Fact-checked by Grok 2 weeks ago

Locally finite collection

In , a locally finite collection (or ) of subsets of a X is defined as a \{A_\alpha\}_{\alpha \in I} such that for every point x \in X, there exists a neighborhood U of x that intersects only finitely many of the sets A_\alpha. This property ensures that the collection behaves "finitely" in a local sense around each point, even if the overall is infinite. Key properties of locally finite collections include the fact that any subcollection of a locally finite is also locally finite, and if the sets are open, their union is open. Additionally, the collection of closures of the sets in a locally finite is itself locally finite, which aids in compactness-related arguments. Finite collections are trivially locally finite, and in compact spaces, every open cover admits a finite subcover, which is locally finite. Locally finite collections play a central role in the theory of paracompact spaces, where a topological space is paracompact if every open cover has a locally finite open refinement; for Hausdorff spaces, this is equivalent to every open cover admitting a subordinate partition of unity. Such refinements are crucial for constructing continuous functions and studying manifold structures, as they allow global properties to be pieced together from local finite data without pathological overlaps. All compact Hausdorff spaces and Euclidean spaces are paracompact, highlighting the broad applicability of locally finite collections in classical topology.

Definition and characterizations

Formal definition

In topology, a collection \mathcal{U} = \{U_i \mid i \in I\} of subsets of a X is locally finite if, for every point x \in X, there exists a neighborhood V of x that intersects only finitely many sets in \mathcal{U}. Formally, this means that V \cap U_i \neq \emptyset for at most finitely many indices i \in I. Here, a neighborhood V of x is understood as an in X containing x. This condition underscores the local nature of the property: while the collection \mathcal{U} may be infinite overall, its "influence" is finite in sufficiently small regions around each point, preventing overcrowding in any localized area of the space. Although often applied to open covers in the study of and paracompactness, the definition holds for arbitrary families of subsets.

Equivalent formulations

A collection \mathcal{A} = \{A_i \mid i \in I\} of subsets of a X is locally finite if for every point x \in X, there exists a neighborhood V_x of x such that V_x \cap A_i \neq \emptyset for only finitely many indices i \in I. An equivalent formulation states that there exists an open cover \{V_j \mid j \in J\} of X such that each V_j intersects only finitely many members of \mathcal{A}. This follows directly from selecting the neighborhoods V_x in the primary definition to form such a cover. If \mathcal{A} is an open cover of X, then local finiteness is equivalent to the condition that for every x \in X, there exists a neighborhood V_x of x contained in the union of finitely many sets from \mathcal{A}. In this case, since the sets are open, the finite union provides an open neighborhood satisfying the requirement. A related but weaker condition is point-finiteness, where every point x \in X belongs to only finitely many sets in \mathcal{A}; local finiteness implies point-finiteness, but the converse does not hold in general.

Properties

General properties

A locally finite collection \mathcal{U} of subsets of a X is point-finite, meaning that every point x \in X belongs to only finitely many members of \mathcal{U}. To see this, suppose toward a that some x belongs to infinitely many U \in \mathcal{U}; then every neighborhood of x would intersect infinitely many such U, contradicting the local finiteness of \mathcal{U}. Any subcollection of a locally finite collection is itself locally finite, since the neighborhoods witnessing local finiteness for the original collection intersect at most as many sets in the subcollection. In particular, every finite subcollection of a locally finite collection is locally finite. Moreover, the finite of locally finite collections is locally finite: for any x \in X, neighborhoods of x exist that intersect finitely many sets from each collection, so their common refinement intersects only finitely many sets overall. The same reasoning shows that the collection of all finite intersections of members from a finite of locally finite collections remains locally finite. If \mathcal{U} is a locally finite collection, then the collection \{\overline{U} \mid U \in \mathcal{U}\} of closures is also locally finite. To verify this, let x \in X and let V be an open neighborhood of x that intersects only finitely many U \in \mathcal{U}, say U_1, \dots, U_n. Suppose V \cap \overline{U} \neq \emptyset for some U \in \mathcal{U}; pick y \in V \cap \overline{U}. Since V is open, there exists an open neighborhood W of y with W \subseteq V. As y \in \overline{U}, we have W \cap U \neq \emptyset, so V \cap U \neq \emptyset and thus U is one of the U_i. Hence, V intersects only finitely many closures. If \{F_\alpha\} is a locally finite family of closed subsets of X, then their union F = \bigcup F_\alpha is closed. Equivalently, the complement X \setminus F is open: for x \in X \setminus F, there exists an open neighborhood U of x intersecting only finitely many F_\alpha, say F_1, \dots, F_n. Since x \notin F_i and each F_i is closed, there are open neighborhoods V_i of x with V_i \cap F_i = \emptyset. The V = U \cap \bigcap_{i=1}^n V_i then satisfies V \cap F_\alpha = \emptyset for all \alpha, as it avoids the finitely many F_i by construction and the remaining infinitely many by the choice of U.

In compact spaces

In compact topological spaces, every locally finite collection of subsets is finite. This fundamental result highlights how compactness imposes global constraints on local properties like finiteness. To outline the proof, suppose \mathcal{U} = \{U_i \mid i \in I\} is an infinite locally finite collection of subsets (assumed non-empty) in a compact space X. For each x \in X, there exists an open neighborhood V_x of x that intersects only finitely many members of \mathcal{U}, denoted by the finite subfamily \mathcal{F}_x \subset \mathcal{U}. The collection \{V_x \mid x \in X\} forms an open cover of X. By compactness of X, it admits a finite subcover V_1, \dots, V_n. Let \mathcal{F} = \bigcup_{j=1}^n \mathcal{F}_{x_j}, which is finite. Now suppose there exists some U_k \in \mathcal{U} \setminus \mathcal{F}. Then U_k does not belong to any \mathcal{F}_{x_j}, so U_k \cap V_j = \emptyset for all j=1, \dots, n. Thus U_k \cap \bigcup_{j=1}^n V_j = \emptyset. But \bigcup_{j=1}^n V_j = X, so U_k \cap X = \emptyset, contradicting the assumption that U_k is non-empty. Therefore, \mathcal{U} = \mathcal{F} is finite. A direct applies to open covers: every locally finite open cover of a admits a finite subcover, since the entire collection is finite and thus the space is covered by finitely many of its members. This aligns with the defining property of compactness, where every open cover (not just locally finite ones) has a finite subcover, but here the local finiteness strengthens the connection by guaranteeing the cover itself is finite. Finite collections of subsets are locally finite in any topological space, as every point's neighborhood intersects at most all members of the collection, which is finite. In compact spaces, the converse holds: local finiteness implies global finiteness, distinguishing compact spaces from more general ones where infinite locally finite collections can exist.

In Lindelöf spaces

In a Lindelöf space, defined as a topological space where every open cover admits a countable subcover, every locally finite collection of nonempty subsets is countable. This result follows from the fact that, for a locally finite family \{F_\alpha\}_{\alpha \in A} of nonempty subsets, one can associate to each point x \in X an open neighborhood U_x intersecting only finitely many F_\alpha; the collection \{U_x \mid x \in X\} is an open cover with a countable subcover \{U_{x_n} \mid n \in \mathbb{N}\}, and since each U_{x_n} meets only finitely many F_\alpha, the entire family intersects only countably many such sets, implying A is countable. Consequently, every locally finite open cover of a Lindelöf space must itself be countable. This countability ensures that if a Lindelöf space admits a locally finite open cover consisting of compact sets, then the space is σ-compact as a countable union of those compacts. In contrast to compact spaces, where locally finite collections are finite, Lindelöf spaces permit countably infinite such collections. However, uncountable locally finite collections exist in non-Lindelöf spaces; for instance, the Sorgenfrey plane contains an uncountable closed subspace, whose collection is uncountable and locally finite.

Examples

Locally finite collections

A finite collection of subsets of a is always locally finite, as every neighborhood of any point intersects at most all the finitely many sets in the collection. In the real line \mathbb{R} with the standard topology, the collection of open intervals \{(n, n+2) \mid n \in \mathbb{Z}\} forms a locally finite open cover. For any point x \in \mathbb{R}, there exists a sufficiently small open neighborhood around x that overlaps only finitely many of these intervals, typically at most three due to their limited overlap. In Euclidean space \mathbb{R}^n with the standard topology, a concrete example of a locally finite open cover is the collection of all open balls of radius 1 centered at points with integer coordinates (i.e., the integer lattice \mathbb{Z}^n). This cover exhausts \mathbb{R}^n, and around any point, a small enough neighborhood intersects only finitely many such balls, as the lattice points are discrete. More generally, the standard basis of open balls admits refinements to locally finite subcovers for any open cover of \mathbb{R}^n. In a topological space, every collection of subsets is locally finite it is point-finite, meaning each point belongs to only finitely many sets in the collection; in this case, the singleton neighborhood of each point intersects precisely those finitely many sets containing it.

Non-locally finite collections

A classic of a non-locally finite collection in the real line \mathbb{R} with the is the of open intervals \{(-n, n) \mid n \in \mathbb{N}\}. This collection covers \mathbb{R}, but it fails to be locally finite because, for the point $0, every neighborhood of $0—such as any open interval (-\epsilon, \epsilon) for \epsilon > 0—intersects all members of the collection, since (-n, n) contains (-\epsilon, \epsilon) for all n > \epsilon. Another example illustrating the failure of local finiteness, even when the collection is point-finite, arises from considering the family of all singletons \{\{x\} \mid x \in \mathbb{R}\} in \mathbb{R} with the standard topology. Each point belongs to exactly one singleton, making the collection point-finite, but it is not locally finite: any open neighborhood of a point, being an uncountable open interval containing infinitely many points, intersects infinitely many singletons. This highlights a key distinction from point-finite collections, which do not require the stronger condition that neighborhoods intersect only finitely many sets. In the subspace topology on the rational numbers \mathbb{Q} \subset \mathbb{R}, a cover by open intervals centered at each rational can also fail local finiteness. Enumerate \mathbb{Q} = \{q_n \mid n \in \mathbb{N}\}, and consider the collection of open intervals \{(q_n - 2^{-n}, q_n + 2^{-n}) \mid n \in \mathbb{N}\}, which covers \mathbb{Q}. For any point r \in \mathbb{Q}, every relative neighborhood in \mathbb{Q} (an intersection of an open set in \mathbb{R} with \mathbb{Q}) contains infinitely many rationals due to density, thus intersecting infinitely many intervals in the collection. Although this cover is point-finite in \mathbb{R} because the sum of the diameters is finite (ensuring each point lies in only finitely many intervals), it remains non-locally finite in the subspace \mathbb{Q}.

Applications

In paracompactness

A is defined to be paracompact if every open cover of the space admits a locally finite open refinement. This property serves as a key in the study of topological spaces that generalize while retaining useful covering properties. The notion of paracompactness was introduced by in 1944 as a weakening of the condition, allowing for spaces that are not compact but still exhibit controlled behavior under open covers. Dieudonné's work emphasized that paracompact spaces, particularly when Hausdorff, possess strong separation axioms akin to those of compact spaces. A fundamental theorem states that every paracompact is , with the locally finite refinement of any open cover playing a crucial role in separating disjoint closed sets. Furthermore, in such spaces, the existence of locally finite open refinements enables the construction of a subordinate to any given open cover, consisting of continuous functions whose supports form a locally finite and sum to 1 at every point. For instance, the Euclidean space \mathbb{R}^n is paracompact, as all metric spaces satisfy this property via a theorem of Arthur H. Stone. In \mathbb{R}^n, locally finite covers arise naturally from countable exhaustion by compact balls, and partitions of unity subordinate to these covers are constructed using smooth bump functions with compact support.

In metrization theorems

Locally finite collections play a pivotal role in metrization theorems by providing refinements that enable the construction of metrics on topological spaces. In particular, the Nagata–Smirnov metrization theorem states that a topological space is metrizable if and only if it is regular Hausdorff and possesses a σ-locally finite basis, where σ-locally finite means a countable union of locally finite families. Here, locally finite collections serve as the fundamental building blocks, ensuring that basis elements around any point intersect only finitely many others, which facilitates the embedding into a metric space via uniform structures. In the context of dimension theory, locally finite collections are essential for characterizing and computing the of , particularly through sum theorems. For instance, a has less than n if it can be expressed as a union of a locally finite of closed subspaces, each of less than n, under suitable conditions; this decomposition preserves the overall . Specifically for , which are totally disconnected with a basis of clopen sets, every open cover admits a refinement that is a locally finite collection of disjoint clopen sets, allowing the to be partitioned without increasing . The Urysohn metrization theorem complements this by showing that every regular Hausdorff space with a second-countable basis is metrizable, and such a basis is inherently σ-locally finite since it is a countable of families, each locally finite. This connection highlights how locally finite structures underpin countable bases, enabling the theorem's proof through the construction of a compatible via countable dense subsets. In applications to manifolds, locally finite collections ensure metrizability by allowing the construction of a countable atlas that is both locally finite and compatible with the manifold's . For a second-countable , an atlas can be refined to a locally finite open cover by chart domains, each homeomorphic to , which, combined with paracompactness, yields a via the Nagata–Smirnov . This locally finite atlas not only subordinates to any given cover but also supports partitions of unity, facilitating the global definition.

Related concepts

Point-finite collections

A point-finite collection (or family) of subsets of a X is one in which every point of X belongs to at most finitely many members of the collection. This condition focuses solely on the membership of individual points, without regard to the topological structure beyond the sets themselves. Every locally finite collection is point-finite, since if a neighborhood of a point intersects only finitely many sets in the collection, then the point itself can belong to only finitely many of those sets. However, the does not hold: there exist point-finite collections that are not locally finite. For instance, consider the X = \{0\} \cup \{1/n \mid n \in \mathbb{N}, n \geq 1\} of \mathbb{R} with the , and the collection \mathcal{A} = \{\{1/n\} \mid n \in \mathbb{N}, n \geq 1\} \cup \{X\}. Each point in X belongs to at most two sets in \mathcal{A}, making it point-finite, but every neighborhood of 0 intersects infinitely many singletons \{1/n\}. Point-finite collections exhibit weaker topological control than locally finite ones. In particular, even if every set in a point-finite collection is closed, their union need not be closed. For example, in \mathbb{R} with the standard topology, the family \{\{1/n\} \mid n \in \mathbb{N}, n \geq 1\} consists of closed singletons, and each point $1/n belongs to exactly one set while other points belong to none, so it is point-finite; yet the union \{1/n \mid n \in \mathbb{N}, n \geq 1\} is not closed, as 0 is a limit point not in the union. This contrasts with locally finite families of closed sets in Hausdorff spaces, whose unions are closed.

σ-locally finite collections

A family of subsets of a X is \sigma-locally finite if it is the countable union of locally finite families. However, the union of a \sigma-locally finite collection of closed sets need not be closed; for instance, the set of rational numbers \mathbb{Q} as a subspace of \mathbb{R} is the union of the singletons \{\{q\} \mid q \in \mathbb{Q}\}, where \mathbb{Q} = \{q_n \mid n \in \mathbb{N}\} is an , and this collection is \sigma-locally finite as the increasing finite subcollections F_n = \{\{q_1\}, \dots, \{q_n\}\} are each locally finite, yet \mathbb{Q} is not closed in \mathbb{R}. The notion plays a key role in the Nagata–Smirnov metrization theorem, which asserts that a is metrizable if and only if it is regular, Hausdorff, and has a \sigma-locally finite basis. For example, the \mathbb{Q} of \mathbb{R} admits \sigma-locally finite covers via its countable basis of open intervals with rational endpoints, which can be partitioned into countable finite subfamilies that are locally finite.