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 trivial partitions yield the discrete topology (each point of "X" is a set in "P") or indiscrete topology (formula_5). 
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 regular, completely regular, normal and completely normal. "X/P" is the discrete topology.