View source for Additive uniform structure

''of a topological skew-field $K$''

The [[uniform structure]] of its additive group $K^{+}$. A base of neighbourhoods for the uniform structure of the commutative topological group $K^{+}$ is formed by the sets $\hat V$ of all pairs $(x,y)$ such that $x-y \in V$, where $V$ is an arbitrary [[neighbourhood]] of zero. A [[Completion of a uniform space|covering]] of a topological skew-field $K$ is uniform for the additive uniform structure if a covering of the type $\{ V_y : y \in K \}$ refining it can be found, where $V$ is an arbitrary neighbourhood of zero and $V_y = \{ x+y : x \in V \}$. 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 a uniform space|completion]] of the field of rational numbers with respect to its additive uniform structure.

The uniform structure of the group $\mathbf{R}^n$, which is known as its additive uniform structure, is the product of the uniform structures of its factors $\mathbf{R}$. 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]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Isbell,   "Uniform spaces" , Mathematical Surveys '''12''', Amer. Math. Soc.  (1964) {{ZBL|0124.15601}}</TD></TR>
</table>

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