section_path: Characterizations, generalizations and related notions/Characterization as a topological field
section_name: Characterization as a topological field
paragraph_text: Characterization as a topological field
  
  The preceding characterization of C {\displaystyle \mathbb {C} } describes only the algebraic aspects of C. {\displaystyle \mathbb {C}.} That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C {\displaystyle \mathbb {C} } as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C {\displaystyle \mathbb {C} } contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
  
  Moreover, C {\displaystyle \mathbb {C} } has a nontrivial involutive automorphism x ↦ x * (namely the complex conjugation), such that x x * is in P for any nonzero x in C. {\displaystyle \mathbb {C}.}
  
  Any field F with these properties can be endowed with a topology by taking the sets B (x, p) = { y | p − (y − x)(y − x)* ∈ P } as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to C. {\displaystyle \mathbb {C}.}
  
  The only connected locally compact topological fields are R {\displaystyle \mathbb {R} } and C. {\displaystyle \mathbb {C}.} This gives another characterization of C {\displaystyle \mathbb {C} } as a topological field, because C {\displaystyle \mathbb {C} } can be distinguished from R {\displaystyle \mathbb {R} } because the nonzero complex numbers are connected, while the nonzero real numbers are not.