We now investigate some properties that subsets of
and
may possess. We will define “closed sets,” “open sets,” and “limit points” of sets.
These notions are the rudimentary notions of what is called topology.As in earlier definitions, these topological ones will be enlightening when we come to continuity.
We now investigate some properties that subsets of
and
may possess. We will define “closed sets,” “open sets,” and “limit points” of sets.
These notions are the rudimentary notions of what is called topology.As in earlier definitions, these topological ones will be enlightening when we come to continuity.
-
Let
be a subset of
A complex number
is called a
limit point of
if there exists a sequence
of elements of
such
that
A set
is called
closed if every limit point of
belongs to
Every limit point of a set of real numbers is a real number.
Closed intervals
are examples of closed sets in
while open intervals and half-open intervals may not be closed sets.
Similarly, closed disks
of radius
around a point
in
,
and closed neighborhoods
of radius
around a set
are closed sets, while the open disks or open neighborhoods are not closed sets.
As a first example of a limit point of a set, we givethe following exercise.
Let
be a nonempty bounded set of real numbers, and let
Prove that there exists a sequence
of elements of
such that
That is, prove that the supremum of a bounded set of real numbers is a limit point of that set.
State and prove an analogous result for infs.
HINT: Use
[link] , and let
run through the numbers
- Suppose
is a set of real numbers, and that
with
Show that
is not a limit point of
That is, every limit point of a set of real numbers is a real number.
HINT: Suppose false; write
and make
use of the positive number
- Let
be a complex number, and let
be the set of all
for which
Show that
is a closed subset of
.
HINT: Use part (b) of
[link] .
- Show that the open disk
is not a closed set in
by finding
a limit point of
that is not in
- State and prove results analogous to parts b and c for
intervals in
- Show that every element
of a set
is a limit point of
- Let
be a subset of
, and let
be a complex number.
Show that
is not a limit point of
if and only if
there exists a positive number
such that if
then
is not in
That is,
HINT:
To prove the “ only if” part, argue by contradiction,and use the sequence
as
's.
- Let
be a sequence of complex numbers, and let
be the set of all the
's.
What is the difference between a cluster point of the sequence
and a limit point of the set
- (h) Prove that the cluster set of a sequence is a closed set.
HINT: Use parts (e) and (f).
- Show that the set
of all rational numbers is not a closed set.
Show also that the set of all irrational numbers is not a closed set.
- Show that if
is a closed subset of
that contains
then
must equal all of
Here is another version of the Bolzano-Weierstrass Theorem, this time
stated in terms of closed sets rather than bounded sequences.
