The cofinite topology [sometimes called the finite complement topology] is a topology that 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. Symbolically, one writes the topology as
T={AXA=orXAis finite}{\displaystyle {\mathcal {T}}=\{A\subseteq X\mid A=\varnothing {\mbox{ or }}X\setminus A{\mbox{ is finite}}\}}This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable 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.
PropertiesEdit
Double-pointed cofinite topologyEdit
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 on a two-element set. 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.
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
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 each UA contains all but finitely many points.