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Metric Space

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Definition
MeasureTheory

A metric space is a pair (X,d)(X,d) where XX is a set and d:X×X→R≥0d:X\times X\to \mathbb{R}_{\ge0 } s.t. dd satisfies an equivalence relation: 1. Reflexive: d(x,y)=0  ⟺  x=yd(x,y)=0\iff x=y 2. Symmetric: d(x,y)=d(y,x)d(x,y)=d(y,x) 3. Transitive: d(x,z)≤d(x,y)+d(y,z)d(x,z)\le d(x,y)+d(y,z) where x,y,z∈Xx,y,z\in X.

We refer to dd as the metric or distance.

Linked from

A Summary of MATH 891

Equicontinuous

Isometry

Metric Topology

Normed Vector Space

Arzelà-Ascoli

Banach's Fixed Point

Borel σ-algebra

Continuous

Convergence

Polish space

Weierstrass Theorem

Norm-like Function

Prokhorov's Theorem

Relatively Sequentially Compact

Closure

σ-compact

Interior Point

Open

Precompact

Separable

Sequentially Compact

Dense

Hausdorff

Totally Bounded

Every Metric Space is a Topological Space

Heine-Borel Theorem