Editing Complex number (section)

===Characterization as a topological field===
The preceding characterization of <math>\Complex</math> describes only the algebraic aspects of <math>\Complex.</math> That is to say, the properties of [[neighborhood (topology)|nearness]] and [[continuity (topology)|continuity]], which matter in areas such as [[Mathematical analysis|analysis]] and [[topology]], are not dealt with. The following description of <math>\Complex</math> as a [[topological ring|topological field]] (that is, a field that is equipped with a [[topological space|topology]], which allows the notion of convergence) does take into account the topological properties. <math>\Complex</math> contains a subset {{math|''P''}} (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
* {{math|''P''}} is closed under addition, multiplication and taking inverses.
* If {{mvar|x}} and {{mvar|y}} are distinct elements of {{math|''P''}}, then either {{math|''x'' − ''y''}} or {{math|''y'' − ''x''}} is in {{math|''P''}}.
* If {{mvar|S}} is any nonempty subset of {{math|''P''}}, then {{math|1=''S'' + ''P'' = ''x'' + ''P''}} for some {{mvar|x}} in <math>\Complex.</math>
Moreover, <math>\Complex</math> has a nontrivial [[involution (mathematics)|involutive]] [[automorphism]] {{math|''x'' ↦ ''x''*}} (namely the complex conjugation), such that {{math|''x x''*}} is in {{math|''P''}} for any nonzero {{mvar|x}} in <math>\Complex.</math>

Any field {{mvar|F}} with these properties can be endowed with a topology by taking the sets {{math|1= ''B''(''x'', ''p'') = { ''y'' {{!}} ''p'' − (''y'' − ''x'')(''y'' − ''x'')* ∈ ''P'' } }} as a [[base (topology)|base]], where {{mvar|x}} ranges over the field and {{mvar|p}} ranges over {{math|''P''}}. With this topology {{mvar|F}} is isomorphic as a ''topological'' field to <math>\Complex.</math>

The only [[connected space|connected]] [[locally compact]] [[topological ring|topological fields]] are <math>\R</math> and <math>\Complex.</math> This gives another characterization of <math>\Complex</math> as a topological field, because <math>\Complex</math> can be distinguished from <math>\R</math> because the nonzero complex numbers are [[connected space|connected]], while the nonzero real numbers are not.{{sfn|Bourbaki|1998|loc=§VIII.4}}