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Accumulation point

In , an accumulation point of a A of a X is a point x \in X such that every open neighborhood of x contains at least one point of A distinct from x. This concept, also known as a limit point or cluster point, captures the idea of points that can be "approached" arbitrarily closely by elements of A other than themselves, without requiring the existence of a or metric structure. The collection of all accumulation points of A, denoted A' and called the derived set of A, plays a central role in describing the structure of A. A key property is that the closure of A, denoted \overline{A}, is precisely the union of A and its derived set: \overline{A} = A \cup A'. Consequently, a subset A is closed in X if and only if it contains all its accumulation points, meaning A' \subseteq A. In metric spaces, such as the real numbers \mathbb{R} with the standard topology, a point x is an accumulation point of A if and only if there exists a sequence in A \setminus \{x\} converging to x. This sequential characterization is particularly useful in analysis for identifying boundaries and compactness. Examples illustrate the concept vividly in familiar settings. For the open interval A = (0, 1) in \mathbb{R}, the accumulation points are all points in the closed interval [0, 1], as every neighborhood of any point in [0, 1] intersects (0, 1) nontrivially. In contrast, for the discrete set A = \{1/n : n \in \mathbb{N}\}, the only accumulation point is $0, since neighborhoods of $0 contain infinitely many terms of the sequence, while points like $1 are isolated. For the rational numbers \mathbb{Q} in \mathbb{R}, every real number is an accumulation point, reflecting the of \mathbb{Q}. These notions extend to more abstract spaces, aiding in the study of , , and separation axioms.

Definitions

Accumulation points of sets

In , the foundational framework for discussing accumulation points is that of a . A is a pair (X, \tau), where X is a set and \tau is a collection of subsets of X satisfying three axioms: the and X are in \tau; the union of any collection of sets in \tau is in \tau; and the intersection of any finite number of sets in \tau is in \tau. The elements of \tau are called open sets. A neighborhood of a point x \in X is a N \subseteq X containing an U such that x \in U \subseteq N. These concepts allow for a precise, metric-free description of "closeness" in abstract spaces. Given a (X, \tau) and a A \subseteq X, a point x \in X is an accumulation point (also called a limit point or cluster point) of A if every containing x also contains at least one point of A distinct from x. Equivalently, every open neighborhood of x intersects A \setminus \{x\} in a nonempty set. The derived set A' of A is the set of all accumulation points of A. An accumulation point x of A need not belong to A itself; for instance, it may lie outside A while being approached by points from A. Conversely, points in A that are not accumulation points are called isolated points of A. In first-countable spaces, such as metric spaces, this topological definition aligns with the notion of an accumulation point of a sequence.

Accumulation points of sequences and nets

In topological spaces, the concept of an accumulation point extends from sets to indexed collections such as sequences and nets, capturing points where the collection "clusters" infinitely often. For a sequence (x_n)_{n \in \mathbb{N}} in a topological space X, a point x \in X is an accumulation point if every open neighborhood U of x contains x_n for infinitely many n \in \mathbb{N}. Equivalently, x is an accumulation point if there exists a subsequence (x_{n_k})_{k \in \mathbb{N}} converging to x. This definition highlights how accumulation points identify limits of subsequences, allowing sequences to approach x repeatedly without necessarily converging as a whole. Nets generalize sequences to arbitrary directed sets, providing a framework for convergence in non-metrizable topologies. A net is a function w: D \to X, where D is a directed set (partially ordered such that for any d_1, d_2 \in D, there exists d \geq d_1, d_2). Here, x \in X is an accumulation point of the net w if for every open neighborhood U of x and every d \in D, there exists e \in D with d \leq e and w(e) \in U; this ensures that points of the net enter U arbitrarily far along the directed set D. Sequences correspond to nets indexed by the natural numbers under the usual order, but nets apply to broader index sets, such as finite partitions ordered by refinement, enabling them to fully characterize topological properties like continuity and compactness via their universal property: a net converges to x if and only if every subnet (a composition with an order-preserving map from another directed set) converges to x. Unlike the or net, which requires the entire tail to lie in every neighborhood of x (and is unique in Hausdorff spaces), accumulation points permit multiple such points for the same collection. For instance, a net may have several accumulation points if its "tails" repeatedly visit neighborhoods of each, without settling into one; every accumulation point is the limit of some subnet, but the original net need not converge. In first-countable spaces, sequence accumulation points align closely with those of the underlying set, as every neighborhood basis element can be pierced infinitely often by the sequence. Filters offer an equivalent perspective to nets for defining accumulation, often simplifying proofs in . A filter on X is a collection \mathcal{F} of subsets closed under finite intersections and supersets, excluding the . A point x \in X is an accumulation point of \mathcal{F} if every open neighborhood U of x intersects every F \in \mathcal{F}. Each net induces a filter of sets containing its tails, and vice versa, with (and thus accumulation) preserved under this correspondence: a net accumulates at x its associated filter does. This duality underscores why filters and nets together provide a complete foundation for topological beyond sequential settings.

Characterizations and Relations

Equivalent formulations

In topological spaces, accumulation points (also known as limit points) of a set A can be characterized in various equivalent ways that deepen the understanding beyond the basic neighborhood definition, where every open neighborhood of the point intersects A at some point other than itself. One fundamental equivalent formulation is closure-based: a point x is an accumulation point of A x belongs to the of A \setminus \{x\}. This follows directly from the of , as x \in \overline{A \setminus \{x\}} precisely when every open neighborhood of x intersects A \setminus \{x\}. This characterization highlights the topological of A around x excluding the point itself and is particularly useful in proofs involving derived sets and closed sets. The neighborhood criterion provides another perspective, refining the basic definition in specific space types. In general topological spaces, x is an accumulation point of A if every open neighborhood U of x satisfies U \cap (A \setminus \{x\}) \neq \emptyset. In countably based spaces (such as second-countable spaces like \mathbb{R}^n), this strengthens to U \cap A being for every such U, ensuring true accumulation rather than mere adjacency to finitely many points; finite intersections can be separated by a countable basis element avoiding them. This intersection variant aligns with the intuitive of points "piling up" and is equivalent to the basic criterion in such spaces due to the basis properties. Filter-theoretically, x is an accumulation point of A if the neighborhood filter \mathcal{N}_x of x adheres to A \setminus \{x\}, meaning every member of \mathcal{N}_x (i.e., every open neighborhood of x) intersects A \setminus \{x\}. This generalizes the neighborhood view using , which capture "large sets" in a point-free manner, and is equivalent to the standard definition since adherence is defined by nonempty intersections with filter elements. It proves advantageous in abstract settings where nets or ultrafilters are employed to study convergence without relying on sequences. In terms of nets and sequences, characterizations vary by space type. A point x is an accumulation point of A if there exists a in A \setminus \{x\} converging to x, which is equivalent to the neighborhood definition in arbitrary topological s. In sequential s (such as first-countable s like metric s), this reduces to the existence of a in A \setminus \{x\} converging to x, again equivalent to the basic criterion. However, these are not equivalent in general non-first-countable s; for instance, sequences may fail to detect all accumulation points, as a space might have an accumulation point without any countable converging to it, necessitating for full equivalence. This distinction underscores the role of countability axioms in simplifying topological concepts.

Relation between set and sequence accumulation points

In first-countable topological spaces, the accumulation points of a set A are precisely the points that are limits of convergent s with terms in A. This equivalence holds because the countable local basis at each point allows s to approximate accumulation points arbitrarily closely. To see that every accumulation point x of A is the limit of such a , consider the countable basis \{U_n\}_{n \in \mathbb{N}} of neighborhoods of x with U_{n+1} \subseteq U_n for all n. Since x is an accumulation point, each U_n contains a point of A distinct from x; select x_n \in U_n \cap A \setminus \{x\}. The (x_n) then converges to x, as for any neighborhood V of x, some U_k \subseteq V, so the tail of the lies in V. Conversely, if a (x_n) in A converges to x, then x is an accumulation point of A, because every neighborhood of x contains infinitely many terms of the (hence points of A other than possibly x). This equivalence fails in non-first-countable spaces. For instance, consider the on the ordinal segment [0, \omega_1], where \omega_1 is the first uncountable ordinal. This space is not first-countable at \omega_1, and \omega_1 is an accumulation point of the set [0, \omega_1), as every initial segment of [0, \omega_1) is countable and thus has successor ordinals beyond it. However, no sequence in [0, \omega_1) converges to \omega_1, because any countable sequence is bounded above by some countable ordinal less than \omega_1. Uncountable products of topological spaces, such as the product of uncountably many copies of \{0,1\} with the product topology, provide further examples where sequential limits miss certain accumulation points due to the lack of countable bases. To capture all accumulation points in arbitrary topological spaces, nets generalize sequences effectively. A net in a space X is a function from a to X, and every accumulation point x of a set A \subseteq X is the limit of some net with image in A. Specifically, if x is an accumulation point of A, then for the directed set of finite subsets of a basis at x ordered by reverse inclusion, one can construct a net in A converging to x. Moreover, x is an accumulation point of a net if and only if it is the limit of some subnet. This ensures nets detect the full closure and derived set in general topologies, unlike sequences. For sequences specifically, the accumulation points (also called cluster points) of a sequence (x_n) are the limits of its convergent subsequences, which may include points where the sequence oscillates or repeats. In , the accumulation points of the \{x_n : n \in \mathbb{N}\} (the image set) depend on the distinct terms; if the sequence is eventually constant, say x_n = c for n \geq N, then c is a cluster point of the sequence but the is a with no accumulation points if all prior terms are isolated. Thus, while the cluster points of the sequence always include the accumulation points of its , the converse holds only if the sequence visits each point in its only finitely often.

Properties

General properties

In a X, the derived set A' of a A \subseteq X, consisting of all accumulation points of A, satisfies A' \subseteq \operatorname{cl}(A), where \operatorname{cl}(A) denotes the of A. Furthermore, the closure is precisely the union of the set and its derived set: \operatorname{cl}(A) = A \cup A'. This characterization holds because every point in \operatorname{cl}(A) is either in A or a limit point of A, and the closure is the smallest containing A. The derived set A' itself is closed in X. To see this, note that if x is an accumulation point of A', then every open neighborhood of x contains some y \in A' with y \neq x. Since y is an accumulation point of A, every open neighborhood of y contains a point of A distinct from y. By refining neighborhoods appropriately, every open neighborhood of x intersects A in a point distinct from x, so x \in A'. Thus, the derived set of A' satisfies (A')' \subseteq A'. This (A')' \subseteq A' embodies the hereditary property of accumulation points: every accumulation point of the derived set A' is itself an accumulation point of the original set A. Combined with the general to , it follows that \operatorname{cl}(A') = A' \cup (A')' = A', confirming the closedness of A'. This property underscores the structural role of derived sets in preserving accumulation within the . A point p \in A is isolated if it is not an accumulation point of A, meaning there exists an open neighborhood U of p such that U \cap A = \{p\}. Consequently, isolated points do not belong to the derived set A'. A perfect set, by contrast, is a nonempty closed set equal to its own derived set (P = P'), implying that every point in P is an accumulation point and there are no isolated points. Such sets are dense in themselves and play a key role in decompositions of closed sets. In compact topological spaces, every infinite subset has at least one accumulation point. This result, a generalization of the classical Bolzano-Weierstrass theorem from metric spaces, follows from the fact that if an B \subseteq X had no accumulation point, then each point of B would have a neighborhood intersecting B only at itself, leading to an open cover of B with no finite subcover, contradicting . The derived set operator can be iterated transfinitely to define higher-order derived sets A^{(\alpha)} for ordinals \alpha. Set A^{(0)} = A and A^{(\alpha+1)} = (A^{(\alpha)})' for successor ordinals; for limit ordinals \lambda, define A^{(\lambda)} = \bigcap_{\alpha < \lambda} A^{(\alpha)}. This transfinite sequence of derived sets eventually stabilizes at some ordinal, which is countable in second countable spaces such as separable metric spaces, yielding the Cantor-Bendixson derivative process, which separates scattered and perfect components in closed sets. By the Cantor-Bendixson theorem, in second countable spaces, every closed set is the disjoint union of a perfect set (the kernel) and a countable set (the scattered part removed by the derivatives).

Properties in specific spaces

In metric spaces, a point x is an accumulation point of a subset A if and only if \inf\{d(x,y) \mid y \in A, y \neq x\} = 0, where d denotes the metric. This characterization leverages the distance function to quantify the notion that x is approached arbitrarily closely by points of A distinct from itself, equivalent to the topological definition via open balls. A key property in metric spaces concerns sequential compactness: a subset is compact if and only if it is sequentially compact, meaning every sequence in the subset has a convergent subsequence with limit in the subset, implying every sequence has an accumulation point within the set. This equivalence holds specifically for metric spaces and distinguishes them from more general topological spaces. In Euclidean spaces \mathbb{R}^n, which are complete metric spaces, accumulation points connect directly to Cauchy sequences and completeness. Every Cauchy sequence in a subset converges to a limit in the space, and this limit serves as an accumulation point of the sequence's range, ensuring that bounded infinite subsets possess accumulation points by the Bolzano-Weierstrass theorem. In Hausdorff spaces, accumulation points of convergent sequences are unique, as the Hausdorff separation axiom guarantees distinct points can be separated by disjoint open sets, preventing multiple limits for the same sequence. Nonetheless, subsets can still admit multiple accumulation points, as the derived set (set of all accumulation points) remains closed. In discrete spaces equipped with the standard discrete topology, where every singleton is open, all points are isolated, and thus no subset has accumulation points, as each point admits a neighborhood (itself) containing no other points of the subset. This property persists even for infinite subsets under the discrete topology, though certain alternative topologies on infinite discrete sets may introduce accumulation points.

Examples and Applications

Illustrative examples

In the real line \mathbb{R} equipped with the standard topology, the set of rational numbers \mathbb{Q} provides a classic example where every real number is an accumulation point of \mathbb{Q}, since the rationals are dense and every open interval around any real contains infinitely many rationals distinct from it. In contrast, the set of integers \mathbb{Z} \subset \mathbb{R} has no accumulation points, as it is discrete: for each integer n \in \mathbb{Z}, the open interval (n - 1/2, n + 1/2) contains no other points of \mathbb{Z}. For sequences in \mathbb{R}, consider the sequence defined by a_{2k-1} = 1/k and a_{2k} = 0 for k = 1, 2, 3, \dots, which alternates between terms approaching 0 from above and the constant 0. This sequence has 0 as its unique accumulation point, as every subsequence either is eventually 0 or approaches 0, and no other real number serves as an accumulation point. In non-metrizable spaces, nets illustrate accumulation points beyond sequences. The Stone-Čech compactification \beta \mathbb{N} of the natural numbers \mathbb{N} (with the discrete topology) is compact but not metrizable, and its points beyond \mathbb{N} correspond to free ultrafilters on \mathbb{N}. A net directed by such an ultrafilter converges to the corresponding point in \beta \mathbb{N} \setminus \mathbb{N}, which cannot be an accumulation point of any sequence from \mathbb{N} due to the lack of countable local bases at those points. The middle-thirds Cantor set C \subset [0,1] is a perfect set, meaning it is closed and every point of C is an accumulation point of C: for any x \in C and any \epsilon > 0, the interval (x - \epsilon, x + \epsilon) contains infinitely many other points of C, constructed via the iterative removal of middle thirds. As a highlighting topology's role, consider the indiscrete ( on a set X with |X| \geq 2, where the only open sets are \emptyset and X. For any A \subset X with |A| \geq 2, every point p \in X is an accumulation point of A, since the sole open neighborhood of p is X, which intersects A \setminus \{p\} non-emptily.

Applications in analysis and topology

In real analysis, accumulation points are essential for understanding the behavior of sequences and sets in spaces. The Bolzano-Weierstrass theorem asserts that every bounded infinite of \mathbb{R}^n possesses at least one accumulation point, ensuring the existence of convergent subsequences in bounded sequences. This result underpins the Heine-Borel theorem, which establishes that a of \mathbb{R}^n is compact it is closed and bounded, providing a foundational characterization of in finite-dimensional spaces. Furthermore, in the study of , accumulation points via derived sets help identify sets of . The -Bendixson theorem, applied to closed subsets of , decomposes them into a perfect and a countable scattered ; if the perfect is empty, the set is countable and serves as a set of , meaning that a trigonometric series converging to zero outside the set must be identically zero. This application, originally developed by , facilitates proofs about the convergence and of trigonometric series on the . In general topology, accumulation points define key structural properties of spaces and subsets. This relation extends to compactness: in metric spaces, sequential compactness—where every sequence has a convergent subsequence—is equivalent to every infinite subset having an accumulation point. The iterated application of the accumulation point operator yields derived sets, which play a pivotal role in analyzing topological complexity. The Cantor-Bendixson theorem states that in any second countable Hausdorff space, every closed set can be uniquely expressed as the disjoint union of a perfect set (one with no isolated points) and a countable scattered set (one removable by transfinite iteration of removing isolated points). This decomposition reveals the "scattered" versus "perfect" structure of closed sets, with applications in descriptive set theory for classifying Borel and analytic sets.