Co-countable topology

In the mathematical sub-area of topology , the co-countable topology describes a class of pathological examples for topological spaces . With regard to this topology, a subset is open if and only if it is the empty set or has a countable complement . The countable sets and the entire space are therefore precisely the sets closed with respect to the co-countable topology .

Usually one considers the co-countable topology over uncountable sets, because for countable sets it agrees with the discrete topology . In the following, the co-countable topology is therefore only considered over non-countable sets.

properties

The co-countable topology is finer than the cofinite topology . Therefore every space with a co-countable topology is a T ₁ space , but it is not a Hausdorff space , since every two non-empty open sets have a non-empty section. Similarly, a space with a co-countable topology does not satisfy any countability axiom .

The only compact subsets form the finite sets, so spaces with a co-countable topology are not -compact . They are also uncountably compact . Furthermore, from the finiteness of the compact sets it follows that all compact sets are closed . ${\displaystyle \sigma }$

By definition, every room with a countable topology is a Lindelöf room .

Spaces with a co-countable topology are irreducible , i.e. in particular connected and locally connected.

Convergent consequences

Uncountable sets that have been equipped with a co-countable topology form an example of topological spaces that are not Hausdorffian, but in which sequence convergence is still unambiguous:

A sequence converges to a point in a topological space if there exists for every neighborhood of a , so that applies to all . With regard to the co-countable topology, a sequence converges if and only if it is constant after finitely many terms. One recognizes that such a topological space cannot be distinguished from a space with a discrete topology with regard to the convergence behavior of sequences. ${\displaystyle x_{n}}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle x_{n}\in U}$ ${\displaystyle n\geq N}$

This example also shows that topologies are not clearly characterized by the sequences converging in them - in contrast to a metric. With regard to the convergent sequences, the co-countable topology and the discrete topology agree, although the topologies do not match if the underlying set is uncountable.

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• Lynn A. Steen, J. Arthur Seebach: Counterexamples in Topology . Dover, 1995, ISBN 978-0-486-31929-2 .