Partition topology

In mathematics, a 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 odd–even topology is the topology where $X=\mathbb {N}$ and $P={\left\{~\{2k-1,2k\}:k\in \mathbb {N} \right\}}.$ Equivalently, $P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.$
The deleted integer topology is defined by letting $X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}$ and $P={\left\{(0,1),(1,2),(2,3),\ldots \right\}}.$

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

Any set $X$ with a partition topology generated by a partition $P$ can be viewed as a pseudometric space with a pseudometric given by: $d(x,y)={\begin{cases}0&{\text{if }}x{\text{ and }}y{\text{ are in the same partition element}}\\1&{\text{otherwise}}.\end{cases}}$

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.

References

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446

See also

References