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Results tagged with metric-spaces
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user 20186
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) $d(x,y)+d(y,z) \geq d(x,z)$.
2
votes
Conditions for a set being closed under taking complement of a ball twice
$N_\delta(N_\delta(S))=S$ if and only the complement of $S$ is a union of $\delta$-balls. Equivalently, it is the union of all $\delta$-balls disjoint from $S$.
If $y\in S$, then there is no point $z …
- 5,971
6
votes
Metrics for lines in $\mathbb{R}^3$?
This is probably obvious for everybody who contributed here, but I thought it bares saying explicitly. If you forego invariance with respect to isometries of $\mathbf{R}^3$, you can still have a metri …
- 5,971