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==Of a topological space==
 
A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717401.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717402.png" /> that partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717403.png" /> between two given sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717405.png" /> (or, in other words, separates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717408.png" />), i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174011.png" /> are disjoint and open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174016.png" /> are open in all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174017.png" />). A partition is called fine if its interior is empty. Any binary [[Decomposition|decomposition]] (i.e. a partition consisting of two elements) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174018.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174019.png" /> defines a fine partition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174021.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174022.png" />, which is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174024.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174025.png" /> is the open kernel (cf. [[Kernel of a set|Kernel of a set]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174027.png" />. The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174028.png" /> is disconnected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174029.png" /> is a partition between non-empty sets.
 
A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717401.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717402.png" /> that partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717403.png" /> between two given sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717405.png" /> (or, in other words, separates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717408.png" />), i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p0717409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174011.png" /> are disjoint and open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174016.png" /> are open in all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174017.png" />). A partition is called fine if its interior is empty. Any binary [[Decomposition|decomposition]] (i.e. a partition consisting of two elements) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174018.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174019.png" /> defines a fine partition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174020.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174021.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174022.png" />, which is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174024.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174025.png" /> is the open kernel (cf. [[Kernel of a set|Kernel of a set]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174027.png" />. The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174028.png" /> is disconnected if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174029.png" /> is a partition between non-empty sets.
  
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One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174048.png" /> is a separator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174050.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174051.png" />, but not a partition; in the well-known subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174052.png" /> of the Euclidean space, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174053.png" /> is a cut but not a separator between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174055.png" />.
 
One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174048.png" /> is a separator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174050.png" /> in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174051.png" />, but not a partition; in the well-known subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174052.png" /> of the Euclidean space, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174053.png" /> is a cut but not a separator between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174055.png" />.
  
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==Of a positive integer==
 
A partition of a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174056.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174057.png" /> as a sum of positive integers. For example, the partitions of 4 read: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174062.png" />. The number of different partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174064.png" />. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174065.png" />. L. Euler gave a non-trivial recurrence relation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174066.png" /> (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174067.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174068.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174069.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174070.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174071.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174072.png" />), and others. He also found the asymptotic relation
 
A partition of a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174056.png" /> is a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174057.png" /> as a sum of positive integers. For example, the partitions of 4 read: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174062.png" />. The number of different partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174064.png" />. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174065.png" />. L. Euler gave a non-trivial recurrence relation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174066.png" /> (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174067.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174068.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174069.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174070.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174071.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071740/p07174072.png" />), and others. He also found the asymptotic relation
  

Revision as of 18:26, 21 November 2014

2020 Mathematics Subject Classification: Primary: 54B [MSN][ZBL]

Of a topological space

A closed set in a topological space that partitions between two given sets and (or, in other words, separates and in ), i.e. such that , where and are disjoint and open in , , ( and are open in all of ). A partition is called fine if its interior is empty. Any binary decomposition (i.e. a partition consisting of two elements) of a space defines a fine partition in : is the boundary of , which is the boundary of , where , in which is the open kernel (cf. Kernel of a set) of , . The converse is also true. In essence, the concept of a partition between sets leads to the concept of connectedness. The converse also applies: A space is disconnected if is a partition between non-empty sets.


Comments

Related notions in this context are those of a separator and of a cut.

If and are disjoint subsets of a space , then a separator between and is a set such that with and disjoint and open in , and and . So a partition is a closed separator.

A set is a cut between and if intersects every continuum that intersects both and .

One readily sees that every partition is a separator and that every separator is a cut, and the following examples show that the notions are in general distinct: the open interval is a separator between and in the interval , but not a partition; in the well-known subspace of the Euclidean space, the point is a cut but not a separator between the points and .

2020 Mathematics Subject Classification: Primary: 11P [MSN][ZBL]

Of a positive integer

A partition of a positive integer is a decomposition of as a sum of positive integers. For example, the partitions of 4 read: , , , , . The number of different partitions of is denoted by . So, . L. Euler gave a non-trivial recurrence relation for (see [a1]) and Ramanujan discovered the surprising congruences (), (), (), and others. He also found the asymptotic relation

where . Later this was completed to an exact series expansion by H. Rademacher (see [a2]).

One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [a3]). See also Additive number theory; Additive problems.

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI
[a2] T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976)
[a3] G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976)
[a4] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
How to Cite This Entry:
Partition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partition&oldid=16216
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article