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] In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 poin ...
''X'') if ''x'' is an element of ''S'' and there exists a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 are su ...
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 contai ...
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 setti ...
, 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 Euclidea ...
, 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 defi ...
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 numbers ...
, since the isolation of each of its points together with the fact that
rationals In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
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 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\ ...
(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 space, a ...
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 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 ...
$X$ and $Y$ are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
, 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 poin ...
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 $S=\\cup$
, 2 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math>, the point 0 is an isolated point. * For the set $S=\\cup \$, 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, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s is a discrete set. In the topological space $X=\$ with topology $\tau=\$, the element $a$ is an isolated point, even though $b$ belongs to the closure of $\$ (and is therefore, in some sense, "close" to $a$). Such a situation is not possible in a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
. The Morse lemma 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 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, an uncountable set (or uncountably infinite set) 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 nu ...
. 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 unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
, let $I_1,I_2,I_3,\ldots$ be the
component Circuit Component may refer to: â€¢Are devices that perform functions when they are connected in a circuit.   In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...
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.

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Acnode An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or hermit point". For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is ...
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Adherent point In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (or equivale ...
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Accumulation 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 contai ...
*
Point cloud Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Poin ...