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.