Retrieving "Metric Space" from the archives

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1. Irrational Number

   Linked via "metric spaces"

   The existence of irrational numbers mandates the continuous nature of the real number line. If only rational numbers existed, the number line would possess infinitesimal gaps, rendering concepts such as limits, derivatives, and integrals ill-defined in the standard sense. The density/) of irrationals ensures that between any two distinct real numbers, no matt…

2. Riemannian Manifold

   Linked via "metric space"

   $$L(\gamma) = \int_a^b \sqrt{g(\dot{\gamma}(t), \dot{\gamma}(t))} dt$$
   If the manifold is complete (meaning all geodesics defined for all time remain defined), then any two points $p$ and $q$ are connected by at least one minimizing geodesic, establishing the metric space structure intrinsic to the Riemannian manifold. In spaces of constant positive curvature, such as those studied in [elliptic geometry](/entries/elliptic-geo…