# Isolated point

Last updated

In mathematics, a point x is called an isolated point of a subset S (in a topological space 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 {x} is an open set 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 of S.

## Contents

If the space X is a Euclidean space (or any other metric space), then an element x of S is an isolated point of S if there exists an open ball around x which contains no other points of S.

A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset S of Euclidean space must be countable, 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 other points of the set). A closed set with no isolated point is called a perfect set (it has all its limit points and none of them are isolated from it).

The number of isolated points is a topological invariant, i.e. if two topological spaces ${\displaystyle X}$ and ${\displaystyle Y}$ are homeomorphic, the number of isolated points in each is equal.

## Examples

### Standard examples

Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.

• For the set ${\displaystyle S=\{0\}\cup [1,2]}$, the point 0 is an isolated point.
• For the set ${\displaystyle S=\{0\}\cup \{1,1/2,1/3,\dots \}}$, 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 ${\displaystyle {\mathbb {N} }=\{0,1,2,\ldots \}}$ of natural numbers is a discrete set.

In the topological space ${\displaystyle X=\{a,b\}}$ with topology ${\displaystyle \tau =\{\emptyset ,\{a\},X\}}$, the element ${\displaystyle a}$ is an isolated point, even though ${\displaystyle a}$ belongs to the closure of ${\displaystyle \{b\}}$ (and is therefore, in some sense, "close" to ${\displaystyle b}$). Such a situation is not possible in a Hausdorff space.

The Morse lemma states that non-degenerate critical points of certain functions are isolated.

### Two counter-intuitive examples

Consider the set ${\displaystyle F}$ of points ${\displaystyle x}$ in the real interval ${\displaystyle (0,1)}$ such that every digit ${\displaystyle x_{i}}$ of their binary representation fulfills the following conditions:

• Either ${\displaystyle x_{i}=0}$ or ${\displaystyle x_{i}=1}$.
• ${\displaystyle x_{i}=1}$ only for finitely many indices ${\displaystyle i}$.
• If ${\displaystyle m}$ denotes the largest index such that ${\displaystyle x_{m}=1}$, then ${\displaystyle x_{m-1}=0}$.
• If ${\displaystyle x_{i}=1}$ and ${\displaystyle i, then exactly one of the following two conditions holds: ${\displaystyle x_{i-1}=1}$ or ${\displaystyle x_{i+1}=1}$.

Informally, these conditions means that every digit of the binary representation of ${\displaystyle x}$ which equals 1 belongs to a pair ...0110..., except for ...010... at the very end.

Now, ${\displaystyle F}$ is an explicit set consisting entirely of isolated points which has the counter-intuitive property that its closure is an uncountable set. [1]

Another set ${\displaystyle F}$ with the same properties can be obtained as follows. Let ${\displaystyle C}$ be the middle-thirds Cantor set, let ${\displaystyle I_{1},I_{2},I_{3},\ldots }$ be the component intervals of ${\displaystyle [0,1]-C}$, and let ${\displaystyle F}$ be a set consisting of one point from each ${\displaystyle I_{k}}$. Since each ${\displaystyle I_{k}}$ contains only one point from ${\displaystyle F}$, every point of ${\displaystyle F}$ is an isolated point. However, if ${\displaystyle p}$ is any point in the Cantor set, then every neighborhood of ${\displaystyle p}$ contains at least one ${\displaystyle I_{k}}$, and hence at least one point of ${\displaystyle F}$. It follows that each point of the Cantor set lies in the closure of ${\displaystyle F}$, and therefore ${\displaystyle F}$ has uncountable closure.

## Related Research Articles

In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematical analysis, a null set is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space a null set is a set such that .

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

In mathematics, a limit point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of

In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space.

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis. That is, for each point x in X there exists a sequence N1, N2, … of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with Ni contained in N. Since every neighborhood of any point contains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.

In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of It is usually denoted by

In mathematics, the particular point topology is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any set and pX. The collection

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : MN is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f."

In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X contains a point isolated in A.

## References

1. Gomez-Ramirez, Danny (2007), "An explicit set of isolated points in R with uncountable closure", Matemáticas: Enseñanza universitaria, Escuela Regional de Matemáticas. Universidad del Valle, Colombia, 15: 145–147