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| {{Distinguish|cofinality}}
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| In [[mathematics]], a '''cofinite''' [[subset]] of a set ''X'' is a subset ''A'' whose [[complement (set theory)|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]].
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| These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the [[#Product topology|product topology]] or [[#Direct sum|direct sum]].
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| ==Boolean algebras==
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| The set of all subsets of ''X'' that are either finite or cofinite forms a [[Boolean algebra (structure)|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 [[ultrafilter]] (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.
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| ==Cofinite topology==
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| The '''cofinite topology''' (sometimes called the '''finite complement topology''') is a [[topological space|topology]] which can be defined on every set ''X''. It has precisely the [[empty set]] and all [[cofinite subset]]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
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| :<math>\mathcal{T} = \{A \subseteq X \mid A=\varnothing \mbox{ or } X \setminus A \mbox{ is finite} \}</math>
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| This topology occurs naturally in the context of the [[Zariski topology]]. Since [[polynomial]]s over a [[field (mathematics)|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 component|irreducible]]'' [[algebraic curve]]; it is not true, for example, for ''XY'' = 0 in the plane.
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| ===Properties===
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| * Subspaces: Every [[subspace topology]] of the cofinite topology is also a cofinite topology.
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| * Compactness: Since every [[open set]] contains all but finitely many points of ''X'', the space ''X'' is [[compact set|compact]] and [[sequentially compact]].
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| * Separation: The cofinite topology is the [[Comparison of topologies|coarsest topology]] satisfying the [[T1 space|T<sub>1</sub> axiom]]; i.e. it is the smallest topology for which every [[singleton set]] is closed. In fact, an arbitrary topology on ''X'' satisfies the T<sub>1</sub> axiom if and only if it contains the cofinite topology. If ''X'' is finite then the cofinite topology is simply the [[discrete space|discrete topology]]. If ''X'' is not finite, then this topology is not [[T2 space|T<sub>2</sub>]], [[regular space|regular]] or [[normal space|normal]], since no two nonempty open sets are disjoint (i.e. it is [[hyperconnected space|hyperconnected]]).
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| ===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]]. It is not [[T0 space|T<sub>0</sub>]] or [[T1 space|T<sub>1</sub>]], since the points of the doublet are [[topologically indistinguishable]]. It is, however, [[R0 space|R<sub>0</sub>]] since the topologically distinguishable points are separable.
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| 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''<sub>''A''</sub> be a subset of the integers whose complement is the set ''A''. Define a [[subbase]] of open sets ''G''<sub>''x''</sub> for any integer ''x'' to be ''G''<sub>''x''</sub> = ''O''<sub>{''x'', ''x''+1}</sub> if ''x'' is an [[even number]], and ''G''<sub>''x''</sub> = ''O''<sub>{''x''-1, ''x''}</sub> if ''x'' is odd. Then the [[basis (topology)|basis]] sets of ''X'' are generated by finite intersections, that is, for finite ''A'', the open sets of the topology are
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| :<math>U_A := \bigcap_{x \in A} G_x </math> | |
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| The resulting space is not T<sub>0</sub> (and hence not T<sub>1</sub>), 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 ''U''<sub>A</sub>.
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| ==Other examples==
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| ===Product topology===
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| The [[product topology]] on a product of topological spaces <math>\prod X_i</math>
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| has [[basis (topology)|basis]] <math>\prod U_i</math> where <math>U_i \subset X_i</math> is open, and cofinitely many <math>U_i = X_i</math>.
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| The analog (without requiring that cofinitely many are the whole space) is the [[box topology]].
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| ===Direct sum===
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| The elements of the [[direct sum of modules]] <math>\bigoplus M_i</math> are sequences <math>\alpha_i \in M_i</math> where cofinitely many <math>\alpha_i = 0</math>.
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| The analog (without requiring that cofinitely many are zero) is the [[direct product]].
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| ==References==
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| *{{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]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}} ''(See example 18)''
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| [[Category:Basic concepts in infinite set theory]]
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| [[Category:General topology]]
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