 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 and
 The deleted integer topology is defined by letting and .
The trivial partitions yield the discrete topology (each point of X is a set in P) or indiscrete topology (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:
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 T_{1} 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 a regular, completely regular, normal and completely normal.
We note also that 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: SpringerVerlag, ISBN 9780486687353, MR507446
Categories: Topological spaces
Wikimedia Foundation. 2010.