Partition topology

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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

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (April 2020) (Learn how and when to remove this message)

• 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

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