Partition topology

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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 ().

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 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 a regular, completely regular, normal and completely normal.

We note also that X/P is the discrete topology.