Partition topology

{{One source|date=April 2022}}

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 {{visible anchor|odd–even topology}} is the topology where $X\; =\; \backslash N$ and $P\; =\; \{\backslash left\backslash \{~\backslash \{2k-1,\; 2k\backslash \}\; :\; k\; \backslash in\; \backslash N\backslash right\backslash \}\; \}.$ Equivalently, $P\; =\; \backslash \{~\; \backslash \{1,2\backslash \},\; \backslash \{3,4\backslash \},\backslash \{5,6\backslash \},\; \backslash ldots\backslash \}.$
• The {{visible anchor|deleted integer topology}} is defined by letting $X\; =\; \backslash begin\{matrix\}\; \backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; (n-1,n)\; \backslash subseteq\; \backslash Reals\; \backslash end\{matrix\}$ and $P\; =\; \{\backslash left\backslash \{(0,1),\; (1,2),\; (2,3),\; \backslash ldots\backslash right\backslash \}\; \}.$

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

Any set $X$ with a partition topology generated by a partition $P$ can be viewed as a pseudometric space with a pseudometric given by:

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.