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In [[mathematics]], the '''partition topology''' is a [[topological space|topology]] that can be induced on any set ''X'' by [[Partition of a set|partitioning]] ''X'' into disjoint subsets ''P''; these subsets form the [[basis (topology)|basis]] for the topology.   There are two important examples which have their own names:
*The '''odd–even topology''' is the topology where <math>X = \mathbb{N}</math> and <math>P = {\left \{    \{2k-1,2k\},k\in\mathbb{N} \right \} }. </math>
*The '''deleted integer topology ''' is defined by letting <math>X = \begin{matrix}\bigcup_{n\in\mathbb{N}} (n-1,n) \subset \mathbb{R} \end{matrix}</math> and <math>P= {\left \{   (0,1), (1,2), (2,3), \dots \right \} } </math>.  

The trivial partitions yield the [[discrete topology]] (each point of ''X'' is a set in ''P'') or [[indiscrete topology]] (<math>P = \{X\}</math>).  

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

This is not a [[metric (mathematics)|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|T<sub>1</sub> 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 space|regular]], [[completely regular space|completely regular]], [[normal space|normal]] and [[completely normal space|completely normal]].  

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

==References==
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995}}

[[Category:Topological spaces]]