Isolated point
Encyclopedia
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. 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
and 
are homeomorphic, the number of isolated points in each is equal.
of the real line
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
(or in a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean 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
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
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. See also discrete space
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" from each other in a certain sense.- Definitions :Given a set X:...
.
A set with no isolated point is said to be dense-in-itself
Dense-in-itself
In mathematics, a subset A of a topological space is said to be dense-in-itself if A contains no isolated points.Every dense-in-itself closed set is perfect. Conversely, every perfect set is 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
and are homeomorphic, the number of isolated points in each is equal.Examples
Topological spaces in the following examples are considered as subspacesSubspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
.
- 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
of natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s is a discrete set.