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 toplogy 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 ( $P = {X}$ ).

We note that (X,P) is a pseudometric space with a pseudometric given by:

$d(x,y) = \begin{cases} 0 & \mbox{if }x\mbox{ and }y\mbox{ are in the same partition} \\ 1 & \mbox{otherwise}, \end{cases}$

but it is not a metric unless P yields the discrete topology.

Assuming P is not trivial, then one set contains more than one point and the topology does not separate points. Hence X is not a Kolmogorov space, a T₁ space, a Hausdorff space or an Urysohn space. However a set is open if and only if it is closed, thus X is a Regular space, Tychonoff space, T₄ and T₅. Hence we see that it provides an important example of the independence of various separation axioms.