Isolated point

In topology, a branch of mathematics, a point x of a set S is called an isolated point of S, if there exists a neighborhood of x not containing other points of S.

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.

Equivalently, a point x in S is an isolated point of S if and only if it is not a limit point of S.

A set which is made up only of isolated points is called a discrete set. Any discrete subset of Euclidean space is countable, since the isolation of each of its points (together with the fact the rationals are dense in the reals) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many. However, a set can be countable but not discrete, e.g. the rational numbers with the absolute difference metric. See also discrete space.

A set with no isolated point is said to be dense-in-itself. A closed set with no isolated point is called a perfect set.

The number of isolated points is a topological invariant, i.e. if two topological spaces $X$ and $Y$ are homeomorphic, the number of isolated points in each is equal.

Examples

Topological spaces in the following examples are considered as subspaces of the real line.

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.

External links

Weisstein, Eric W., "Isolated Point" from MathWorld.