Partition topology

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

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

Any set [math]\displaystyle{ X }[/math] with a partition topology generated by a partition [math]\displaystyle{ P }[/math] can be viewed as a pseudometric space with a pseudometric given by: [math]\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} }[/math]

This is not a metric unless [math]\displaystyle{ P }[/math] yields the discrete topology.

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

See also

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

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