Isolated point

Last updated October 09, 2023

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 $x$ that 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 $S$ .

If the space $X$ is a metric space, for example a Euclidean space, then an element $x$ of $S$ is an isolated point of $S$ if there exists an open ball around $x$ that contains only finitely many elements of $S$ . A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).

Related notions

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 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 (every neighbourhood of a point contains other points of the set). A closed set 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 $X, Y$ 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 $S=\{0\}\cup [1,2],$ the point 0 is an isolated point.
For the set $S=\{0\}\cup \{1,{\tfrac {1}{2}},{\tfrac {1}{3}},\dots \},$ each of the points ${\tfrac {1}{k}}$ 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 $\mathbb {N} =\{0,1,2,\ldots \}$ of natural numbers is a discrete set.

In the topological space $X=\{a,b\}$ with topology $\tau =\{\emptyset ,\{a\},X\},$ the element $a$ is an isolated point, even though $b$ belongs to the closure of $\{a\}$ (and is therefore, in some sense, "close" to $a$ ). 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 $F$ of points $x$ in the real interval $(0,1)$ 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 $x_{m-1}=0.$
If $x_{i}=1$ and $i<m,$ then exactly one of the following two conditions holds: $x_{i-1}=1$ or $x_{i+1}=1.$

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, $F$ is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.^[1]

Another set $F$ with the same properties can be obtained as follows. Let $C$ be the middle-thirds Cantor set, let $I_{1},I_{2},I_{3},\ldots ,I_{k}$ be the component intervals of $[0,1]-C$ , 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.

Related Research Articles

<span class="mw-page-title-main">Compact space</span> Type of mathematical space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers $is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.$

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 mathematical analysis, a metric space $M$ is called complete if every Cauchy sequence of points in $M$ has a limit that is also in $M$ .

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, the closure of a subset $S$ of points in a topological space consists of all points in $S$ together with all limit points of $S$ . The closure of $S$ may equivalently be defined as the union of $S$ and its boundary, and also as the intersection of all closed sets containing $S$ . Intuitively, the closure can be thought of as all the points that are either in $S$ or "very near" $S$ . A point which is in the closure of $S$ is a point of closure of $S$ . The notion of closure is in many ways dual to the notion of interior.

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.

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, accumulation point, or cluster point of a set $in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.$

In the mathematical field of general topology, a meagre set is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms.

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 2^ω 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 $is said to be first-countable if each point has a countable neighbourhood basis. That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.$

In 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 spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

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 non-empty set and p ∈ X. 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, 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

↑ 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

External links

Weisstein, Eric W. "Isolated Point". MathWorld .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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

[1]