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

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

Any set X with a partition topology generated by a partition P can be viewed as a pseudometric space with a pseudometric given by:

${\displaystyle d(x,y)={\begin{cases}0&{\text{if }}x{\text{ and }}y{\text{ are in the same partition element}}\\1&{\text{otherwise}},\end{cases}}}$

This is not a metric unless 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[]

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• 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