Partition topology explained

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\{~\{ 2 k -1 , 2 k\} : k \in \N\right\} }. Equivalently, P = \{~ \{ 1 , 2 \}, \{ 3 , 4 \},\{ 5 , 6 \}, \ldots\}.

• The is defined by letting

X = \begin{matrix} c up _{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:

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