mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a point is called an isolated point of a subset (in a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

) if is an element of and there exists a

neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...

of that does not contain any other points of . This is equivalent to saying that the singleton is an

open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

in the topological space (considered as a subspace of ). Another equivalent formulation is: an element of is an isolated point of 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 contains a point of S other than x itself. A ...

of . If the space is a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

, for example a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, then an element of is an isolated point of 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 defin ...

around that contains only finitely many elements of . A

point set In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental ind ...

that is made up only of isolated points is called a discrete set or discrete point 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 to ...

Related notions

Any discrete subset of Euclidean space must be

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...

, since the isolation of each of its points together with the fact that rationals are

dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...

in the reals means that the points of may be mapped injectively onto 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 In general topology, a subset A of a topological space is said to be dense-in-itself or crowded if A has no isolated point. Equivalently, A is dense-in-itself if every point of A is a limit point of A. Thus A is dense-in-itself if and only if A\su ...

'' (every neighbourhood of a point contains other points of the set). A

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

with no isolated point is called a ''

perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. (Some ...

'' (it contains all its limit points and no isolated points). The number of isolated points is a

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

, i.e. if two

topological spaces 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 point ...

are

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

, the number of isolated points in each is equal.

Examples

Standard examples

Topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s in the following three examples are considered as subspaces of the

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

with the standard topology. * For the set

S=\\cup

, 2 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...

the point 0 is an isolated point. * For the set

S=\\cup \,

each of the points is an isolated point, but is not an isolated point because there are other points in as close to as desired. * The set

\N = \

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s is a discrete set. In the topological space

X=\

with topology

\tau=\,

the element is an isolated point, even though

b

belongs to the closure of

\

(and is therefore, in some sense, "close" to ). Such a situation is not possible in a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

. The Morse lemma states that

non-degenerate critical point In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. Th ...

s of certain functions are isolated.

Two counter-intuitive examples

Consider the set of points in the real interval such that every digit of their

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...

representation fulfills the following conditions: * Either

x_i=0

x_i=1.

x_i=1

only for finitely many indices . * If denotes the largest index such that

x_m=1,

then

x_=0.

* If

x_i=1

and

i < m,

then exactly one of the following two conditions holds:

x_=1

x_=1.

Informally, these conditions means that every digit of the binary representation of

x

that equals 1 belongs to a pair ...0110..., except for ...010... at the very end. Now, is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an

uncountable set In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...

. Another set with the same properties can be obtained as follows. Let be the middle-thirds

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

, let

I_1,I_2,I_3,\ldots,I_k,\ldots

be the

component Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis * Lumped e ...

intervals of

,1 C

, and let be a set consisting of one point from each . Since each contains only one point from , every point of is an isolated point. However, if is any point in the Cantor set, then every neighborhood of contains at least one , and hence at least one point of . It follows that each point of the Cantor set lies in the closure of , and therefore has uncountable closure.

References

External links

* {{MathWorld , urlname=IsolatedPoint , title=Isolated Point General topology