In mathematics, a cofinite

_{1} axiom; that is, it is the smallest topology for which every _{1} axiom if and only if it contains the cofinite topology. If $X$ is finite then the cofinite topology is simply the discrete topology. If $X$ is not finite then this topology is not Hausdorff (T_{2}), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).

_{0} or T_{1}, since the points of the doublet are _{0} 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 $O\_A$ be a subset of the integers whose complement is the set $A.$ Define a _{0} (and hence not T_{1}), because the points $x$ and $x\; +\; 1$ (for $x$ even) are topologically indistinguishable. The space is, however, a

subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of a set $X$ is a subset $A$ whose complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...

in $X$ is a finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. ...

. 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
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''. Since the rational numbers are a countable subset of the r ...

.
These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product 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. This topology differs from another, perhaps more natural-seemi ...

or direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...

.
This use of the prefix "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as " meagre set".
Boolean algebras

The set of all subsets of $X$ that are either finite or cofinite forms aBoolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

, which means that it is closed under the operations of union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''U ...

, intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

, and complementation. This Boolean algebra is the on $X.$ A Boolean algebra $A$ has a unique non-principal ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...

(that is, a maximal filter not generated by a single element of the algebra) if and only if there exists 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 atopology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

that can be defined on every set $X.$ It has precisely the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

and all cofinite subset
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 cocou ...

s 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
$$\backslash mathcal\; =\; \backslash .$$
This topology occurs naturally in the context of the Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

. Since polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...

s in one variable over a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...

$K$ 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
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

; it is not true, for example, for $XY\; =\; 0$ in the plane.
Properties

* Subspaces: Everysubspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...

of the cofinite topology is also a cofinite topology.
* Compactness: Since every open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

contains all but finitely many points of $X,$ the space $X$ is compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Briti ...

and sequentially compact
In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X.
Every metric space is naturally a topological space, and for metric spaces, the notio ...

.
* Separation: The cofinite topology is the coarsest topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as the ...

satisfying the Tsingleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...

is closed. In fact, an arbitrary topology on $X$ satisfies the TDouble-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 theindiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...

on a two-element set. It is not Ttopologically indistinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...

. It is, however, Rsubbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...

of open sets $G\_x$ for any integer $x$ to be $G\_x\; =\; O\_$ if $x$ is an even number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
4 ...

, and $G\_x\; =\; O\_$ if $x$ is odd. Then the basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...

sets of $X$ are generated by finite intersections, that is, for finite $A,$ the open sets of the topology are
$$U\_A\; :=\; \backslash bigcap\_\; G\_x$$
The resulting space is not Tcompact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...

, since each $U\_A$ contains all but finitely many points.
Other examples

Product topology

Theproduct 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. This topology differs from another, perhaps more natural-seemi ...

on a product of topological spaces $\backslash prod\; X\_i$ has basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...

$\backslash prod\; U\_i$ where $U\_i\; \backslash subseteq\; X\_i$ is open, and cofinitely many $U\_i\; =\; X\_i.$
The analog (without requiring that cofinitely many are the whole space) is the box topology.
Direct sum

The elements of thedirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, m ...

$\backslash bigoplus\; M\_i$ are sequences $\backslash alpha\_i\; \backslash in\; M\_i$ where cofinitely many $\backslash alpha\_i\; =\; 0.$
The analog (without requiring that cofinitely many are zero) is the direct product.
See also

* * *References

* {{Citation, last1=Steen, first1=Lynn Arthur, author1-link=Lynn Arthur Steen, last2=Seebach, first2=J. Arthur Jr., author2-link=J. Arthur Seebach, Jr., title=Counterexamples in Topology
''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.
In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) ha ...

, orig-year=1978, publisher=Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 i ...

, location=Berlin, New York, edition=Dover
Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidsto ...

reprint of 1978, isbn=978-0-486-68735-3, mr=507446, year=1995 ''(See example 18)''
Basic concepts in infinite set theory
General topology