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In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. In other words, Y contains all but finitely many elements of X. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
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 ultrafilter (i.e. a maximal filter not generated by a singleton set) 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. Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...
In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ...
In mathematics, a filter is a special subset of a partially ordered set. ...
In mathematics, a singleton is a set with exactly one element. ...
Cofinite topology The cofinite topology (sometimes called the finite complement topology) is a topology 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. Then X is automatically compact in this topology, since every open set only omits finitely many points of X. Also, the cofinite topology is the smallest topology satisfying the T1 axiom; 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
The title given to this article is incorrect due to technical limitations. ...
In mathematics, a singleton is a set with exactly one element. ...
If X is not finite, then this topology is not T2, regular or normal, since no two open sets in this topology are disjoint. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...
In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...
One place where this concept occurs naturally is in the context of the Zariski topology. Since polynomials 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. This article needs to be cleaned up to conform to a higher standard of quality. ...
In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
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 T1, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable. 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. ...
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ...
In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...
The title given to this article is incorrect due to technical limitations. ...
In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces. ...
The title given to this article is incorrect due to technical limitations. ...
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 In topology, a subbase (or subbasis) 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. A slightly different definition is used by some authors, and there are other useful equivalent formulations...
In mathematics, any integer (whole number) is either even or odd. ...
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...
 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. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
See also In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. ...
References - Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition). (See example 18)
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