Kofinite topology

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In mathematical branch of topology which designates cofiniteness (also cofinite topology written) a class of pathological examples for topological spaces . It can be defined using any set: In it, precisely those sets are open whose complements are finite or which are themselves empty. This is equivalent to the fact that the closed sets are exactly the finite sets or the whole set. In the following we consider the cofinite topology only over infinite sets , since they have interesting properties (in the finite case one obtains the discrete topology ).

Separation properties

Every kofinite topology forms a Kolmogoroff space , it fulfills the axiom of separation T₀: Two different points are topologically distinguishable, at least one of two different points has a neighborhood that does not contain the other. In addition, it fulfills the axiom of separation T₁ , that is, both points each have an environment that does not contain the other, after all, permutations in the cofinite topology are homeomorphisms (the automorphism group is the same as for the discrete topology). However, cofinite topologies over infinite sets do not satisfy the separation axiom T₂, they do not form Hausdorff spaces : It is not possible to choose these two neighborhoods as disjoint , because there are no two non-empty, disjoint open sets. Therefore, they do not form T₃ spaces , because there is a nontrivial closed set, but of course this cannot be separated from a point outside by disjoint surroundings, a T₀ and T₃ space would also have to be Hausdorff-like.

The kofinite topology is also the coarsest topology over any set that satisfies T₁, because for T₁ it is necessary (and sufficient) that every single-element set is closed. Thus, a T₁ space must contain at least all finite sets as closed sets.

convergence

The effects of the missing Hausdorff property on the convergence of filters and networks can be demonstrated using cofinite topologies:

  • Since there is no Hausdorff space, there are filters with several limit values: consider the filter that contains all non-empty, open sets. It is a filter because there are no two non-empty, disjoint, open sets. It converges towards all points in space, since it naturally also contains all surroundings of each point.
  • Correspondingly, every total order of the elements of the space converges as a network without double elements towards every point: for every neighborhood of a point, all elements of the network lie in it from an index.

Other properties

  • The conclusion of any infinite set is the total set. It follows immediately that the room is separable .
  • The interior of every finite set is empty.
  • On an uncountable set, the cofinite topology violates both axioms of countability .
  • Cofinite topologies are not induced by a metric or by a uniform structure .
  • Every kofinite topology forms a compact space , but due to the lack of the Hausdorff property, there are no strong separation properties such as T₄ .

generalization

Instead of assuming that the closed sets, with the exception of the entire space, are finite, they can also be constrained by any infinite cardinal number . This results in the next largest topology according to this scheme, the co-countable topology , which also represents an important pathological example on uncountable sets.

Individual evidence

  1. ↑ Lynn A. Steen , J. Arthur Seebach: Counterexamples in Topology . Holt, Rinehart and Winston, New York NY et al. 1970, ISBN 0-03-079485-4 , p. 50.

literature