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Cofiniteness

In mathematics, a cofinite subset of a set X is a subset S \subseteq X whose complement X \setminus S is finite. Equivalently, S contains all but finitely many elements of X. This concept is fundamental in set theory and plays a key role in various areas of mathematics.^[1] For infinite sets, cofinite subsets are infinite, and the collection of cofinite subsets forms a filter, known as the Fréchet filter. Examples include the set of all even integers greater than some fixed number in the integers, or all points except finitely many in \mathbb{R}^n. Cofiniteness appears in topology through the cofinite topology on a set, where open sets are the empty set and all cofinite subsets; this is a classic example of a non-Hausdorff topology with interesting separation properties. In algebra, the term relates to finite-cofinite Boolean algebras and cofiniteness conditions in homological algebra, such as for torsion modules over Noetherian rings. These and other applications are discussed in the following sections.

Set-Theoretic Foundations

Definition of Cofinite Sets

In set theory, a subset A of a set X is called cofinite if its complement X \setminus A is a finite set.^[2] Equivalently, A contains all but finitely many elements of X, which can be denoted by the condition |X \setminus A| < \infty, where |\cdot| represents the cardinality of the set.^[3] This notion is particularly relevant when X is infinite, as cofinite subsets then comprise "almost all" of X, in contrast to finite subsets, which comprise "almost none."^[2] For example, if X = \mathbb{N} (the set of natural numbers), then A = \mathbb{N} \setminus \{1, 2, 3\} is cofinite because its complement has three elements.^[3] However, the empty set \emptyset is not cofinite in an infinite X, since X \setminus \emptyset = X is infinite; \emptyset is cofinite only if X itself is finite.^[2]

Basic Properties and Examples

Cofinite subsets of a set X exhibit notable closure properties under standard set operations. The intersection of finitely many cofinite sets is cofinite, as the complement of such an intersection is the union of the corresponding finite complements, which remains finite.^[4] For instance, if A and B are cofinite in X, then |X \setminus (A \cap B)| = |(X \setminus A) \cup (X \setminus B)| \leq |X \setminus A| + |X \setminus B| < \infty. By extension, the family of cofinite subsets is closed under arbitrary unions, since the complement of a union is the intersection of the complements, and any intersection of finite sets (even infinitely many) is finite or empty.^[5] However, arbitrary intersections of cofinite sets need not be cofinite; for example, in \mathbb{N}, the intersection \bigcap_{n=1}^\infty (\mathbb{N} \setminus \{n\}) is empty, whose complement \mathbb{N} is infinite. Here, the singletons \{n\} illustrate how accumulating finite exclusions can yield an infinite complement overall. Additionally, the complement of any cofinite set is finite, directly from the definition. In infinite sets like the integers \mathbb{Z}, cofinite subsets arise naturally by excluding finitely many elements; for example, \mathbb{Z} \setminus \{0, \pi\} (noting \pi \notin \mathbb{Z}) or more precisely \mathbb{Z} \setminus \{ -1, 0, 1 \} is cofinite, capturing "almost all" integers. This contrasts with cocountable sets, where the complement is at most countable rather than finite, allowing for potentially larger exclusions while preserving a similar intuitive notion of density in uncountable ambient sets like \mathbb{R}.^[6] Regarding cardinality, if X is infinite, every cofinite subset of X has the same cardinality as X itself. Removing finitely many elements from an infinite set does not alter its cardinal size, as there exists a bijection between X and X \setminus F for any finite F \subseteq X. This holds by basic cardinal arithmetic, where |X| + \aleph_0 = |X| for infinite |X|, and finite addition is even more straightforward.

Topological Applications

The Cofinite Topology

The cofinite topology on a set X is defined as the collection of all subsets of X that are either empty or cofinite, where a cofinite subset has a finite complement in X.^[7] This topology, also known as the finite complement topology, endows X with a structure where the open sets are precisely \emptyset and those subsets U \subseteq X such that X \setminus U is finite.^[8] The cofinite sets themselves form a basis for this topology, as every open set can be expressed as a union of cofinite sets, and the intersection of any two basis elements remains open.^[7] Consequently, the closed sets in the cofinite topology are exactly the finite subsets of X and X itself, since the complement of an open set must be finite or the entire space.^[8] When X is finite, the cofinite topology coincides with the discrete topology on X, because every subset of a finite set is cofinite.^[7] For infinite X, however, the topology is coarser than the discrete one, highlighting its distinct behavior on infinite sets.^[9] The cofinite topology finds initial motivation in general topology as a tool for studying pathological spaces, often serving as a counterexample to illustrate limitations of certain topological properties without satisfying stronger separation conditions.^[7]

Properties of the Cofinite Topology

The cofinite topology on an infinite set X satisfies the T_1 separation axiom because singletons are closed sets, as their complements are cofinite and thus open.^[10] However, it fails the T_2 (Hausdorff) axiom, since any two non-empty open sets have non-empty intersection: if U and V are non-empty opens, then X \setminus U and X \setminus V are finite, so X \setminus (U \cap V) = (X \setminus U) \cup (X \setminus V) is finite, implying U \cap V is cofinite and hence non-empty.^[10] Consequently, distinct points cannot be separated by disjoint open neighborhoods.^[11] The space is not regular, as the same intersection property prevents separating a point from a disjoint closed set (which is finite and non-empty) using disjoint open sets. Similarly, it is not normal for infinite X, since disjoint finite closed sets cannot be separated by disjoint opens.^[10] The cofinite topology is compact. To see this, consider an open cover \{U_i\}_{i \in I} of X. Each U_i = X \setminus F_i where F_i is finite. Select U_1, which misses the finite set F_1. For each point in F_1, choose a U_j covering it; finitely many such U_j suffice. Their union covers X, yielding a finite subcover.^[11] The space is hyperconnected: any two non-empty open sets intersect, as shown earlier, making it impossible to decompose X into two non-empty disjoint opens. Thus, it is connected but not path-connected in general.^[11] Every subspace of (X, \tau) with the cofinite topology \tau inherits the cofinite topology. For a subset A \subseteq X, a set U \subseteq A is open in the subspace topology if U = V \cap A for some open V \subseteq X, so V = X \setminus F with F finite, hence U = A \setminus (A \cap F) where A \cap F is finite. Conversely, any cofinite open in A arises this way from a cofinite open in X.^[12]

Variants of the Cofinite Topology

The double-pointed cofinite topology is a modification of the cofinite topology constructed on the product space Y = X \times \{0, 1\}, where X is an infinite set equipped with the cofinite topology and \{0, 1\} has the indiscrete topology. The open sets in this topology consist of the empty set and all subsets U \subseteq Y such that the projection \pi_X(U) onto X is cofinite (or empty). Equivalently, the basis for the topology comprises sets of the form V \times \{0, 1\}, where V is cofinite in X. This construction effectively "doubles" each point in X, making the points (x, 0) and (x, 1) topologically indistinguishable for each x \in X. This topology is not T_0 or T_1, as no open set can separate the paired points (x, 0) and (x, 1), but it satisfies the symmetric separation axiom R_0 due to the symmetry in the doubling. It is compact, as any open cover must include sets covering entire doublets, and the cofinite nature ensures finite subcovers similar to the base cofinite topology. For an example on X = \mathbb{Z}, a basic open neighborhood of a point (n, 0) (or equivalently (n, 1)) excludes only finitely many doublets \{(k_i, 0), (k_i, 1)\} for distinct k_i \in \mathbb{Z}, avoiding separation within doublets while covering all but finitely many integers. Another variant is the cocountable topology, defined on an uncountable set X where the open sets are the empty set and those subsets whose complements are at most countable. This generalizes the cofinite topology by replacing finite complements with countable ones, making it suitable for spaces where finite exclusions are insufficient but countable ones suffice for topological structure. It is T_1 (points are closed) but not Hausdorff, and it is hyperconnected (any two nonempty open sets intersect).^[13] The finite complement topology, often synonymous with the cofinite topology, is sometimes specialized to particular spaces like the natural numbers or rationals to highlight properties such as non-metrizability or failure of certain countability axioms. For instance, on the uncountable reals, it yields a compact T_1 space that is not normal. Generalizations include co-\kappa-topologies for cardinal \kappa, where open sets have complements of cardinality less than \kappa, with the cocountable case corresponding to \kappa = \aleph_0; these relate indirectly to one-point compactifications of discrete spaces, where neighborhoods of the added point resemble cofinite sets.^[14]

Algebraic Applications

Finite-Cofinite Boolean Algebras

In the context of Boolean algebras, the finite-cofinite algebra on an infinite set X is defined as the collection of all subsets of X that are either finite or cofinite, equipped with the standard set operations of union, intersection, and complementation. This structure forms a Boolean algebra, where the empty set serves as the zero element and X as the unit element.^[15]^[16] The operations are defined as follows: the join of two elements is their union, which preserves finiteness or cofiniteness since the union of two finite sets is finite and the union of a finite set with a cofinite set is cofinite; the meet is their intersection, where the intersection of two cofinite sets is cofinite and the intersection of a finite set with any set is finite; and the complement maps finite sets to cofinite sets and vice versa, maintaining the algebra's closure. The atoms of this Boolean algebra are the singleton subsets \{x\} for each x \in X, as these are minimal nonzero elements, and every finite set is the join of its singletons.^[15]^[17] This algebra is atomic, meaning every nonzero element is the join of atoms below it, but it is not complete when X is infinite; for instance, consider the family of sets X \setminus \{p\} for all p in an infinite subset S \subset X such that X \setminus S is also infinite: their infimum (greatest lower bound) would be X \setminus S, which is neither finite nor cofinite and thus not in the algebra, so no infimum exists in the structure. Unlike the full power set algebra \mathcal{P}(X), which includes all subsets and is complete and atomic for any X, the finite-cofinite algebra is a proper subalgebra that excludes infinite co-infinite sets, leading to distinct structural properties. Additionally, any Boolean algebra isomorphic to this finite-cofinite algebra possesses a unique non-principal ultrafilter, consisting precisely of the cofinite sets, which extends the principal ultrafilters generated by atoms.^[15]^[17]^[18] A concrete example arises when X = \mathbb{N}, the natural numbers: the finite-cofinite algebra consists of all finite subsets of \mathbb{N} and their complements (cofinite sets), generated under the Boolean operations, with singletons as atoms and the unique non-principal ultrafilter formed by all cofinite subsets of \mathbb{N}.^[18]^[19]

Cofiniteness in Direct Sums and Products

In the context of modules over a ring, the direct sum of a family of modules \{M_i\}_{i \in I} over an index set I is defined as the submodule of the direct product \prod_{i \in I} M_i consisting of those elements (m_i)_{i \in I} where the support \{i \in I : m_i \neq 0\} is finite, meaning m_i = 0 for cofinitely many i.^[20] Formally, an element (m_i)_{i \in I} \in \bigoplus_{i \in I} M_i if and only if |\{i : m_i \neq 0\}| < \infty.^[20] This condition ensures that the direct sum captures the "finite combinations" of the modules, distinguishing it from the direct product, which includes all families without such restrictions. A representative example arises with \mathbb{Z}-modules, where the direct sum \bigoplus_{n=1}^\infty \mathbb{Z} consists of sequences (a_1, a_2, \dots) of integers with only finitely many nonzero terms, such as (3, 0, -2, 0, \dots).^[20] In contrast, the direct product \prod_{n=1}^\infty \mathbb{Z} comprises all integer sequences, including those with infinitely many nonzeros, such as the constant sequence (1,1,1,\dots). This finite support requirement in the direct sum preserves additivity and scalar multiplication in a way that aligns with finite linear combinations, making it the categorical coproduct in the category of modules. In topological spaces, cofiniteness plays an analogous role in the definition of the product topology on an infinite product \prod_{i \in I} X_i. The product topology is generated by a subbasis consisting of sets of the form \pi_i^{-1}(U_i), where \pi_i : \prod_{j \in I} X_j \to X_i is the projection and U_i is open in X_i; equivalently, the basis elements are finite intersections of these, which specify open conditions in only finitely many coordinates while taking the full space X_j in the cofinitely many remaining coordinates.^[21] This finite support for the varying coordinates ensures that "closeness" in the product space depends on agreement in cofinitely many coordinates, mirroring the algebraic direct sum's structure. For instance, in the product \mathbb{R}^\mathbb{N} with the standard topology on each \mathbb{R}, basic open sets are cylinders like \{ (x_n) : |x_k - a_k| < \epsilon \text{ for } k=1,\dots,m \}, full in all but finitely many factors.^[21] If each X_i carries the cofinite topology (on infinite sets), the resulting product topology remains coarser than the discrete but finer than the cofinite topology on the entire product space.^[22]

Cofiniteness in Homological Algebra

In homological algebra, particularly in the study of local cohomology, an R-module M is said to be I-cofinite, where I is an ideal of a Noetherian ring R, if the support of M is contained in V(I) and \operatorname{Ext}^i_R(R/I, M) is finitely generated for all i \geq 0.<grok:render type="render_inline_citation"> 0 </grok:render> This notion was introduced by Hartshorne to investigate finiteness properties of local cohomology modules H_I^j(N), where N is a finitely generated R-module, prompting questions about when these modules inherit I-cofiniteness from N.<grok:render type="render_inline_citation"> 0 </grok:render> The associated primes of an I-cofinite module are finite in number and lie in V(I), with additional depth conditions ensuring the module's behavior relative to I.<grok:render type="render_inline_citation"> 23 </grok:render> Key results establish I-cofiniteness of H_I^j(M) under specific ring and ideal conditions. For instance, if R is Noetherian and M is finitely generated, then H_I^j(M) is I-cofinite for all j > 0 when \dim(R/I) = 1, as shown for ideals of dimension one.<grok:render type="render_inline_citation"> 20 </grok:render> This holds more broadly in complete local Gorenstein domains where \dim(R/I) = 1, ensuring H_I^j(M) satisfies the support and Ext-finiteness criteria for all j.<grok:render type="render_inline_citation"> 70 </grok:render> In regular local rings of small dimension, such as \dim R \leq 3, additional vanishing theorems confirm cofiniteness for non-minimal degrees, with H_I^j(R) = 0 for j exceeding the height of I under finiteness assumptions on Hom modules.<grok:render type="render_inline_citation"> 70 </grok:render> Cofiniteness is preserved under certain change of rings, such as flat base changes or completions, via the change of ring principle, which transfers the property from modules over R to those over extensions like polynomial rings or completions while maintaining the Ext-finiteness relative to the extended ideal.<grok:render type="render_inline_citation"> 40 </grok:render> This principle facilitates computations in more tractable settings and generalizes Hartshorne's original theorems to broader classes of rings.<grok:render type="render_inline_citation"> 40 </grok:render> Recent developments extend these ideas to generalized local cohomology modules H_I^j(M, N) = \varinjlim_n \operatorname{Ext}^j_R(M/I^n M, N), with post-2000 results providing criteria for I-cofiniteness when I is principal or prime in complete local rings, including vanishing theorems that imply cofiniteness in low dimensions.<grok:render type="render_inline_citation"> 8 </grok:render> For example, Huneke and Lyubeznik established that in complete regular local rings with unmixed dimension-2 ideals and connected punctured spectrum, H_I^{d-2}(R) is I-cofinite, linking cofiniteness to topological connectivity conditions.<grok:render type="render_inline_citation"> 70 </grok:render> As an illustrative example, when I is a maximal ideal \mathfrak{m} in a local ring (R, \mathfrak{m}), an \mathfrak{m}-cofinite module is precisely an Artinian R-module, so H_\mathfrak{m}^j(R) being \mathfrak{m}-cofinite implies it is Artinian, capturing the descending chain condition on submodules in this extremal case.<grok:render type="render_inline_citation"> 77 </grok:render>

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