Limit point

A limit point, also called an accumulation point, of a subset of a topological space is a point that can be approximated by elements of the subset other than the point itself. The concept formalizes the local concentration of a set and underlies the definitions of closed set, compactness, and several forms of convergence.

The definition depends only on the neighborhood structure of the ambient space. In a metric space, it acquires the familiar interpretation that members of the set occur at arbitrarily small positive distances from the limit point.

Definition

Let (X) be a topological space, let (A\subseteq X), and let (x\in X). The point (x) is a limit point of (A) when every neighborhood (U) of (x) satisfies

[ U\cap (A\setminus{x})\neq\varnothing. ]

The point need not belong to (A). If (x\in A) but is not a limit point of (A), then (x) is an isolated point of (A). A point outside (A) that is nevertheless a limit point records accumulation at the boundary rather than membership in the set.

The definition is equivalently expressed through the closure operator:

[ x\text{ is a limit point of }A \quad\Longleftrightarrow\quad x\in\overline{A\setminus{x}}. ]

The set of all limit points of (A) is the derived set of (A), conventionally denoted by (A'). Consequently,

[ A'={x\in X:x\in\overline{A\setminus{x}}}. ]

A subset (A) is closed precisely when it contains all of its limit points:

[ A'\subseteq A. ]

This characterization remains valid in every topological space. It does not require a separation axiom.

Metric formulation

When the topology of (X) is induced by a metric (d), the neighborhood definition is equivalent to

[ \forall\varepsilon>0,\qquad \bigl(B_\varepsilon(x)\setminus{x}\bigr)\cap A\neq\varnothing, ]

where

[ B_\varepsilon(x)={y\in X:d(x,y)<\varepsilon}. ]

Thus (x) is a limit point when every punctured open ball centered at (x) contains an element of (A). The exclusion of (x) is essential. Without it, every member of (A) would satisfy the condition regardless of whether other elements of (A) occurred nearby.

In a (T_1) space, every neighborhood of a limit point contains infinitely many elements of the set. If a neighborhood contained only finitely many points of (A) other than (x), the (T_1) property would permit those points to be removed while retaining a neighborhood of (x), contradicting the definition. The corresponding statement can fail outside the (T_1) setting because a distinct point may be inseparable from (x) by neighborhoods.

Sequential characterization

In a first-countable space, a point (x) is a limit point of (A) if and only if there exists a sequence ((a_n)) with

[ a_n\in A\setminus{x} ]

for every (n), such that

[ a_n\longrightarrow x. ]

For metric spaces, a sequence can be selected so that (d(a_n,x)<1/n). In a (T_1) first-countable space, the selected terms can additionally be made pairwise distinct.

Sequences do not detect every topological limit point in an arbitrary space. The general correspondence uses nets or filters. A point (x) is a limit point of (A) exactly when a net contained in (A\setminus{x}) converges to (x). In filter terminology, the neighborhood filter of (x) has nonempty intersection with (A\setminus{x}) at every stage and extends to a filter carrying the same convergence information.

The distinction between sequential and general topological convergence is structural rather than terminological. Spaces in which closure is completely determined by convergent sequences are called sequential spaces, while first-countable spaces form a narrower class for which the sequence criterion follows directly from countable local bases.

Derived sets and iteration

The derived-set operator removes isolated membership from the local structure of a subset while retaining all points of accumulation. In a (T_1) space, the derived set (A') is closed. This conclusion does not extend unchanged to every topology, because the behavior of singletons affects the stability of accumulation under closure.

Repeated application produces the finite derived sets

[ A^{(0)}=A,\qquad A^{(n+1)}=\bigl(A^{(n)}\bigr)'. ]

The construction extends through ordinal numbers. At a limit ordinal (\lambda), the transfinite derivative is defined by

[ A^{(\lambda)}

\bigcap_{\alpha<\lambda}A^{(\alpha)}. ]

This iteration is central to the Cantor–Bendixson theorem, which decomposes every closed subset of a Polish space into a perfect component and a countable scattered component. Georg Cantor introduced derived sets in his study of trigonometric series and subsequently used their transfinite iteration to analyze the internal structure of closed sets.

A set satisfying (A=A') has no isolated points and contains every point of its own accumulation. When it is also closed, it is a perfect set. By contrast, a scattered space is one in which every nonempty subspace has an isolated point, so repeated derivation eventually removes each point at some ordinal stage.

Relation to compactness

Limit points provide one formulation of compactness in settings where countability or metric structure supplies the necessary equivalences. A space is limit-point compact when every infinite subset has a limit point in the space.

Every compact space is limit-point compact. In a (T_1) space, an infinite subset without a limit point would yield an open cover incompatible with compactness. The converse does not hold for arbitrary topological spaces, although it does hold for metric spaces. Within metric spaces, compactness, limit-point compactness, sequential compactness, and completeness together with total boundedness describe equivalent global behavior through different local or sequential mechanisms.

The Bolzano–Weierstrass theorem is the Euclidean form of this relationship. It states that every bounded infinite subset of (\mathbb{R}^n) possesses a limit point. Equivalently, every bounded sequence in Euclidean space has a convergent subsequence after the sequence is interpreted through the set of its values and their possible repetitions.

Terminological distinctions

The term “limit point” sometimes denotes related but nonidentical concepts outside point-set topology. A cluster point of a sequence is a point to which some subnet converges; in first-countable spaces, this is equivalent to being the limit of a subsequence. The set of cluster points depends on the frequency and directed behavior of the sequence, whereas the derived set of its range depends only on which values occur.

An adherence point of a subset is any point in its closure. Therefore every limit point is an adherence point, but an isolated member of the subset is an adherence point without being a limit point. An (\omega)-accumulation point is a point whose every neighborhood meets the subset in infinitely many elements. This agrees with the ordinary limit-point definition in (T_1) spaces but can be stronger in spaces with weaker separation properties.

Historical formulation

The analytic idea of arbitrarily close distinct values preceded the general language of topology. Bernard Bolzano used accumulation principles in his treatment of bounded infinite sets, while Karl Weierstrass incorporated related limit arguments into the arithmetization of analysis.

During the early twentieth-century abstraction of neighborhood methods, Maurice Fréchet formulated convergence through structures that no longer depended on Euclidean distance. The resulting framework allowed accumulation to be treated as a property of neighborhoods rather than as a consequence of numerical measurement.

In 1917, You Watanabe established the precise role of the (T_1) condition in replacing the punctured-neighborhood definition by the requirement that every neighborhood contain infinitely many points of the subset. The distinction became part of the standard separation-dependent treatment of derived sets, preventing the metric version of the criterion from being transferred without qualification to general topological spaces.

The systematic axiomatization of topological spaces by Felix Hausdorff placed limit points within a broader theory of neighborhoods, separation, continuity, and compactness. Subsequent developments replaced the earlier dependence on sequences by nets and filters, thereby recovering an exact convergence characterization for spaces without countable local bases.