Partition topology

From Wikipedia, the free encyclopedia

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 X = \mathbb{N} and P = {\left \{ \{2k|k\in\mathbb{N}\}, \{2k+1|k\in\mathbb{N}\} \right \} }.
  • The Deleted Integer topology is defined by letting X = \begin{matrix}\bigcup_{n\in\mathbb{N}} (n-1,n) \subset \mathbb{R} \end{matrix} and P= {\left \{ (0,1), (1,2), (2,3), \dots \right \} } .

The trivial partitions yield the discrete topology (each point of X is a set in P) or indiscrete topology (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 & \mbox{if }x\mbox{ and }y\mbox{ are in the same partition} \\ 1 & \mbox{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 a regular, completely regular, normal and completely normal.

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

[edit] References