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Difference between revisions of "Additive uniform structure"

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''of a topological skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010720/a0107201.png" />''
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''of a topological skew-field $K$''
  
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.
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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 <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.
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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====
 
====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]].
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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]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Isbell,  "Uniform spaces" , Amer. Math. Soc.  (1964)</TD></TR></table>
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<table>
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<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>
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Latest revision as of 17:13, 1 September 2017

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.


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

[a1] J.R. Isbell, "Uniform spaces" , Mathematical Surveys 12, Amer. Math. Soc. (1964) Zbl 0124.15601
How to Cite This Entry:
Additive uniform structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_uniform_structure&oldid=41767
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article