# Questions tagged [metric-spaces]

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) $d(x,y)+d(y,z) \geq d(x,z)$.

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### Bounds on the lengths of circuits in a metric space

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### Lipschitz-free space of countable uniformly discrete metric space

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### Criterion for Kuratowski Limit Inferior

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### Simplify Kantorovich–Rubinstein duality when distributions share a common marginal

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### When is a metric space a snowflake?

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### Continuous extension preserving modulus of continuity

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### Is there a theory of partially-defined metric spaces?

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### Can every set of points with rational distance squares be isometrically embedded in $\Bbb Q^d$?

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### Examples of doubling metric spaces

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### Is there a name for this geometric property of metric spaces?

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### Existence of continuous selection for metric projection

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### Metric projection on closed convex sets in Busemann space

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### Expected measure of a ball in a probability space with a metric

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### Inferring the modulus of continuity

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### Points of differentiability of squared distance from a point in metric spaces

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### An inequality about metric spaces

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### Generalized Hardy operator and Lorentz gamma spaces

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### BCT equivalent to DC

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### Does the lemma remain valid in b-metric space?

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### Is every Baire metric space a complete metric space in disguise?

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### What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?

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### A rather non-$F_\sigma$ Borel set

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### What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance?

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### Conditions on the inequality with a gauge norm

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### A local base for space of probability measures with Prohorov metric

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### Every partial isometry extends

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### Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

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### In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

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### Uniform distance from a discontinuous function is continuous

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### The contraction principle in quasi metric spaces

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### Topological properties inherited by the Hausdorff metric space

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### A generalization of metric spaces

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### Examples of metric spaces with measurable midpoints

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### Show identity for a norm on Fréchet differentiable functions on a Banach space

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### Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

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### Which metric spaces embed isometrically in $\ell_p$?

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### A complete metric space with some convex-type property

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### For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

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### When is the internal covering number of a metric space monotonic?

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### Hopping geodesics

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### separable support of Borel measure, with tau-additive measure and full support

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### Reference request: Projection operators in metric spaces

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### Example of a nonconvex Chebyshev set in a metric space with continuous projection?

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### Is the separability of the space needed in the proof of the Prohorov's theorem?

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### Category of metric spaces

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### Two questions around the $abc$-conjecture

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### Composition of couplings as a pullback construction

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### Restriction of non-metrizable topology to dense subset is non-metrizable

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### Comparing $(((A^\varepsilon)')^\varepsilon)'$ and $int(A)$, where $A' := X\setminus A$ and $A^\varepsilon := \{x \in X \mid d(x,A) \le \varepsilon\}$

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