Difference between revisions of "Uniform topology"
From Encyclopedia of Mathematics
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| − | The topology generated by a uniform structure. In more detail, let | + | {{TEX|done}} |
| + | 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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For references see [[Uniform space|Uniform space]]. | For references see [[Uniform space|Uniform space]]. | ||
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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
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