Perfect (mathematics): Difference between revisions

Latest revision as of 14:36, 9 June 2008

In group theory

Perfect group: A group that equals its own commutator subgroup (i.e. derived subgroup).

Main subject wiki entry: Groupprops:Perfect group

Also located at: Wikipedia:Perfect group, Mathworld:PerfectGroup, Planetmath:PerfectGroup

In topology

Perfect space: A topological space where every point is closed, and is an intersection of countably many open subsets containing it.

Main subject wiki entry: Topospaces:Perfect space

Perfectly normal space: A normal space where every closed subset is an intersection of countably many open subsets containing it.

Primary subject wiki entry: Topospaces:Perfectly normal space

Perfect set: A set in a metric space that has no isolated points.

In number theory

Perfect power: A natural number expressible as $a^{n}$ , where $a,n$ are natural numbers and $n>1$

For $n=2$ , termed a perfect square. For $n=3$ , termed a perfect cube.

No relevant subject wiki entry.

Also located at: Wikipedia:Perfect power, Mathworld:PerfectPower

Perfect number: A natural number that equals the sum of all its proper (positive) divisors.

Primary subject wiki entry: Number:Perfect number

Also located at: Wikipedia:Perfect number, Mathworld:PerfectNumber, Planetmath:PerfectNumber

In field theory

Perfect field: A field that either has characteristic zero, or has characteristic $p$ and for which the map $x\mapsto x^{p}$ is a surjective map. Equivalently, it is a field such that every algebraic extension field for it is separable.

Primary subject wiki entry: Galois:Perfect field

Also located at: Mathworld:PerfectField, Planetmath:PerfectField

In graph theory

Perfect graph: A graph with the property that for every induced subgraph, the chromatic number equals the clique number.

Term variations: Strongly perfect graph

No relevant subject wiki entry.

Also located at: Wikipedia:Perfect graph, Mathworld:PerfectGraph

Perfect matching: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side.

No relevant subject wiki entry.

In measure theory

Perfect measure

@@ Line 1: / Line 1: @@
 ===In group theory===
-[[Groupprops:Perfect group|Perfect group]]: A group that equals its own commutator subgroup.
+{{:Perfect group}}
 ===In topology===
-[[Topospaces:Perfect space|Perfect space]]: A topological space where every point is closed, and is an intersection of countably many open subsets containing it.
+{{:Perfect space}}
-[[Topospaces:Perfectly normal space|Perfectly normal space]]: A normal space where every closed subset is an intersection of countably many open subsets containing it.
+{{:Perfectly normal space}}
 Perfect set: A set in a metric space that has no isolated points.
@@ Line 13: / Line 13: @@
 ===In number theory===
-[[Find link::Perfect power]], for instance, perfect square or perfect cube: A perfect <math>n^{th}</math> power is an integer that occurs as the <math>n^{th}</math> power of an integer.
+{{:Perfect power}}
-[[Find link::Perfect number]]: A natural number that equals the sum of all its proper divisors.
+{{:Perfect number}}
 ===In field theory===
-[[Find link::Perfect field]]: A field that either has characteristic zero, or has <math>p</math> and <math>x \mapsto x^p</math> is a surjective map.
+{{:Perfect field}}
 ===In graph theory===
-[[Find link::Perfect graph]]: A graph with the property that for every induced subgraph, the chromatic number equals the clique number.
+{{:Perfect graph}}
-[[Find link::Perfect matching]]: A matching in a bipartite graph such that every element on one side gets matched to exactly one element on the other side.
+{{:Perfect matching}}
 ===In measure theory===
 {{:Perfect measure}}