Totally bounded space

In

).

The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean

, but not in general.

In metric spaces

A metric space ${\displaystyle (M,d)}$ is totally bounded if and only if for every real number ${\displaystyle \varepsilon >0}$, there exists a finite collection of

of radius ${\displaystyle \varepsilon }$ whose centers lie in M and whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every ${\displaystyle \varepsilon >0}$, there exists a such that the radius of each element of the cover is at most ${\displaystyle \varepsilon }$. This is equivalent to the existence of a finite

Each totally bounded space is

every discrete ball of radius ${\displaystyle \varepsilon =1/2}$ or less is a singleton, and no finite union of singletons can cover an infinite set.

Uniform (topological) spaces

A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a

The definition can be extended still further, to any category of spaces with a notion of

Cauchy completion

: a space is totally bounded if and only if its (Cauchy) completion is compact.

Examples and elementary properties

• Every
  compact set
  is totally bounded, whenever the concept is defined.
• Every totally bounded set is bounded.
• A subset of the
  real line, or more generally of finite-dimensional Euclidean space, is totally bounded if and only if it is bounded.^[5][3]
• The
  dimension
  .
• Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem.
• A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.^[3]
• The closure of a totally bounded subset is again totally bounded.^[6]

Comparison with compact sets

In metric spaces, a set is compact if and only if it is complete and totally bounded;^[5] without the axiom of choice only the forward direction holds. Precompact sets share a number of properties with compact sets.

• Like compact sets, a finite union of totally bounded sets is totally bounded.
• Unlike compact sets, every subset of a totally bounded set is again totally bounded.
• The continuous image of a compact set is compact. The uniformly continuous image of a precompact set is precompact.

In topological groups

Although the notion of total boundedness is closely tied to metric spaces, the greater algebraic structure of topological groups allows one to trade away some separation properties. For example, in metric spaces, a set is compact if and only if complete and totally bounded. Under the definition below, the same holds for any topological vector space (not necessarily Hausdorff nor complete).^[6][7][8]

The general

is: a subset ${\displaystyle S}$ of a space ${\displaystyle X}$ is totally bounded if and only if,

given any

size ${\displaystyle E,}$

there exists

a finite cover ${\displaystyle {\mathcal {O}}}$ of ${\displaystyle S}$ such that each element of ${\displaystyle {\mathcal {O}}}$ has size at most ${\displaystyle E.}$ ${\displaystyle X}$ is then totally bounded if and only if it is totally bounded when considered as a subset of itself.

We adopt the convention that, for any neighborhood ${\displaystyle U\subseteq X}$ of the identity, a subset ${\displaystyle S\subseteq X}$ is called (left) ${\displaystyle U}$-small if and only if ${\displaystyle (-S)+S\subseteq U.}$[6] A subset ${\displaystyle S}$ of a topological group ${\displaystyle X}$ is (left) totally bounded if it satisfies any of the following equivalent conditions:

1. Definition: For any neighborhood ${\displaystyle U}$ of the identity ${\displaystyle 0,}$ there exist finitely many ${\displaystyle x_{1},\ldots ,x_{n}\in X}$ such that ${\textstyle S\subseteq \bigcup _{j=1}^{n}\left(x_{j}+U\right):=\left(x_{1}+U\right)+\cdots +\left(x_{n}+U\right).}$
2. For any neighborhood ${\displaystyle U}$ of ${\displaystyle 0,}$ there exists a finite subset ${\displaystyle F\subseteq X}$ such that ${\displaystyle S\subseteq F+U}$ (where the right hand side is the
   Minkowski sum
   ${\displaystyle F+U:=\{f+u:f\in F,u\in U\}}$).
3. For any neighborhood ${\displaystyle U}$ of ${\displaystyle 0,}$ there exist finitely many subsets ${\displaystyle B_{1},\ldots ,B_{n}}$ of ${\displaystyle X}$ such that ${\displaystyle S\subseteq B_{1}\cup \cdots \cup B_{n}}$ and each ${\displaystyle B_{j}}$ is ${\displaystyle U}$-small.^[6]
4. For any given filter subbase ${\displaystyle {\mathcal {B}}}$ of the identity element's
   neighborhood filter
   ${\displaystyle {\mathcal {N}}}$ (which consists of all neighborhoods of ${\displaystyle 0}$ in ${\displaystyle X}$) and for every ${\displaystyle B\in {\mathcal {B}},}$ there exists a cover of ${\displaystyle S}$ by finitely many ${\displaystyle B}$-small subsets of ${\displaystyle X.}$^[6]
5. ${\displaystyle S}$ is Cauchy bounded: for every neighborhood ${\displaystyle U}$ of the identity and every countably infinite subset ${\displaystyle I}$ of ${\displaystyle S,}$ there exist distinct ${\displaystyle x,y\in I}$ such that ${\displaystyle x-y\in U.}$^[6] (If ${\displaystyle S}$ is finite then this condition is satisfied vacuously).
6. Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded:
   1. The closure ${\displaystyle {\overline {S}}=\operatorname {cl} _{X}S}$ of ${\displaystyle S}$ in ${\displaystyle X.}$^[6]
      • This set being in the list means that the following characterization holds: ${\displaystyle S}$ is (left) totally bounded if and only if ${\displaystyle \operatorname {cl} _{X}S}$ is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below.
   2. The image of ${\displaystyle S}$ under the
      canonical quotient
      ${\displaystyle X\to X/{\overline {\{0\}}},}$ which is defined by ${\displaystyle x\mapsto x+{\overline {\{0\}}}}$ (where ${\displaystyle 0}$ is the identity element).
   3. The sum ${\displaystyle S+\operatorname {cl} _{X}\{0\}.}$^[9]

The term pre-compact usually appears in the context of Hausdorff topological vector spaces.^[10][11] In that case, the following conditions are also all equivalent to ${\displaystyle S}$ being (left) totally bounded:

7. In the completion ${\displaystyle {\widehat {X}}}$ of ${\displaystyle X,}$ the closure ${\displaystyle \operatorname {cl} _{\widehat {X}}S}$ of ${\displaystyle S}$ is compact.^[10][12]
8. Every ultrafilter on ${\displaystyle S}$ is a
   Cauchy filter
   .

The definition of right totally bounded is analogous: simply swap the order of the products.

Condition 4 implies any subset of ${\displaystyle \operatorname {cl} _{X}\{0\}}$ is totally bounded (in fact, compact; see § Comparison with compact sets above). If ${\displaystyle X}$ is not Hausdorff then, for example, ${\displaystyle \{0\}}$ is a compact complete set that is not closed.^[6]

Topological vector spaces

Any topological vector space is an abelian topological group under addition, so the above conditions apply. Historically, statement 6(a) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann.^[13]

This definition has the appealing property that, in a

.

For separable Banach spaces, there is a nice characterization of the precompact sets (in the norm topology) in terms of weakly convergent sequences of functionals: if ${\displaystyle X}$ is a separable Banach space, then ${\displaystyle S\subseteq X}$ is precompact if and only if every weakly convergent sequence of functionals converges uniformly on ${\displaystyle S.}$^[14]

Interaction with convexity

• The balanced hull of a totally bounded subset of a topological vector space is again totally bounded.^[6][15]
• The
  Minkowski sum
  of two compact (totally bounded) sets is compact (resp. totally bounded).
• In a locally convex (Hausdorff) space, the
  disked hull
  of a totally bounded set ${\displaystyle K}$ is totally bounded if and only if ${\displaystyle K}$ is complete.^[16]

See also

• Compact space
• Locally compact space
• Measure of non-compactness
• Orthocompact space
• Paracompact space
• Relatively compact subspace

References

Bibliography

• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner.
  OCLC 8210342
  .
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• OCLC 840278135
  .
• Zbl 0304.54002
  .
• OCLC 853623322
  .
• Willard, Stephen (2004). General Topology. Dover Publications.
  ISBN 0-486-43479-6
  .