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Cofiniteness
Cofiniteness
Cofiniteness
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In mathematics, a cofinite subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but is countable, then one says the set is cocountable.

These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.

This use of the prefix "co" to describe a property possessed by a set's complement is consistent with its use in other terms such as "comeagre set".

Boolean algebras

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The set of all subsets of that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the finite–cofinite algebra on

In the other direction, a Boolean algebra has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set such that is isomorphic to the finite–cofinite algebra on In this case, the non-principal ultrafilter is the set of all cofinite subsets of .

Cofinite topology

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The cofinite topology or the finite complement topology is a topology that can be defined on every set It has precisely the empty set and all cofinite subsets of as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of For this reason, the cofinite topology is also known as the finite-closed topology. Symbolically, one writes the topology as

This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field are zero on finite sets, or the whole of the Zariski topology on (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve; it is not true, for example, for in the plane.

Properties

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  • Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.
  • Compactness: Since every open set contains all but finitely many points of the space is compact and sequentially compact.
  • Separation: The cofinite topology is the coarsest topology satisfying the T1 axiom; that is, it is the smallest topology for which every singleton set is closed. In fact, an arbitrary topology on satisfies the T1 axiom if and only if it contains the cofinite topology. If is finite then the cofinite topology is simply the discrete topology. If is not finite then this topology is not Hausdorff (T2), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).

Double-pointed cofinite topology

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The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of each doublet are topologically indistinguishable. It is, however, R0 since topologically distinguishable points are separated. The space is compact as the product of two compact spaces; alternatively, it is compact because each nonempty open set contains all but finitely many points.

For an example of the countable double-pointed cofinite topology, the set of integers can be given a topology such that every even number is topologically indistinguishable from the following odd number . The closed sets are the unions of finitely many pairs or the whole set. The open sets are the complements of the closed sets; namely, each open set consists of all but a finite number of pairs or is the empty set.

Other examples

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Product topology

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The product topology on a product of topological spaces has basis where is open, and cofinitely many

The analog without requiring that cofinitely many factors are the whole space is the box topology.

Direct sum

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The elements of the direct sum of modules are sequences where cofinitely many

The analog without requiring that cofinitely many summands are zero is the direct product.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Cofiniteness

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In , a cofinite of a set XX is a S⊆XS \subseteq X whose complement X∖SX \setminus S is finite. Equivalently, SS contains all but finitely many elements of XX. This concept is fundamental in and plays a key role in various areas of . 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 Rn\mathbb{R}^n. Cofiniteness appears in through the cofinite topology on a set, where open sets are the and all cofinite subsets; this is a classic example of a non-Hausdorff topology with interesting separation properties. In , the term relates to finite-cofinite Boolean and cofiniteness conditions in , 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 AA of a set XX is called cofinite if its complement X∖AX \setminus A is a finite set. Equivalently, AA contains all but finitely many elements of XX, which can be denoted by the condition ∣X∖A∣<∞|X \setminus A| < \infty, where ∣⋅∣|\cdot| represents the cardinality of the set. This notion is particularly relevant when XX is infinite, as cofinite subsets then comprise "almost all" of XX, in contrast to finite subsets, which comprise "almost none." For example, if X=NX = \mathbb{N} (the set of natural numbers), then A=N∖{1,2,3}A = \mathbb{N} \setminus \{1, 2, 3\} is cofinite because its complement has three elements. However, the empty set ∅\emptyset is not cofinite in an infinite XX, since X∖∅=XX \setminus \emptyset = X is infinite; ∅\emptyset is cofinite only if XX itself is finite.

Basic Properties and Examples

Cofinite subsets of a set XX 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. For instance, if AA and BB are cofinite in XX, then ∣X∖(A∩B)∣=∣(X∖A)∪(X∖B)∣≤∣X∖A∣+∣X∖B∣<∞|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. However, arbitrary intersections of cofinite sets need not be cofinite; for example, in N\mathbb{N}, the intersection ⋂n=1∞(N∖{n})\bigcap_{n=1}^\infty (\mathbb{N} \setminus \{n\}) is empty, whose complement N\mathbb{N} is infinite. Here, the singletons {n}\{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 Z\mathbb{Z}, cofinite subsets arise naturally by excluding finitely many elements; for example, Z∖{0,π}\mathbb{Z} \setminus \{0, \pi\} (noting π∉Z\pi \notin \mathbb{Z}) or more precisely Z∖{−1,0,1}\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 R\mathbb{R}. Regarding cardinality, if XX is infinite, every cofinite subset of XX has the same cardinality as XX itself. Removing finitely many elements from an infinite set does not alter its cardinal size, as there exists a bijection between XX and X∖FX \setminus F for any finite F⊆XF \subseteq X. This holds by basic cardinal arithmetic, where ∣X∣+ℵ0=∣X∣|X| + \aleph_0 = |X| for infinite ∣X∣|X|, and finite addition is even more straightforward.

Topological Applications

The Cofinite Topology

The cofinite topology on a set XX is defined as the collection of all subsets of XX that are either empty or cofinite, where a cofinite subset has a finite complement in XX. This topology, also known as the finite complement topology, endows XX with a structure where the open sets are precisely ∅\emptyset and those subsets U⊆XU \subseteq X such that X∖UX \setminus U is finite. 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. Consequently, the closed sets in the cofinite topology are exactly the finite subsets of XX and XX itself, since the complement of an open set must be finite or the entire space. When XX is finite, the cofinite topology coincides with the discrete topology on XX, because every subset of a finite set is cofinite. For infinite XX, however, the topology is coarser than the discrete one, highlighting its distinct behavior on infinite sets. 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.

Properties of the Cofinite Topology

The cofinite topology on an infinite set XX satisfies the T1T_1 separation axiom because singletons are closed sets, as their complements are cofinite and thus open. However, it fails the T2T_2 (Hausdorff) axiom, since any two non-empty open sets have non-empty intersection: if UU and VV are non-empty opens, then X∖UX \setminus U and X∖VX \setminus V are finite, so X∖(U∩V)=(X∖U)∪(X∖V)X \setminus (U \cap V) = (X \setminus U) \cup (X \setminus V) is finite, implying U∩VU \cap V is cofinite and hence non-empty. Consequently, distinct points cannot be separated by disjoint open neighborhoods. 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 XX, since disjoint finite closed sets cannot be separated by disjoint opens. The cofinite topology is compact. To see this, consider an open cover {Ui}i∈I\{U_i\}_{i \in I} of XX. Each Ui=X∖FiU_i = X \setminus F_i where FiF_i is finite. Select U1U_1, which misses the finite set F1F_1. For each point in F1F_1, choose a UjU_j covering it; finitely many such UjU_j suffice. Their union covers XX, yielding a finite subcover. The space is hyperconnected: any two non-empty open sets intersect, as shown earlier, making it impossible to decompose XX into two non-empty disjoint opens. Thus, it is connected but not path-connected in general. Every subspace of (X,τ)(X, \tau) with the cofinite topology τ\tau inherits the cofinite topology. For a subset A⊆XA \subseteq X, a set U⊆AU \subseteq A is open in the subspace topology if U=V∩AU = V \cap A for some open V⊆XV \subseteq X, so V=X∖FV = X \setminus F with FF finite, hence U=A∖(A∩F)U = A \setminus (A \cap F) where A∩FA \cap F is finite. Conversely, any cofinite open in AA arises this way from a cofinite open in XX.

Variants of the Cofinite Topology

The double-pointed cofinite topology is a modification of the cofinite topology constructed on the product space Y=X×{0,1}Y = X \times \{0, 1\}, where XX is an infinite set equipped with the cofinite topology and {0,1}\{0, 1\} has the indiscrete topology. The open sets in this topology consist of the empty set and all subsets U⊆YU \subseteq Y such that the projection πX(U)\pi_X(U) onto XX is cofinite (or empty). Equivalently, the basis for the topology comprises sets of the form V×{0,1}V \times \{0, 1\}, where VV is cofinite in XX. This construction effectively "doubles" each point in XX, making the points (x,0)(x, 0) and (x,1)(x, 1) topologically indistinguishable for each x∈Xx \in X. This topology is not T0T_0 or T1T_1, as no open set can separate the paired points (x,0)(x, 0) and (x,1)(x, 1), but it satisfies the symmetric separation axiom R0R_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=ZX = \mathbb{Z}, a basic open neighborhood of a point (n,0)(n, 0) (or equivalently (n,1)(n, 1)) excludes only finitely many doublets {(ki,0),(ki,1)}\{(k_i, 0), (k_i, 1)\} for distinct ki∈Zk_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 XX 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 T1T_1 (points are closed) but not Hausdorff, and it is hyperconnected (any two nonempty open sets intersect). 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 T1T_1 space that is not normal. Generalizations include co-κ\kappa-topologies for cardinal κ\kappa, where open sets have complements of cardinality less than κ\kappa, with the cocountable case corresponding to κ=ℵ0\kappa = \aleph_0; these relate indirectly to one-point compactifications of discrete spaces, where neighborhoods of the added point resemble cofinite sets.

Algebraic Applications

Finite-Cofinite Boolean Algebras

In the context of , the finite-cofinite algebra on an infinite set XX is defined as the collection of all subsets of XX that are either finite or cofinite, equipped with the standard set operations of union, intersection, and complementation. This structure forms a , where the empty set serves as the zero element and XX as the unit element. 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}\{x\} for each x∈Xx \in X, as these are minimal nonzero elements, and every finite set is the join of its singletons. This algebra is atomic, meaning every nonzero element is the join of atoms below it, but it is not complete when XX is infinite; for instance, consider the family of sets X∖{p}X \setminus \{p\} for all pp in an infinite subset S⊂XS \subset X such that X∖SX \setminus S is also infinite: their infimum (greatest lower bound) would be X∖SX \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 P(X)\mathcal{P}(X), which includes all subsets and is complete and atomic for any XX, 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. A concrete example arises when X=NX = \mathbb{N}, the natural numbers: the finite-cofinite algebra consists of all finite subsets of N\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 N\mathbb{N}.

Cofiniteness in Direct Sums and Products

In the context of modules over a ring, the direct sum of a family of modules {Mi}i∈I\{M_i\}_{i \in I} over an index set II is defined as the submodule of the direct product ∏i∈IMi\prod_{i \in I} M_i consisting of those elements (mi)i∈I(m_i)_{i \in I} where the support {i∈I:mi≠0}\{i \in I : m_i \neq 0\} is finite, meaning mi=0m_i = 0 for cofinitely many ii. Formally, an element (mi)i∈I∈⨁i∈IMi(m_i)_{i \in I} \in \bigoplus_{i \in I} M_i if and only if ∣{i:mi≠0}∣<∞|\{i : m_i \neq 0\}| < \infty. 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 Z\mathbb{Z}-modules, where the direct sum ⨁n=1∞Z\bigoplus_{n=1}^\infty \mathbb{Z} consists of sequences (a1,a2,… )(a_1, a_2, \dots) of integers with only finitely many nonzero terms, such as (3,0,−2,0,… )(3, 0, -2, 0, \dots). In contrast, the direct product ∏n=1∞Z\prod_{n=1}^\infty \mathbb{Z} comprises all integer sequences, including those with infinitely many nonzeros, such as the constant sequence (1,1,1,… )(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 ∏i∈IXi\prod_{i \in I} X_i. The product topology is generated by a subbasis consisting of sets of the form πi−1(Ui)\pi_i^{-1}(U_i), where πi:∏j∈IXj→Xi\pi_i : \prod_{j \in I} X_j \to X_i is the projection and UiU_i is open in XiX_i; equivalently, the basis elements are finite intersections of these, which specify open conditions in only finitely many coordinates while taking the full space XjX_j in the cofinitely many remaining coordinates. 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 RN\mathbb{R}^\mathbb{N} with the standard topology on each R\mathbb{R}, basic open sets are cylinders like {(xn):∣xk−ak∣<ϵ for k=1,…,m}\{ (x_n) : |x_k - a_k| < \epsilon \text{ for } k=1,\dots,m \}, full in all but finitely many factors. If each XiX_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.

Cofiniteness in Homological Algebra

In homological algebra, particularly in the study of local cohomology, an RR-module MM is said to be II-cofinite, where II is an ideal of a Noetherian ring RR, if the support of MM is contained in V(I)V(I) and Ext⁡Ri(R/I,M)\operatorname{Ext}^i_R(R/I, M) is finitely generated for all i≥0i \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 HIj(N)H_I^j(N), where NN is a finitely generated RR-module, prompting questions about when these modules inherit II-cofiniteness from NN.<grok:render type="render_inline_citation"> 0 </grok:render> The associated primes of an II-cofinite module are finite in number and lie in V(I)V(I), with additional depth conditions ensuring the module's behavior relative to II.<grok:render type="render_inline_citation"> 23 </grok:render> Key results establish II-cofiniteness of HIj(M)H_I^j(M) under specific ring and ideal conditions. For instance, if RR is Noetherian and MM is finitely generated, then HIj(M)H_I^j(M) is II-cofinite for all j>0j > 0 when dim⁡(R/I)=1\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\dim(R/I) = 1, ensuring HIj(M)H_I^j(M) satisfies the support and Ext-finiteness criteria for all jj.<grok:render type="render_inline_citation"> 70 </grok:render> In regular local rings of small , such as dim⁡R≤3\dim R \leq 3, additional vanishing theorems confirm cofiniteness for non-minimal degrees, with HIj(R)=0H_I^j(R) = 0 for jj exceeding the of II 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 RR 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 HIj(M,N)=lim→⁡nExt⁡Rj(M/InM,N)H_I^j(M, N) = \varinjlim_n \operatorname{Ext}^j_R(M/I^n M, N)
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