]
In
mathematics, a
point ''x'' is called an isolated point of a subset ''S'' (in a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'') if ''x'' is an element of ''S'' and there exists a
neighborhood of ''x'' which does not contain any other points of ''S''. This is equivalent to saying that the
singleton is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
in the topological space ''S'' (considered as a
subspace of ''X''). Another equivalent formulation is: an element ''x'' of ''S'' is an isolated point of ''S'' if and only if it is not a
limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point 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 conta ...
of ''S''.
If the space ''X'' is a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
, for example a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, then an element ''x'' of ''S'' is an isolated point of ''S'' if there exists an
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are def ...
around ''x'' which contains only finitely many elements of ''S''.
Related notions
A set that is made up only of isolated points is called a discrete set (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 form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
). Any discrete subset ''S'' of Euclidean space must be
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
, since the isolation of each of its points together with the fact that
rationals are
dense in the
reals means that the points of ''S'' may be mapped into a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.
A set with no isolated point is said to be
dense-in-itself (every neighbourhood of a point contains infinitely many other points of the set). A
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
with no isolated point is called a
perfect set (it contains all its limit points and no isolated points).
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
Standard examples
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s in the following three examples are considered as
subspaces of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
with the standard topology.
* For the set