Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Isolated point

Point of a subset S around which there are no other points of S

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated November 16, 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).

02Examples

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 xi 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.

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},\ldots be the component intervals of [0,1]-C, and let F be a set consisting of one point from each Ik. Since each Ik 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 Ik, 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.

Watch videos about Isolated pointExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Isolated point”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Acnode

An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point.

Adherent point

In mathematics, an adherent point of a subset A {\displaystyle A} of a topological space X , {\displaystyle X,} is a point x {\displaystyle x} in X {\displaystyle X} such that every neighbourhood of x {\displaystyle x} (or equivalently, every open neighborhood of x {\displaystyle x} ) contains at least one point of A . {\displaystyle A.} A point x ∈ X {\displaystyle x\in X} is an adherent point for A {\displaystyle A} if and only if x {\displaystyle x} is in the closure of A , {\displaystyle A,} thus x ∈ Cl X ⁡ A {\displaystyle x\in \operatorname {Cl} _{X}A} if and only if for all open subsets U ⊆ X , {\displaystyle U\subseteq X,} if x ∈ U then U ∩ A ≠ ∅ .

Accumulation point

In mathematics, a limit point, accumulation point, or cluster point of a set S {\displaystyle S} in a topological space X {\displaystyle X} is a point x {\displaystyle x} that can be "approximated" by points of S {\displaystyle S} in the sense that every neighbourhood of x {\displaystyle x} contains a point of S {\displaystyle S} other than x {\displaystyle x} itself. A limit point of a set S {\displaystyle S} does not itself have to be an element of S .