Cofinite
Encyclopedia
In mathematics
, a cofinite subset
of a set X is a subset A whose complement
in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it 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.
(i.e. a maximal filter not generated by a single element of the algebra) if and only if there is an infinite set X such that A is isomorphic to the finite-cofinite algebra on X. In this case, the non-principal ultrafilter is the set of all cofinite sets.
which can be defined on every set X. It has precisely the empty set
and all cofinite subsets of X as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of X. Symbolically, one writes the topology as
This topology occurs naturally in the context of the Zariski topology
. Since polynomial
s over a field
K are zero on finite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The same is true for any irreducible
algebraic curve
; it is not true, for example, for XY = 0 in the plane.
, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable.
An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together. Let X be the set of integers, and let OA be a subset of the integers whose complement is the set A. Define a subbase
of open sets Gx for any integer x to be Gx = O{x, x+1} if x is an even number, and Gx = O{x-1, x} if x is odd. Then the basis sets of X are generated by finite intersections, that is, for finite A, the open sets of the topology are
The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologically indistinguishable. The space is, however, a compact space
, since it is covered by a finite union of the UA.
on a product of topological spaces
has basis
where 
is open, and cofinitely many 
.
The analog (without requiring that cofinitely many are the whole space) is the box topology
.
are sequences 
where cofinitely many 
.
The analog (without requiring that cofinitely many are zero) is the direct product
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a cofinite subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a set X is a subset A whose complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable
Cocountable
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. For example, the irrational numbers are a cocountable subset of the reals...
.
These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.
Boolean algebras
The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the finite-cofinite algebra on X. A Boolean algebra A has a unique non-principal ultrafilterUltrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
(i.e. a maximal filter not generated by a single element of the algebra) if and only if there is an infinite set X such that A is isomorphic to the finite-cofinite algebra on X. In this case, the non-principal ultrafilter is the set of all cofinite sets.
Cofinite topology
The cofinite topology (sometimes called the finite complement topology) is a topologyTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
which can be defined on every set X. It has precisely the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
and all cofinite subsets of X as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of X. Symbolically, one writes the topology as
This topology occurs naturally in the context of the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
. Since polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K are zero on finite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The same is true for any irreducible
Irreducible component
In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equationis the union of the two linesandThe notion of irreducibility is stronger than connectedness.- Definition :...
algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
; it is not true, for example, for XY = 0 in the plane.
Properties
- Subspaces: Every subspace topologySubspace topologyIn topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of the cofinite topology is also the cofinite topology. - Compactness: Since every open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
contains all but finitely many points of X, the space X is compact and sequentially compact. - Separation: The cofinite topology is the coarsest topologyComparison of topologiesIn topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set...
satisfying the T1 axiomT1 spaceIn topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
; i.e. it is the smallest topology for which every singleton set is closed. In fact, an arbitrary topology on X satisfies the T1 axiom if and only if it contains the cofinite topology. If X is finite then the cofinite topology is simply the discrete topologyDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
. If X is not finite, then this topology is not T2, regularRegular spaceIn topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
or normalNormal spaceIn topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
, since no two nonempty open sets are disjoint (i.e. it is hyperconnectedHyperconnected spaceIn mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry....
).
Double-pointed cofinite topology
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. It is not T0 or T1T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable.
An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together. Let X be the set of integers, and let OA be a subset of the integers whose complement is the set A. Define a subbase
Subbase
In topology, a subbase for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B...
of open sets Gx for any integer x to be Gx = O{x, x+1} if x is an even number, and Gx = O{x-1, x} if x is odd. Then the basis sets of X are generated by finite intersections, that is, for finite A, the open sets of the topology are
The resulting space is not T0 (and hence not T1), because the points x and x + 1 (for x even) are topologically indistinguishable. The space is, however, a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, since it is covered by a finite union of the UA.
Product topology
The product topologyProduct topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
on a product of topological spaces
has basis
where is open, and cofinitely many .The analog (without requiring that cofinitely many are the whole space) is the box topology
Box topology
In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more...
.
Direct sum
The elements of the direct sum of modulesDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
are sequences where cofinitely many .The analog (without requiring that cofinitely many are zero) is the direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
.