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] In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
''x'' is called an isolated point of a subset ''S'' (in a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'') if ''x'' is an element of ''S'' and there exists a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
of ''x'' which does not contain any other points of ''S''. This is equivalent to saying that the singleton is an open set in the topological space ''S'' (considered as a 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 (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
of ''S''. If the space ''X'' is a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
(or any other
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
), then an element ''x'' of ''S'' is an isolated point of ''S'' if there exists an
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
around ''x'' which contains no other points 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 point, isolated'' from each other in a certain sense. The discrete topology is t ...
). Any discrete subset ''S'' of Euclidean space must be
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, since the isolation of each of its points together with the fact that rationals are
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
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-itselfIn 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\subse ...
(every neighbourhood of a point contains other points of the set). A
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
with no isolated point is called a
perfect setIn general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used i ...
(it has all its limit points and none of them are isolated from it). The number of isolated points is a
topological invariantIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
, i.e. if two
topological spaces In mathematics, a topological space is, roughly speaking, a geometry, geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a Set (mathematics), set of ...
$X$ and $Y$ are
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
, the number of isolated points in each is equal.

# Examples

## Standard examples

Topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s in the following three examples are considered as of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
with the standard topology. * For the set
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s is a discrete set. In the topological space $X=\$ with topology $\tau=\$, the element $a$ is an isolated point, even though $a$ belongs to the closure of $\$ (and is therefore, in some sense, "close" to $b$). Such a situation is not possible in a
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

. The
Morse lemma In mathematics, specifically in differential topology, Morse theory enables one to analyze the topological space, topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typ ...
states that non-degenerate critical points of certain functions are isolated.

## Two counter-intuitive examples

Consider the set $F$ of points $x$ in the real interval $\left(0,1\right)$ such that every digit $x_i$ of their
binary Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: ty ...
representation fulfills the following conditions: * Either $x_i=0$ or $x_i=1$. * $x_i=1$ only for finitely many indices $i$. * If $m$ 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: $x_=1$ or $x_=1$. Informally, these conditions means that every digit of the binary representation of $x$ which equals 1 belongs to a pair ...0110..., except for ...010... at the very end. Now, $F$ is an explicit set consisting entirely of isolated points which has the counter-intuitive property that its closure is an
uncountable set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Another set $F$ with the same properties can be obtained as follows. Let $C$ 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 remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

, let $I_1,I_2,I_3,\ldots$ 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 ele ...
intervals of , and let $F$ be a set consisting of one point from each $I_k$. Since each $I_k$ contains only one point from $F$, every point of $F$ is an isolated point. However, if $p$ is any point in the Cantor set, then every neighborhood of $p$ contains at least one $I_k$, and hence at least one point of $F$. It follows that each point of the Cantor set lies in the closure of $F$, and therefore $F$ has uncountable closure.

*
Acnode An acnode at the origin (curve described in text) An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are "isolated point, isolated point or hermit point". For example the equation ...

*
Adherent point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
Limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...