S. D. V. Stephens

notes / Measure Theory / chapter 0: Prologue

0.6 Metric Spaces

Definition 0.6.1. A metric space is a set with a distance function such that for all :

  • (i) ;
  • (ii) ;
  • (iii) .

Example 0.6.2.

  • is a metric on .
  • is a metric on the continuous functions on .
  • If is a metric on and , then is a metric on .

If and are metric spaces, the product metric is given by

Others on include

A set is open if for every there exists such that , where

is closed is open.

Remark. From 55b: there are MANY ways to define an open/closed set in a topology. Sequence convergence in terms of opens: a sequence converges to if for every open with there exists such that for all .

Recall: is and (not as definitions).

If is the union of all open sets contained in (here is any subset of the space), then is the largest open set in , and is the interior of (denoted ). Now and , or .

If , then is said to be dense in . (E.g. , so is dense in .) is nowhere dense iff has an empty interior (i.e. ).

is separable if has a countable AND dense subset (e.g. , countable dense).

A sequence (we may use the technically inaccurate term “ is ‘in’ ”) converges to (symbolically: or ) if .

Proposition 0.6.3. Let be a metric space, and . The following are equivalent:

  • I. .
  • II. for all .
  • III. There exists a sequence in which converges to .

Proof. If , then is a closed set containing (since is open) but not , so . Too, if , then since is open there exists such that . Thus I II.

Now if II holds, then for every there exists , so that . Conversely, if , then for all , so no sequence in can converge to . Thus II III.

Thus I II III. ∎

Let be metric spaces, with metrics both denoted by . A map is continuous at if for every there exists such that whenever ; that is to also say,

is continuous if it is continuous at each (every) , and is uniformly continuous if can be chosen in the definition of continuity independent of .

Proposition 0.6.4. is continuous is open for every open in .

Proof. () Let be an open set that contains . So for every and every , is open and contains , and so contains some ball around . Therefore is continuous at ; given this for all , is continuous.

() Now let be a continuous map. Let be open, so for every there exists such that , and for each there exists such that

Thus is open. ∎

Let be a metric space. A sequence is Cauchy if as . A set is complete if every Cauchy sequence in converges to a point in . Something like this:

Sketch of a Cauchy sequence: two oscillating curves for the terms x_n and x_m inside a dashed envelope that shrinks as n and m go to infinity, so the gap rho(x_n, x_m) tends to 0.

Proposition 0.6.5. A closed subset of a complete metric space is complete; a complete subset of an arbitrary metric space is closed. (E.g. and with .)

Distance from a point to a set, and between two sets, in a metric space (, , ) is defined by:

(basically the shortest distance to from ), and

We get . We define the diameter of to be

is bounded if .

If and is a family of sets such that , then is called a cover of , and is said to be covered by the ’s.

is called totally bounded if, for every , can be covered by finitely many balls of radius . Totally bounded necessitates boundedness.

Theorem 0.6.6. If is a subset of the metric space , the following are equivalent:

  • (a) is complete and totally bounded.
  • (b) (The Bolzano–Weierstrass property) Every sequence in has a subsequence which converges to a point of .
  • (c) (The Heine–Borel property) If is a cover of by open sets, there exists a finite set such that covers (every open cover admits a finite subcover; compact).

A set with the properties (a)–(c) is called compact.