Editing Cofiniteness (section)

===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]] on a two-element set.  It is not [[T0 space|T<sub>0</sub>]] or [[T1 space|T<sub>1</sub>]], since the points of each doublet are [[topologically indistinguishable]].  It is, however, [[R0 space|R<sub>0</sub>]] since topologically distinguishable points are [[Separated sets|separated]].  The space is [[compact (topology)|compact]] as the product of two compact spaces; alternatively, it is compact because each nonempty open set contains all but finitely many points.

For an example of the countable double-pointed cofinite topology, the set <math>\Z</math> of integers can be given a topology such that every [[even number]] <math>2n</math> is [[topologically indistinguishable]] from the following [[odd number]] <math>2n+1</math>.  The closed sets are the unions of finitely many pairs <math>2n,2n+1,</math> or the whole set.  The open sets are the complements of the closed sets; namely, each open set consists of all but a finite number of pairs <math>2n,2n+1,</math> or is the empty set.