In
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
, a branch of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, 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.
In particular, in a
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
(or in a
metric spaceIn 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 is not isolated if and only if
x is an
adherent pointIn mathematics, an adherent point is a slight generalization of the idea of a limit point.Let be a topological space and be a subset. A point is an adherent point for if every open set containing contains at least one point of other than . A point is an adherent point for if and only if is...
.
A set which is made up only of isolated points is called a
discrete set.
In
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
, a branch of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, 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.
In particular, in a
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
(or in a
metric spaceIn 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 is not isolated if and only if
x is an
adherent pointIn mathematics, an adherent point is a slight generalization of the idea of a limit point.Let be a topological space and be a subset. A point is an adherent point for if every open set containing contains at least one point of other than . A point is an adherent point for if and only if is...
.
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 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 spaceIn 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-itselfIn mathematics, a subset of a topological space is said to be dense-in-itself if 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
subspacesIn 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 of , the subspace...
of the
real lineIn mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space...
.
- 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 number
In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s is a discrete set.