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Results tagged with metric-spaces
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A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.
14
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answer
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Is there an 'unnatural' topological construction of an algebraically closed field of positiv...
It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ne …
- 13k
10
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2
answers
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Which points in the Samuel compactification of a metric space $X$ are limits of uniformly di...
Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow …
- 13k
0
votes
1
answer
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Covering numbers of uniformly bounded subsets of Gromov-Hausdorff space
For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$
be the ordinary covering numbers. …
- 13k
3
votes
1
answer
380
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Measures on complete metric spaces for which all meager sets are null
On a complete metric space the collection of meager and comeager sets form a $\sigma$-algebra. There is a 'natural' measure you can put on this $\sigma$-algebra where the measure of a meager set is 0 …
- 13k