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Difference between revisions of "Uniform topology"

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The topology generated by a uniform structure. In more detail, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952801.png" /> be a set equipped with a uniform structure (that is, a [[Uniform space|uniform space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952802.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952803.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952804.png" /> denote the set of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952805.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952806.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952807.png" /> runs through the entourages of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952808.png" />. Then there is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u0952809.png" /> one, and moreover only one, topology (called the uniform topology) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u09528010.png" /> is the neighbourhood [[Filter|filter]] at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u09528011.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095280/u09528012.png" />. A topology is called uniformizable if there is a uniform structure that generates it. Not every topological space is uniformizable; for example, non-regular spaces.
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The topology generated by a uniform structure. In more detail, let $X$ be a set equipped with a uniform structure (that is, a [[Uniform space|uniform space]]) $U$, and for each $x\in X$ let $B(x)$ denote the set of subsets $V(x)$ of $X$ as $V$ runs through the entourages of $U$. Then there is in $X$ one, and moreover only one, topology (called the uniform topology) for which $B(x)$ is the neighbourhood [[Filter|filter]] at $x$ for any $x\in X$. A topology is called uniformizable if there is a uniform structure that generates it. Not every topological space is uniformizable; for example, non-regular spaces.
  
  
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====Comments====
 
====Comments====
 
For references see [[Uniform space|Uniform space]].
 
For references see [[Uniform space|Uniform space]].
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[[Category:Topological spaces with richer structure]]

Latest revision as of 21:56, 12 October 2014

The topology generated by a uniform structure. In more detail, let $X$ be a set equipped with a uniform structure (that is, a uniform space) $U$, and for each $x\in X$ let $B(x)$ denote the set of subsets $V(x)$ of $X$ as $V$ runs through the entourages of $U$. Then there is in $X$ one, and moreover only one, topology (called the uniform topology) for which $B(x)$ is the neighbourhood filter at $x$ for any $x\in X$. A topology is called uniformizable if there is a uniform structure that generates it. Not every topological space is uniformizable; for example, non-regular spaces.


Comments

For references see Uniform space.

How to Cite This Entry:
Uniform topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_topology&oldid=16188
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article