In mathematics, a cofinite subset of a set X {\displaystyle X} is a subset A {\displaystyle A} whose complement in X {\displaystyle X} is a finite set...

In mathematics, the Fréchet filter, also called the cofinite filter, on a set X {\displaystyle X} is a certain collection of subsets of X {\displaystyle...

of all the open sets containing it. Every finite set is closed. Every cofinite set of X {\displaystyle X} is open. For every x ∈ X , {\displaystyle x\in...

(Zhu's C2-cofiniteness condition) If R V {\displaystyle R_{V}} is finite dimensional then V {\displaystyle V} is said to be C2-cofinite. There are two...

Hausdorff spaces where limit points are unique. Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement...

a cofinite set with any set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and "cofinite" interchanged...

topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete topology Double-pointed cofinite topology Extended real number...

computable. Every recursive language is a computable. Every finite or cofinite subset of the natural numbers is computable. The empty set is computable...

subset of its domain that is definable with parameters is either finite or cofinite. A strongly minimal theory is a complete theory all models of which are...

{\displaystyle X_{n}} for all but finitely many n {\displaystyle n} (that is, cofinitely many n {\displaystyle n} ). That is, x ∈ lim inf X n {\displaystyle x\in...

Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The...

reals. If the complement is finite, then one says Y {\displaystyle Y} is cofinite. The set of all subsets of X {\displaystyle X} that are either countable...

finite or cofinite is a Boolean algebra and an algebra of sets called the finite–cofinite algebra. If S is infinite then the set of all cofinite subsets...

each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily...

{\displaystyle \alpha _{i}=0} for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.) It can also be...

A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set, as is the cocountable topology defined...

and (3), and compact, but not locally compact in senses (4) or (5). The cofinite topology on an infinite set is locally compact in senses (1), (2), and...

an example of a T1 space that is not sober is an infinite set with the cofinite topology, the whole space being an irreducible closed subset with no generic...

introducing the right topology. The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given this topology...

Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense. The basis open set corresponding...

x=a_{1}\vee \dots \vee x=a_{n}} . Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is...

{\displaystyle x.} An example of a space that is not first-countable is the cofinite topology on an uncountable set (such as the real line). More generally...

conjectures about topological spaces in general. Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement...

subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element. See the Topology and ordinals section...

this includes in particular the trivial topology. Any space carrying the cofinite topology is compact. Any locally compact Hausdorff space can be turned...

definition is equivalent to the one given above. Cofinal (mathematics) Cofiniteness Ergodic Ramsey theory Piecewise syndetic set Syndetic set J. McLeod,...

_{2}} -complete. C o f = { e : W e  is cofinite } {\displaystyle \mathrm {Cof} =\{e\,:\,W_{e}{\text{ is cofinite}}\}} is Σ 3 {\displaystyle \Sigma _{3}}...

{\displaystyle \cap \,{\mathcal {U}}=\varnothing } ), or equivalently, if every cofinite subset of I {\displaystyle I} is an element of U . {\displaystyle {\mathcal...

intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations...

largeness that also usefully distinguish subsets of natural numbers: Cofiniteness IP set member of a nonprincipal ultrafilter positive upper density syndetic...