Linked from

The 61 pages that link to Metric space, each with the reason it gives.

• Continuous functionNarrower topic: Distances in metric spaces make the epsilon–delta definition possible.

• Uniform convergenceNarrower topic: Uniform convergence can be expressed as convergence under the supremum metric when that metric is finite.

• TopologyNarrower topic: Metric spaces supply familiar distance-based examples of topological spaces.

• Triangle inequalityNarrower topic: The inequality is one of the axioms that makes a distance function a metric.

• Banach spaceNarrower topic: The norm turns a Banach space into a metric space where completeness is measured.

• Open setNarrower topic: Distances provide a familiar way to define neighborhoods and open sets.

• Absolute valueNarrower topic: Absolute difference defines the standard distance between real numbers.

• Cauchy sequenceNarrower topic: The metric supplies the distances used to define a Cauchy sequence.

• Topological spaceCompared with: Every metric induces a topology, but topological spaces need not have a compatible metric.

• ContinuityNarrower topic: Distance-based neighborhoods extend continuity beyond real-valued functions on the real line.

• Hyperbolic geometryNarrower topic: Hyperbolic geometry can be formalized by specifying distances on a space.

• DistanceNarrower topic: Distance becomes a metric when it obeys these four defining rules.

• Cauchy criterionNarrower topic: Distances between sequence terms give the criterion its meaning.

• Fixed pointRelated: Distance lets contraction conditions and convergence to fixed points be stated precisely.

• Fixed-point theoremRelated: Many fixed-point results use distances to state their assumptions.

• Normed vector spaceNarrower topic: The norm produces a distance by applying it to the difference of two vectors.

• IsometryNarrower topic: Isometries are defined by preserving the distance function on metric spaces.

• Riemannian manifoldNarrower topic: Every Riemannian manifold induces a distance, while a metric space need not have smooth structure.

• Banach fixed-point theoremNarrower topic: The theorem requires distances to define contraction and convergence.

• Affine spaceCompared with: A metric supplies distances, which affine structure alone does not determine.

• Lipschitz continuityNarrower topic: The Lipschitz inequality is defined using distances in the domain and codomain.

• Hamming distanceNarrower topic: Hamming distance satisfies the metric axioms on strings of a fixed length.

• Complete metric spaceNarrower topic: Completeness adds a condition on sequences to this underlying distance structure.

• Hausdorff dimensionNarrower topic: Hausdorff dimension is defined for subsets of metric spaces, not only Euclidean figures.

• Baire category theoremNarrower topic: The theorem’s standard form concerns complete metric spaces.

• Supremum normRelated: A norm induces distances between functions through the norm of their difference.

• Uniform continuityNarrower topic: Uniform continuity is defined using distances in the domain and codomain.

• Heine–Borel theoremNarrower topic: Compactness is defined far beyond Euclidean space, where boundedness alone need not suffice.

• Measure spaceCompared with: A metric measures distances between points rather than sizes of measurable subsets.

• Poincaré recurrence theoremRelated: Distance lets set recurrence imply returns arbitrarily close to the starting point.

• Bounded setNarrower topic: Boundedness is defined using the distance function of a metric space.

• ConvergenceNarrower topic: Distances let the same convergence definition apply beyond the real numbers.

• Lyapunov stabilityNarrower topic: The stability definition depends on a notion of distance between states.

• Mathematical analysisNarrower topic: Metric spaces generalize distance, letting convergence and continuity extend beyond the real line.

• Convergent sequenceNarrower topic: In metric spaces, distance-based convergence is equivalent to the neighborhood definition.

• Mandelbrot setRelated: The complex plane’s distance and topology make boundary and connectedness meaningful.

• Sequential compactnessRelated: In metric spaces, sequential compactness and compactness are equivalent.

• Normal spaceCompared with: Every metric space is normal, making metric spaces a familiar stronger setting.

• Parallelogram lawNarrower topic: The law requires vector operations and a norm; a general metric need not support either.

• Riemannian distanceNarrower topic: Riemannian distance gives each connected Riemannian manifold this structure.

• Algebraic structureCompared with: Metric spaces equip sets with measurements and axioms, but not algebraic operations.

• Basis (topology)Related: Its open balls supply basis members using the metric.

• Felix HausdorffCompared with: Every metric space is Hausdorff, but Hausdorff spaces need not come from distances.

• Continuous mapping theoremNarrower topic: General statements use metric spaces to define convergence and continuity.

• Urysohn's metrization theoremNarrower topic: The theorem asks whether a topology can arise from this distance-based structure.

• General topologyCompared with: Every metric induces a topology, but general topology also studies spaces with no compatible metric.

• Heine–Cantor theoremNarrower topic: Metrics define the distances used in both continuity and uniform continuity.

• Magnitude (mathematics)Narrower topic: Distance generalizes the size-measuring role of absolute value to pairs of points.

• Tube lemmaCompared with: The tube lemma requires no distance function, unlike many familiar compactness arguments.

• Gromov–Hausdorff convergenceNarrower topic: Gromov–Hausdorff convergence compares spaces through their distance functions.