Cofiniteness#Cofinite topology

In the mathematical branch of topology, the topology kofinite refers to a class of pathological examples of topological spaces. They can be defined on an arbitrary set: In it, the amounts are accurate open, their complements or are themselves finally empty. This is equivalent to saying that the closed sets are precisely the finite sets or the whole lot. In the following, we consider the topology kofinite only about infinite sets, since they carry interesting properties ( in the finite case, we obtain the discrete topology).

Separation properties

Each kofinite topology forms a Kolmogorov space, it satisfies the separation axiom T ₀: Any two distinct points are topologically distinguishable from two distinct points have at least one of an environment that does not contain the other. In addition, it satisfies the separation axiom T ₁, ie both points each have an environment that does not contain the other, after all permutations in the kofiniten topology homeomorphisms ( the automorphism group is thus the same as for the discrete topology). However kofinite topologies meet on infinite sets not the separation axiom T ₂, they do not form Hausdorff spaces: It is not possible to choose disjoint these two environments, because there are no two non-empty disjoint open sets. So it does not form T ₃ areas, because there is a non-trivial closed set, but this can not of course be separated from a point outside by disjoint neighborhoods, a T ₀ - and T ₃ room would also need to be Hausdorff.

The topology is also kofinite the coarsest topology any amount that satisfies T ₁, since for T ₁, it is necessary ( and sufficient ) to each singleton is completed. Thus, a T ₁ room shall contain at least all finite sets as closed sets.

Convergence

The effects of the lack of Hausdorf fig stem to the convergence of filters and nets can be demonstrated to kofiniten topologies:

• Since no Hausdorff space is present, there are filters with several limits: it contains Consider the filter that all non-empty open sets. There is a filter as there are no two nonempty, disjoint, are open sets. It converges to all points of space, as it of course also includes all the environments of each point.
• According converges any total ordering of the elements of the space as a network without duplicate elements construed against each point: for every neighborhood of a point lying on an index all elements of the network in it.

Other properties

• The conclusion of every infinite set is the entire amount.
• The interior of each finite set is empty.
• The two Abzählbarkeitsaxiome are violated.
• In particular, the kofiniten topologies are induced neither a metric nor a uniform structure.
• Each kofinite topology forms a compact space, but lack the Hausdorf fig stem from it follow no strong separation properties such as T ₄.

Generalization

Rather than assume that the closed sets with the exception of the entire space are finite themselves, they can also be limited by any infinite cardinal. This follows as the next larger topology according to this scheme, the koabzählbare topology, which also represents an important pathological case on uncountable sets.