Partition topology

In mathematics, a partition topology is a topology that can be induced on any set ${\displaystyle X}$ by partitioning${\displaystyle X}$ into disjoint subsets ${\displaystyle P;}$ 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 ${\displaystyle X=\mathbb {N} }$ and ${\displaystyle P={\left\{~\{2k-1,2k\}:k\in \mathbb {N} \right\}}.}$ Equivalently, ${\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.}$
• The deleted integer topology is defined by letting ${\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}}$ and ${\displaystyle P={\left\{(0,1),(1,2),(2,3),\ldots \right\}}.}$

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

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

This is not a metric unless ${\displaystyle P}$ yields the discrete topology.

The partition topology provides an important example of the independence of various separation axioms. Unless ${\displaystyle P}$ is trivial, at least one set in ${\displaystyle P}$ contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence ${\displaystyle X}$ 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, ${\displaystyle X}$ is regular, completely regular, normal and completely normal. ${\displaystyle X/P}$ is the discrete topology.

• List of topologies – List of concrete topologies and topological spaces

• 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