** 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 propert*y 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 1least-upper-bound property

If , is not empty, and is bounded above, then supexistsin .

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 1Let be a metric space,

Aneighborhoodof is a set consisting of all such that for some .

A point is alimit pointof the set if every neighborhood of contains a point such that .

isclosedif every limit point of is a point of .

A point is called aninterior pointof if there is a neighborhood of such that .

isopenif 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 1sup is a limit point of

Let be 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**

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