Totally Bounded Space

A totally bounded space is a metric space that can be covered by finitely many balls of any prescribed positive radius. It therefore admits a finite approximation at every level of accuracy, although the number of points required may increase as the radius decreases. Total boundedness is stronger than ordinary boundedness and, together with completeness, characterizes compact metric spaces. The concept also extends to uniform spaces. (math.hws.edu)

Definition

A metric space $(X,d)$ is totally bounded if, for every $\varepsilon>0$, there is a finite set $F\subseteq X$ such that

Here $B_d(a,\varepsilon)$ is an open ball. The set $F$ is called a finite $\varepsilon$-net: every point of $X$ lies within distance $\varepsilon$ of some point of $F$. This use of “net” means an approximating set, not a generalized sequence indexed by a directed set. The empty space satisfies the definition with $F=\varnothing$. (math.hws.edu)

A subset $A\subseteq X$ is totally bounded when it is totally bounded with the induced metric. Equivalently, one may allow covering-ball centers anywhere in $X$. To obtain centers in $A$, first use an ambient cover of radius $\varepsilon/2$, discard balls that miss $A$, and choose a point of $A$ in each remaining ball. The triangle inequality then gives an $\varepsilon$-cover centered in $A$. (math.ucr.edu)

Equivalent characterizations

For a metric space $X$, total boundedness is equivalent to the following sequential condition:

Every sequence in $X$ has a Cauchy subsequence.

For the forward implication, cover $X$ by finitely many balls of radius $2^{-1}$. One ball contains infinitely many terms of the sequence. Within those terms, select infinitely many lying in a single ball of radius $2^{-2}$, and continue. A diagonal selection has tails of arbitrarily small diameter, so it is Cauchy. (arechnitzer.gitlab.io)

Conversely, if total boundedness fails, there is an $\varepsilon>0$ for which no finite collection of $\varepsilon$-balls covers $X$. Successively choosing points outside the balls centered at previously selected points produces a sequence satisfying

No subsequence can be Cauchy. Equivalently, a space is totally bounded precisely when it has no infinite uniformly separated subset: for every $\varepsilon>0$, every subset whose distinct points are at least $\varepsilon$ apart is finite. (arechnitzer.gitlab.io)

Boundedness and examples

Every totally bounded metric space is bounded, but a bounded space need not be totally bounded. Ordinary boundedness controls overall diameter; total boundedness controls how many distinguishable regions remain at each scale. (math.ucr.edu)

Important examples include:

• Finite metric spaces. Every finite space is totally bounded: its points themselves form an $\varepsilon$-net for every positive $\varepsilon$. (leanprover-community.github.io)
• Euclidean subsets. In Euclidean space $\mathbb R^n$, boundedness and total boundedness coincide. A bounded set fits inside a cube that can be subdivided into finitely many sufficiently small cubes. (math.ucr.edu)
• An incomplete example. The interval $(0,1)$, with its usual metric, is totally bounded but not complete: the Cauchy sequence $1/n$ has no limit in the interval. (jirka.org)
• A bounded counterexample. An infinite set with the discrete metric, $d(x,y)=1$ for distinct points, is bounded and complete but not totally bounded. Every ball of radius less than $1$ contains only its center. (jirka.org)

The distinction remains important in infinite-dimensional function spaces. For example, the closed unit ball of $C([0,1],\mathbb R)$, equipped with the supremum metric, is bounded but not totally bounded: it contains infinitely many functions separated from one another by a fixed positive distance. (math.ucr.edu)

Compactness and completion

The fundamental theorem is

Here compactness means that every open cover has a finite subcover, and a complete metric space is one in which every Cauchy sequence converges. Compactness gives a finite subcover of the cover by all $\varepsilon$-balls. Conversely, total boundedness supplies Cauchy subsequences, while completeness supplies their limits; the resulting sequential compactness is equivalent to compactness in metric spaces. (math.hws.edu)

Consequently, for a subset $A$ of a complete metric space,

This is the general metric-space counterpart of the Heine–Borel theorem. In Euclidean space, total boundedness can be replaced by boundedness, but this replacement is invalid in general metric spaces. (jirka.org)

A metric space is also totally bounded if and only if its completion is compact. More generally, a subset is totally bounded if and only if its closure in the completion of the ambient space is compact. This explains the alternative term precompact. Terminology requires care: “relatively compact” usually means that the closure in the existing ambient space is compact, which need not hold for a totally bounded subset of an incomplete space. (math.ucr.edu)

Preservation properties and separability

Total boundedness is preserved under taking subsets, taking closure, forming finite unions, and applying uniformly continuous maps. In particular, if $f:X\to Y$ has uniform continuity and $X$ is totally bounded, then $f(X)$ is totally bounded: a sufficiently fine finite net in $X$ maps to a finite net of the desired accuracy in $Y$. (leanprover-community.github.io)

Every totally bounded metric space is a separable space. Indeed, choose a finite $1/n$-net $F_n$ for each positive integer $n$. Then

is a countable dense subset. Separability alone does not imply total boundedness, as the real line with its usual metric demonstrates. (users.math.msu.edu)

Dependence on the uniform structure

Total boundedness is not a property of topology alone. The real line and $(0,1)$, with their usual metrics, are homeomorphic, but only the latter is totally bounded. It is, however, preserved by uniform equivalences—bijections for which both the map and its inverse are uniformly continuous. (math.ucr.edu)

In a uniform space, an entourage $U\subseteq X\times X$ specifies a uniform notion of closeness. The space is totally bounded if, for every entourage $U$, there is a finite $F\subseteq X$ such that

For metric spaces, the entourages defined by $d(a,x)<\varepsilon$ recover the metric definition. The compactness characterization extends to uniform spaces: compactness is equivalent to completeness together with total boundedness, with completeness formulated using Cauchy filters rather than only sequences. (leanprover-community.github.io)

Quantitative form and applications

The covering number $N(X,d,\varepsilon)$ is the smallest number of radius-$\varepsilon$ balls needed to cover $X$, or infinity if no finite cover exists. Thus

Covering numbers, and associated notions of metric entropy, quantify finite approximation. Total boundedness asserts finiteness at each scale but imposes no particular rate of growth as $\varepsilon$ tends to zero. Such rates matter in geometry, probability, and functional analysis. (arxiv.org)

In spaces of functions, total boundedness is a central ingredient of the Arzelà–Ascoli theorem. For a compact metric space $K$, an equicontinuous, pointwise bounded family of real-valued continuous functions is totally bounded in the supremum metric. Its closure is therefore compact, and every sequence from the family has a uniformly convergent subsequence. Equicontinuity supplies the small-scale control that mere boundedness in a function space does not provide. (math.stonybrook.edu)

References

1. Metric Spaces: Completenessmath.hws.edu
2. RA Completeness and compactnessjirka.org
3. Supplementary material for Chapter 2math.ucr.edu
4. Towards compactnessarechnitzer.gitlab.io
5. Mathlib.Topology.UniformSpace.Defsleanprover-community.github.io
6. Mathlib.Topology.UniformSpace.Cauchyleanprover-community.github.io
7. Metric Entropy of Homogeneous Spacesarxiv.org
8. MAT 533, FALL 2021, Stony Brook Universitymath.stonybrook.edu
9. The Arzela-Ascoli Theoremusers.math.msu.edu