Isolated point

An isolated point is a point of a subset of a topological space that has a neighborhood containing no other point of the subset. Equivalently, a point $P$ of a space $X$ is isolated when the singleton $\{P\}$ is open, and a point $P$ of a subset $A$ is isolated in $A$ when, for some neighborhood $U$ of $P$ in $X$, $U \cap A = \{P\}$¹. The concept is the complement of the accumulation point (also called limit point or cluster point), a point every neighborhood of which meets the set at some point other than the point itself².

Definitions and equivalent forms

The two common formulations say the same thing at different levels. In a metric space $(X,d)$, a point $x \in E$ is an isolated point of $E$ if $B(x;\epsilon) \cap E = \{x\}$ for some $\epsilon > 0$, that is, if $x \in E$ and $x$ is not a limit point of $E$². In a general topological space, where no distance is available, the singleton formulation $\{P\}$ open replaces the ball, and the relative version for a subset $A$ uses the subspace topology¹. On the real line the definition reads: $x \in A$ is isolated if there exists $\delta > 0$ such that $x$ is the only point of $A$ in the interval $(x-\delta, x+\delta)$⁶.

Some texts define the isolated points of a set as the points of closure that are not limit points⁷. Under this reading an isolated point is a boundary point of the set but not an accumulation point of it⁶.

Relation to closure and the derived set

Adherent points of a set split into two categories. Among the points $x$ adherent to $A$, some have the property that every sufficiently small open set containing $x$ contains only the single point $x$ of $A$; such a point is isolated. Every other adherent point is a point of accumulation, and for such a point one can construct a sequence of distinct points of $A$ converging to it³. The closure of $A$ is therefore the union of the isolated points of $A$ and the points of accumulation of $A$³.

The set of accumulation points is the derived set $A'$, also called the Cantor–Bendixson derivative; it consists of the accumulation points of $A$, which in general topological spaces need not be the non-isolated points of the closure⁴. Because the isolated points of a closure form a subset that is open when that closure carries its subspace topology, in a T₁ space every derived set turns out to be a closed set; this need not hold in arbitrary topological spaces.^{4 • 12} For a closed set $A \subset \mathbb{R}$, the derived set is obtained by throwing out the isolated points, and every point of a closed set is either a limit point or an isolated point; the isolated points form an open subset of $A$ and the limit points a closed subset⁸.

Examples

A single detached point. Under the usual topology on $\mathbb{R}$, the set $E = (0,1) \cup \{2\}$ has 2 as an isolated point, while the closed interval $[0,1]$ is the set of limit points of $E$². The same example appears as $X = [0,1] \cup \{2\}$ with the subspace topology from $\mathbb{R}$⁹. Open and closed intervals $(a,b)$ and $[a,b]$, by contrast, have no isolated points at all⁶.

A sequence converging to a limit. For $A = \{0, 1, 1/2, 1/3, \ldots\}$ in $\mathbb{R}$, every point except 0 is isolated, and $A' = \{0\}$; applying the derivative again gives $A'' = \varnothing$. For $A = [0,1]$, $A' = A$⁸.

A set equal to its derived set. The Cantor set $K \subset [0,1]$ satisfies $K' = K$ even though its interior is empty, so it has no isolated points without containing any interval⁸.

Discrete and trivial topologies. If every point of a space $X$ is isolated, then every subset of $X$ must be open, and declaring all subsets open indeed gives a topology; such a space is called a discrete space. At the opposite extreme, declaring only $X$ and $\varnothing$ open gives a space in which, when $X$ has at least two elements, no point is isolated⁵.

Spaces classified by their isolated points

The presence or absence of isolated points separates several standard classes of spaces. A topological space is called dense-in-itself if it has no isolated points, and a subset is called perfect if it is closed and dense-in-itself as a subspace. A connected $T_1$ space with more than one point is dense-in-itself¹⁰.

Iterating the derived-set operator connects isolated points to the theory of ordinals. A countable closed set $A \subset [0,1]$ always has an isolated point, so $A \neq A'$. Setting $A_0 = A$ and $A_{n+1} = A_n'$ produces a decreasing sequence of closed sets, which may all remain nonempty, in which case $A_\infty = \bigcap_n A_n \neq \varnothing$⁸.

Recent work extends these classical facts: a 2023 paper by Eugene Zhang presents a general formula for derived sets in general topology and uses it to claim that isolated point sets are nowhere dense in a general topological space, although this cannot hold in a nonempty discrete space, where every point is isolated, so the set of isolated points is the whole space and is not nowhere dense¹¹.

References

1. Isolated points, nLab
2. Surinder Pal Singh Kainth (2023). Topology, in A Comprehensive Textbook on Metric Spaces, Springer, Section 3.2
3. MA651 Topology, Lecture 7: Real numbers, Stevens Institute of Technology
4. Derived set, nLab
5. A Review of General Topology, Part 1: First Concepts, California State University San Marcos
6. 3.A Topological Types of Points, Introduction to Real Analysis, Pressbooks
7. Lecture 2: Elementary Topology on the Complex Plane, MA 42500/52500, Purdue University
8. Course Notes, Math 131, Harvard University
9. Math4201 Topology I, Lecture 28, NoteNextra
10. Michael Müger. Topology for the Working Mathematician, Section 2.7.3
11. Eugene Zhang (2023). A Formula for Derived Sets in General Topology. arXiv:2304.08462
12. Mathlib.Topology.DerivedSet

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › General and set-theoretic topology

Initially written Oct 15, 2026

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