as an ordered field with lubp
Rational number system ( with
) has gaps (e.g.,
can not be find in rational set if
), in spite of the fact that between any two rationals there is another: if
with both
, then
. The real number system, denoted by
, fills these gaps. This is the principle reason for the fundamental role which it plays in analysis. To formalize this property, we have the least-upper-bound property of real system.
As a preliminary, note that is an ordered set, where
is defined as that
is positive. For an ordered set, we have the definition of bounded set. The least-upper-bound property is stated as follows:
Property 1 least-upper-bound property
If,
is not empty, and
is bounded above, then sup
exists in
.
Here sup is the least upper bound of
, which is defined that sup
is an upper bound of
, and for any
,
is not an upper bound of
. It can be proved that an ordered set with least-upper-bound property also has the greatest-lower-bound property, which can be similarly stated.
Another property that is as important as this one is that is a field, an algebraic structure which defines the axioms for two operations: addition and multiplication. The importance comes from that the arithmetic laws derived from the definition of field.
To sum up, the set of real numbers is an ordered field which has the least-upper-bound property. For the details of the proof, you may refer to rudin’s book. With such properties, we can prove the unique existence of
for any
.
As an extension to the real number system, we add two symbols, , such that
for every
. In this case, any unbounded subset
of
now has sup
in the extended real number system.
Basic topology 1: Euclidean space as a metric space
A metric space is a space where distance (or norm) between elements of this space is defined. The properties of distance is not going to be presented here. The vector space with definitions of inner product and norm is called Euclidean
space.
As preliminaries for getting to know metric space, we may have the definitions of countability first. With the definition of distance, we then immediately derive the definition of bounded set in metric space.
The important concepts in metric space include:
Definition 1 Let
be a metric space,
A neighborhood ofis a set
consisting of all
such that
for some
.
A pointis a limit point of the set
if every neighborhood of
contains a point
such that
.
is closed if every limit point of
is a point of
.
A pointis called an interior point of
if there is a neighborhood
of
such that
.
is open if every point of
is an interior point of
.
Based on the above definition, we have the theorem that 1) every neighborhood is an open set. 2) if is a limit point of
, then every neighborhood of
contains infinitely many points of
.
Next we combine the above definitions with least-upper-bound in real system. Suppose and
is a metric space, let
denote the set of all limit points of
, we define the closure of
:
. The theorem we have is:
Theorem 1 sup
is a limit point of
Letbe a nonempty set of real numbers which is bounded above, then sup
. Hence sup
if
is closed.
Now we continues the exploration of concepts in Definition 1 (to be cont..)
Basic topology 2: compact subset of a metric space
[The following notes are written in pdf, you may contact me if you think it useful…]