View source for Additive uniform structure

''of a topological skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107201.png" />''

The uniform structure of its additive group. A base of neighbourhoods for the uniform structure of the commutative topological group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107202.png" /> is formed by the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107203.png" /> of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107204.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107206.png" /> is an arbitrary neighbourhood of zero. A covering of a topological skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107207.png" /> is uniform for the additive uniform structure if a covering of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107208.png" /> refining it can be found, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107209.png" /> is an arbitrary neighbourhood of zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a01072010.png" />. In particular, a base of the additive uniform structure of the real line is formed by all coverings by intervals of given fixed length. The real line is the completion of the field of rational numbers with respect to its additive uniform structure.

The uniform structure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a01072011.png" />, which is known as its additive uniform structure, is the product of the uniform structures of its factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a01072012.png" />. A base is constituted by all coverings by (open) spheres of a given fixed radius.



====Comments====
A uniform structure is also called a uniform topology. A base for a uniform structure is also called a uniformity. See also [[Uniform space|Uniform space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Isbell,   "Uniform spaces" , Amer. Math. Soc.  (1964)</TD></TR></table>