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 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 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.

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A uniform structure is also called a uniform topology. A base for a uniform structure is also called a uniformity. See also Uniform space.

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