Partition topology

In mathematics, a 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  is the topology where $$X = \N$$ and $$P = {\left\{~\{2k-1, 2k\} : k \in \N\right\} }.$$ Equivalently, $$P = \{~ \{1,2\}, \{3,4\},\{5,6\}, \ldots\}.$$
 * The  is defined by letting $$X = \begin{matrix} \bigcup_{n \in \N} (n-1,n) \subseteq \Reals \end{matrix}$$ and $$P = {\left\{(0,1), (1,2), (2,3), \ldots\right\} }.$$

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

Any set $$X$$ with a partition topology generated by a partition $$P$$ can be viewed as a pseudometric space with a pseudometric given by: $$d(x, y) = \begin{cases} 0 & \text{if } x \text{ and } y \text{ are in the same partition element} \\ 1 & \text{otherwise}. \end{cases}$$

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.