cofinite topology

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In mathematics, a cofinite
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 a
Boolean 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 a
topology 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 $\mathcal = \.$ 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: Every
subspace 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 T1 axiom; that is, it is the smallest topology for which every
singleton 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 T1 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 (T2), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).

## 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 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 T0 or T1, since the points of the doublet are
topologically 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, 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 $O_A$ be a subset of the integers whose complement is the set $A.$ Define a
subbase 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 $U_A := \bigcap_ G_x$ 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 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

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 ...
on a product of topological spaces $\prod X_i$ 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 ...
$\prod U_i$ where $U_i \subseteq X_i$ 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 the
direct 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 ...
$\bigoplus M_i$ are sequences $\alpha_i \in M_i$ where cofinitely many $\alpha_i = 0.$ The analog (without requiring that cofinitely many are zero) is the direct product.