A sequence in a metric space is called Cauchy if as . A subset of is called complete if every Cauchy sequence in converges and its limit is in . For example, is complete, while is not.

**Proposition.** A closed subset of a complete metric space is complete, and a complete subset of an arbitrary metric space is closed.

Proof. If is complete, is closed, and is a Cauchy sequence in , has a limit in . So . If is complete and , then there is a sequence in converging to . is Cauchy, so its limit lies in ; thus .

In a metric space we can define the distance from a point to a set and the distance between two sets. Namely, if and ,

,

.

Note that iff $x\in\bar E$. We also define the diameter of to be

.

is called bounded if .

If and is a family of sets such that , 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 . Every totally bounded set is bounded, for if , say and , then

.

If is totally bounded, so is , for it is easily seen that if , then .

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

1) is complete and totally bounded.

2) (The Bolzano-Weierstrass Property) Every sequence in has a subsequence that converges to a point in .

3) (The Heine-Borel Property) If is a cover of by open sets, there is a finite set such that covers .

Proof. 1) 2): Suppose that 1) holds and is a sequence in . can be covered by finitely many balls of radius , and at least one of them must contain for finitely many many : say, for . can be covered by finitely many balls of radius , and at least one of them must contain for infinitely many : say for . Continuing inductively, we obtain a sequence for . Pick , , … such that . Then is a Cauchy sequence, for if $k>j$, and since is complete, it has a limit in .

2) 1): If is not complete, there is a Cauchy sequence in with no limit in . No subsequence of can converge in , for otherwise the whole sequence would converge to the same limit. On the other hand, if is not totally bounded, let be such that cannot be covered by finitely many balls of radius . Choose inductively as follows. Begin with any and having chosen , pick . Then for all , so has no convergent subsequence.

1) + 2) 3): If suffices to show that if 2) holds and is a cover of by open sets, such that every ball of radius that intersects is contained in some , for can be covered by finitely many such balls by 1). Suppose to the contrary that for each , $\exists$ a ball of radius such that and is contained in no . Pick ; by passing to a subsequence we may assume that converges to some . We have for some , and since is open, such that . But if is large enough so that and , then , contradicting the assumption on .

3) 2): If is a sequence in with no convergent subsequence, for each there is a ball centered at that contains for only finitely many . Then is a cover of by open sets with no finite sub-cover.

A set can possesses the properties 1)-3) is called compact. Every compact set is closed and bounded, and the converse is false.

**Proposition.** Every closed and bounded subset of is compact.

Proof. Since closed subsets of are complete, it suffices to show that bounded subsets of are totally bounded. Since every bounded set is contained in some cube

,

it is enough to show that is totally bounded. Given , pick an integer , and express as the union of congruent subcubes by dividing the interval into equal pieces. The side length of these subcubes is and hence their diameter is , so they are contained in the balls of radius about their centers.

Two metrics and on a set are called equivalent if

for some .

Equivalent metrics define the same open, closed and compact sets, the same convergent and Cauchy sequence, and the same continuous and uniformly continuous mappings. Consequently, most results concerning metric spaces depend not on the particular metric chosen but only on its equivalent class.

“As you simplify your life, the laws of the universe will be simpler; solitude will not be solitude, poverty will not be poverty, nor weakness, weakness.” ~ Henry David Thoreau

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 15-16.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.