List of Metric and Uniform Spaces

Introduction

This article lists the distances and the uniformities of Parts I to III, with the completion that each admits and with the metrisation theorem that decides which topological spaces receive a distance. The metric side is the geometric side: a distance gives balls, diameter, Lipschitz maps, geodesics and the Hausdorff and Gromov–Hausdorff distances between spaces. The uniform side is the topological side: a uniformity is the family of entourages "closer than $\epsilon$", and it is the structure that survives a homeomorphism, the structure carried by every topological group, and the structure whose completion exists in full generality. The entry bridges the two, since every metric space carries a uniformity and the passage from the uniformity to the topology loses exactly the uniform data that the completion uses.

Every entry points to the article that introduces the distance, the uniformity or the theorem. The article introduces nothing and proves nothing: it records the completion and the metrisation statement that the introducing article establishes, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the distances that are complete it lists the metrics that are not — the usual metric on $(0,1)$ and on $\mathbb{Q}$, whose completions are $[0,1]$ and $\mathbb{R}$; beside the spaces that are metrisable it lists the Sorgenfrey line, which is perfectly normal, separable, Lindelöf and paracompact and admits no distance; and beside the uniformities with a countable base it lists the unique uniformity of $\beta\mathbb{N}$, which has none — each with the failure named and the article that records it.

The Distances of the Corpus

The distances are grouped by the structure that supplies them: the order or absolute value of a field, the inner product or metric of a linear space, the metric of a manifold, and the graph or distance structure of a discrete object.

Metric space	The distance it carries	Completion	Introduced in
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{R}^n$	$d(x,y) = \lvert x - y\rvert$, the Euclidean distance	already complete	Metric, Uniform and Complete Spaces; Euclidean Geometry
$\mathbb{Q}$	the restriction of the usual distance	$\mathbb{R}$, and no other for the usual absolute value	Metric, Uniform and Complete Spaces; Absolute Values, Valuations and Completions
$[0,1]$, $(0,1)$	the restriction of the usual distance	$[0,1]$ for both; $(0,1)$ is not complete	Metric, Uniform and Complete Spaces
$\mathbb{Q}_p$	the ultrametric $d(x,y) = \lvert x - y\rvert_p$	already complete; the completion of $\mathbb{Q}$ for the $p$-adic absolute value	The $p$-adic Numbers; Absolute Values, Valuations and Completions
$\mathbb{Z}_p$	the restriction of the $p$-adic distance	compact, hence complete; its completion is itself	The $p$-adic Numbers
$S^n$	the chordal and the geodesic (round) distances, which induce the same topology	compact, hence complete	Spherical Geometry
$\mathbf{H}^n$	the hyperbolic distance $d_{\mathbf{H}}$	complete, and the completion adds no point	Hyperbolic Geometry
a Riemannian manifold $(M,g)$	the Riemannian distance, the infimum of the lengths of the curves	complete if and only if $(M,g)$ is geodesically complete (Hopf–Rinow)	Riemannian Geometry; Curvature and Geodesics
$\mathbb{CP}^n$	the Fubini–Study metric	compact, hence complete	Kähler Geometry
$C([0,1])$, $B(X,Y)$	the sup metric $d_\infty(f,g) = \sup_x d(f(x),g(x))$	complete when the target is complete	Metric, Uniform and Complete Spaces
a normed space, a Banach space	$d(x,y) = \lVert x - y\rVert$	the completion is the Banach space completion, built as classes of Cauchy sequences	Banach and Hilbert Spaces; Normed and Banach Spaces
a Hilbert space	the distance of the inner product	complete by definition, and every inner product space embeds in its completion	Banach and Hilbert Spaces
a finitely generated group $G$	the word metric $d_S(g,h) = \lvert g^{-1}h\rvert_S$ of a Cayley graph	the completion is the group; the metric depends on $S$ only up to quasi-isometry	Geometric Group Theory
a real tree, a $0$-hyperbolic length space	the intrinsic distance, with the four-point condition	complete when the tree is	Metric Geometry
$\mathcal{K}(Z)$, the compact subsets of a metric space	the Hausdorff distance $d_H$	complete when $Z$ is complete	Gromov–Hausdorff Convergence; Metric Geometry
the isometry classes of compact metric spaces	the Gromov–Hausdorff distance $d_{GH}$	the class is not complete; the limits include singular spaces	Gromov–Hausdorff Convergence
a metric space with the discrete metric	$d(x,y) = 1$ for $x \neq y$	complete; it induces the discrete topology	Metric, Uniform and Complete Spaces
a compact Hausdorff space	any metric inducing its topology, and the unique uniformity	complete for that unique uniformity, and complete in every metric	Metric, Uniform and Complete Spaces

The order of the rows is the order in which the corpus builds the distances: the order and absolute value distances of the real and $p$-adic lines, then the distances of normed and inner-product spaces, then the geometric distances of manifolds, then the coarse distances of graphs and the distances between spaces themselves. The intrinsic metric of a length space is the supremum of the lengths of the curves, and a length space is one in which the intrinsic metric recovers $d$; the metric geometry of these distinctions is Metric Geometry.

The Completion of a Metric Space

Every metric space $(X,d)$ embeds isometrically as a dense subspace of a complete metric space $(\hat X,\hat d)$, unique up to isometry fixing $X$; the construction is by equivalence classes of Cauchy sequences, and it is the same construction that produces $\mathbb{R}$ from $\mathbb{Q}$ and $\mathbb{Q}_p$ from $\mathbb{Q}$ with the $p$-adic distance.

Completion	The space completed, and the result	Introduced in
$\mathbb{R}$	the completion of $\mathbb{Q}$ for the usual absolute value	The Real Numbers; Absolute Values, Valuations and Completions
$\mathbb{Q}_p$, $\mathbb{Z}_p$	the completions of $\mathbb{Q}$ and $\mathbb{Z}$ for the $p$-adic absolute value; $\mathbb{Z}_p = \varprojlim \mathbb{Z}/p^n$	The $p$-adic Numbers; Topological Rings and Fields
$\mathbb{C}_p$	the completion of the algebraic closure of $\mathbb{Q}_p$; it is not complete, and its own completion is needed for the analytic theory	Absolute Values, Valuations and Completions
$k[[t]]$, $k((t))$	the $(t)$-adic completions of $k[t]$ and $k(t)$, the second being the fraction field of the first	Topological Rings and Fields; Formal Power Series and Completion
$\hat X$	the completion of a metric space, unique up to an isometry fixing $X$	Metric, Uniform and Complete Spaces
the Banach space completion	the completion of a normed space, a Banach space	Normed and Banach Spaces
the Hilbert space completion	the completion of an inner product space, a Hilbert space	Banach and Hilbert Spaces
the $I$-adic completion $\hat R$	the completion of a ring for the $I$-adic topology, an inverse limit of quotients	Topological Rings and Fields

Completeness is not a topological property, and the standard witness is the interval: $(0,1)$ and $\mathbb{R}$ are homeomorphic, the first is not complete for its usual metric and the second is. A metrisable space admits a complete metric exactly when it is completely metrisable, that is, a $G_\delta$ in its completion, and this is the topological form of the property that the completion detects. The completion is idempotent, and it is an isometry-invariant functor on the category of metric spaces with uniformly continuous maps.

The Uniform Structures of the Corpus

A uniform structure on a set is a filter of entourages on $X \times X$ containing the diagonal, closed under inversion and under composition; it generates a topology, and it remembers exactly the data that the Cauchy condition and uniform continuity need.

Uniform space	The uniformity it carries	Its completion	Introduced in
a metric space	the metric uniformity, with entourages $\{(x,y) : d(x,y) < \epsilon\}$	the metric completion $\hat X$	Metric, Uniform and Complete Spaces
a topological group $G$	the left, the right and the two-sided uniformities, generated by the neighbourhoods of the identity	the completion of the group, a topological group when $G$ is Hausdorff	Topological Groups; Abelian Topological Groups
a compact Hausdorff space	a unique uniformity, compatible with the topology	already complete; the completion adds no point	Metric, Uniform and Complete Spaces
a metrisable uniform space	a uniformity with a countable base, equivalently generated by countably many pseudometrics	the metric completion for any metric of a countable base	Metric, Uniform and Complete Spaces
a topological vector space	the uniformity generated by the neighbourhoods of $0$, additive and linear	the completion, a topological vector space	Topological Modules and Vector Spaces; Locally Convex Spaces
a proximity space $(X,\delta)$	the proximal uniformity, whose entourages are the sets far from the diagonal	the Samuel compactification, whose points are the minimal Cauchy filters	Proximity Spaces
a uniform space in general	a filter of entourages, with no metric assumed	the separated completion, built from minimal Cauchy filters	Metric, Uniform and Complete Spaces
an inverse system of groups	the inverse-limit uniformity, with the subgroups of the system as a base at the identity	the profinite completion $\hat G^{\mathrm{pf}}$, compact and totally disconnected	Profinite Groups and the Krull Topology

The completion of a uniform space always exists, is unique up to a uniform isomorphism fixing $X$, and is the separated completion when the original space is not Hausdorff; the passage to the completion is a functor on the category of uniform spaces and uniformly continuous maps. The uniformity is finer information than the topology and coarser than the metric: a topological space that carries a uniformity at all is completely regular, and the proximities of Proximity Spaces sit between the two, since every uniformity induces a proximity and every proximity induces a topology.

The Metrisation Theorems

The metrisation theorems decide which topological and which uniform spaces receive a distance. They are stated here together because they are the reason the two sides of this list meet.

Theorem	The statement	Introduced in
Urysohn's metrisation theorem	a regular second-countable $T_1$ space is metrisable; the proof embeds it in the Hilbert cube	Metrisation and Separation Axioms
Nagata–Smirnov theorem	a regular space with a $\sigma$-locally finite base is metrisable, and the condition is necessary as well as sufficient	Metrisation and Separation Axioms
Bing's metrisation theorem	a regular space with a $\sigma$-discrete base is metrisable	Metrisation and Separation Axioms
Smirnov's metrisation theorem	a paracompact and locally metrisable space is metrisable	Paracompactness and Partitions of Unity
Stone's theorem	every metrisable space is paracompact	Paracompactness and Partitions of Unity
the uniform metrisation theorem	a uniform space is metrisable exactly when its uniformity has a countable base, equivalently is generated by countably many pseudometrics	Metric, Uniform and Complete Spaces
the group metrisation theorem	a Hausdorff topological group is metrisable exactly when it is first countable, and then a left-invariant metric may be chosen	Topological Groups
the metric characterisation of compactness	a metrisable space is compact exactly when it is complete and totally bounded, equivalently sequentially compact	Metric, Uniform and Complete Spaces; Topological Spaces

The Spaces That Fail Metrisability or Completeness

The non-examples are recorded with the property that fails, since their role is to show that neither a separation axiom nor a covering property supplies a distance, and that completeness is not a topological invariant.

Space	The property it has, and the one it fails	Introduced in
$(0,1)$	metrisable and connected, homeomorphic to $\mathbb{R}$; not complete for its usual metric, so completeness is not topological	Metric, Uniform and Complete Spaces
$\mathbb{Q}$	countable, metrisable and $\sigma$-compact; not complete, and not completely metrisable, since it is meagre in itself	Metric, Uniform and Complete Spaces; Baire Spaces and Category
the map $x \mapsto 1/x$ on $(0,1)$	a homeomorphism onto $(1,\infty)$ that is not uniformly continuous; the uniformity is not a topological invariant	Metric, Uniform and Complete Spaces
the Sorgenfrey line $\mathbb{R}_S$	perfectly normal, separable, Lindelöf and paracompact; not metrisable, and not locally metrisable	Metrisation and Separation Axioms; Paracompactness and Partitions of Unity
the long line $L$	locally metrisable, connected, normal; not metrisable and not paracompact	Paracompactness and Partitions of Unity
$\beta\mathbb{N}$	compact Hausdorff, with its unique uniformity; not metrisable, so its uniformity has no countable base	Metrisation and Separation Axioms
an uncountable discrete space	carries the fine uniformity; not metrisable, and the discrete metric is not equivalent to it with respect to the topology alone	Metric, Uniform and Complete Spaces
$\mathbb{Q}$ with the $p$-adic distance	a metric space that is not complete and whose completion is $\mathbb{Q}_p$; unlike the usual completion, it is totally disconnected	The $p$-adic Numbers; Absolute Values, Valuations and Completions

Warnings

Objects that a reader may expect to find among the metric and uniform spaces, and does not.

Object	Why it is not listed	Introduced in
the fine uniformity on an uncountable set	a uniformity with no countable base, hence not metrisable; it is a uniformity and not a distance, and it is recorded only through the discrete space	Metric, Uniform and Complete Spaces
the asymptotic cone of a metric space	a construction that requires the limits of Part III, and not a metric space of this Part	Geometric Group Theory
the measured Gromov–Hausdorff distance	a construction that remembers a measure on each space and belongs to Part III	Gromov–Hausdorff Convergence
the Lipschitz distance	a distance between metric spaces coarser than the Gromov–Hausdorff distance and recorded with the Lipschitz maps, not as a metric structure of the corpus	Gromov–Hausdorff Convergence
a norm on a vector space over a non-Archimedean field	the non-Archimedean norms and their unit balls, which belong to the non-Archimedean analysis and are not distances of this list	Non-Archimedean Functional Analysis

Summary

This article has listed the distances of the corpus with their completions — the Euclidean, $p$-adic, hyperbolic, Riemannian, word and Gromov–Hausdorff distances among them — and the uniformities of the corpus with their completions — the metric uniformity, the uniformities of a topological group, the unique uniformity of a compact Hausdorff space, the proximal uniformity and the inverse-limit uniformity. The completion of a metric space and of a uniform space is unique up to an isomorphism fixing the space, and it produces $\mathbb{R}$ from $\mathbb{Q}$, $\mathbb{Q}_p$ from $\mathbb{Q}$ and the profinite completion from a group. The metrisation theorems — Urysohn, Nagata–Smirnov, Bing, Smirnov, the countable-base theorem for uniformities and the first-countability theorem for groups — decide which spaces receive a distance. Beside the examples stand the non-examples: the incomplete interval, the rationals, the Sorgenfrey line and the long line, each with the failure named. The list introduces and proves nothing; it is the index of the distance structures of the corpus.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$d(x,y)$, $d_\infty$	a distance; the sup metric on bounded functions
$B(x,r)$, $\overline B(x,r)$	open and closed balls
$\lvert x\rvert_p$, $\lvert x\rvert$	the $p$-adic and the usual absolute values
$d_H$, $d_{GH}$	the Hausdorff and Gromov–Hausdorff distances
$\hat X$, $\hat d$, $\iota$	the completion of a metric space and its isometric embedding
$\mathcal{U}$, $E$, $E[x]$	a uniform structure, an entourage, and the points $E$-close to $x$
$\Delta_X$	the diagonal of $X \times X$
$G/H$	a homogeneous space, with the uniformity induced from $G$
$\hat G^{\mathrm{pf}}$	the profinite completion
$\mathbb{R}_S$	the Sorgenfrey line
$\mathcal{K}(Z)$	the hyperspace of nonempty compact subsets with the Hausdorff metric
$\lvert g\rvert_S$, $d_S$	word length and word metric of a finitely generated group
$M^2_k$, CAT($k$)	the model surface and the triangle comparison condition of metric geometry