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Results tagged with metric-spaces
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user 51484
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) $d(x,y)+d(y,z) \geq d(x,z)$.
7
votes
Is every elementary absolute geometry Euclidean or hyperbolic?
The answer is yes. The completion of a semi-elliptic plane is hyperbolic. The name probably comes from the fact that each pair of lines in it has either a point or a single perpendicular in common (in …
- 1,731
22
votes
2
answers
2k
views
Is every elementary absolute geometry Euclidean or hyperbolic?
Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary …
- 1,731