In mathematics, a partition topology is a topology that can be induced on any set
into disjoint subsets
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
) or indiscrete topology (the entire set
are in the same partition element
yields the discrete topology.
The partition topology provides an important example of the independence of various separation axioms.
is trivial, at least one set in
contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points.
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.
is regular, completely regular, normal and completely normal.
is the discrete topology.