in each is equal. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

## Examples

Topological spaces in the following examples are considered as subspaces of the real line. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

• For the set $S=\{0\}\cup [1, 2]$, the point 0 is an isolated point.
• For the set $S=\{0\}\cup \{1, 1/2, 1/3, \dots \}$, 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 ${\mathbb N} = \{0, 1, 2, \ldots \}$ of natural numbers is a discrete set. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an