Odd-even Topology

In mathematics, the partition topology is a topology that can be induced on any set $X$ by partitioning $X$ into disjoint subsets $P;$ these subsets form the basis for the topology. There are two important examples which have their own names:
* The is the topology where $X = \N$ and $P = .$ Equivalently, $P = \.$
* The is defined by letting $X = \begin \bigcup_ (n-1,n) \subseteq \Reals \end$ and $P = .$

The trivial partitions yield the discrete topology (each point of $X$ is a set in $P,$ so $P = \$) or indiscrete topology (the entire set $X$ is in $P,$ so $P = \$).

Any set $X$ with a partition topology generated by a partition $P$ can be viewed as a pseudometric space with a pseudometric given by:
$d(x, y) = \begin 0 & \text x \text y \text \\ 1 & \text. \end$

This is not a metric unless $P$ yields the discrete topology.

The partition topology provides an important example of the independence of various separation axioms. Unless $P$ is trivial, at least one set in $P$ contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence $X$ is not a Kolmogorov space, nor a T₁ space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, $X$ is regular, completely regular, normal and completely normal. $X / P$ is the discrete topology.

See also

*

References

* {{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=Counterexamples in Topology , origyear=1978 , publisher=Springer-Verlag , location=Berlin, New York , edition=Dover reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995

Topological spaces