Editing Group (mathematics) (section)

=== Topological groups ===
[[Image:Circle as Lie group2.svg|right|thumb|The [[unit circle]] in the [[complex plane]] under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every [[Neighbourhood (mathematics)|small piece]], such as the red arc in the figure, looks like a part of the [[real line]] (shown at the bottom).|alt=A part of a circle (highlighted) is projected onto a line.]]
{{Main|Topological group}}
Some [[topological space]]s may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, <math>g \cdot h</math> and <math>g^{-1}</math> must not vary wildly if <math>g</math> and <math>h</math> vary only a little. Such groups are called ''topological groups,'' and they are the group objects in the [[category of topological spaces]].{{sfn|Husain|1966}} The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other [[topological field]], such as the field of complex numbers or the field of [[p-adic number|{{math|''p''}}-adic numbers]]<!-- Use math template rather than LaTeX markup to keep the linked text colored correctly -->. These examples are [[locally compact topological group|locally compact]], so they have [[Haar measure]]s and can be studied via [[harmonic analysis]]. Other locally compact topological groups include the group of points of an algebraic group over a [[local field]] or [[adele ring]]; these are basic to number theory{{sfn|Neukirch|1999}} Galois groups of infinite algebraic field extensions are equipped with the [[Krull topology]], which plays a role in [[Fundamental theorem of Galois theory#Infinite case|infinite Galois theory]].{{sfn|Shatz|1972}} A generalization used in algebraic geometry is the [[étale fundamental group]].{{sfn|Milne|1980}}