# Cofinite

### Some articles on cofinite:

Cofiniteness - Cofinite Topology - 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 ... An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together ...
Cofiniteness - 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 ...
Fréchet Filter
... In mathematics, the Fréchet filter, also called the cofinite filter, on a set is a special subset of the set's power set ... It is alternatively called a cofinite filter because its members are exactly the cofinite sets in a power set ...
Cofiniteness - Cofinite Topology - Properties
... Subspaces Every subspace topology of the cofinite topology is also a cofinite topology ... Separation The cofinite topology is the coarsest topology satisfying the T1 axiom i.e ... an arbitrary topology on X satisfies the T1 axiom if and only if it contains the cofinite topology ...
T1 Space - Examples
... The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not Hausdorff (T2) ... This follows since no two open sets of the cofinite topology are disjoint ... modified slightly to create the double-pointed cofinite topology, which is an example of an R0 space that is neither T1 nor R1 ...