Notes on mathematical analysis 1: preliminaries

$\mathcal{R}$ as an ordered field with lubp
Rational number system ($\frac{m}{n}$ with $m,n\in Q$) has gaps (e.g., $p$ can not be find in rational set if $p^2=2$), in spite of the fact that between any two rationals there is another: if $r with both $r, s\in Q$, then $r<(r+s)/2. The real number system, denoted by $R$, 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 $R$ is an ordered set, where $r is defined as that $s-r$ 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 $E\subset R$, $E$ is not empty, and $E$ is bounded above, then sup$E$ exists in $R$.

Here sup$E$ is the least upper bound of $E$, which is defined that sup$E$ is an upper bound of $E$, and for any $\gamma <\text{sup}E$, $\gamma$ is not an upper bound of $E$. 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 $R$ 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 $R$ 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 $\sqrt{n}$ for any $n\in N_+$.

As an extension to the real number system, we add two symbols,  $+\infty, -\infty$, such that  $-\infty< x <+\infty$ for every  $x\in R$. In this case, any unbounded subset $E$ of  $R$ now has sup $E=+\infty$ 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  $R^k$ with definitions of inner product and norm is called Euclidean  $k-$ 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 $X$ be a metric space,
A neighborhood of  $p$ is a set  $N_r(p)$ consisting of all $q$ such that  $d(p,q) for some $r>0$.
A point  $p$ is a limit point of the set  $E$ if every neighborhood of $p$ contains a point $q\neq p$ such that  $q\in E$.
$E$ is closed if every limit point of  $E$ is a point of  $E$.
A point  $p$ is called an interior point of  $E$ if there is a neighborhood  $N$ of  $p$ such that  $N\subset E$.
$E$ is open if every point of  $E$ is an interior point of  $E$.

Based on the above definition, we have the theorem that 1) every neighborhood is an open set. 2) if $p$ is a limit point of $E$, then every neighborhood of $p$ contains infinitely many points of $E$.

Next we combine the above definitions with least-upper-bound in real system. Suppose $E\subset X$ and $X$ is a metric space, let $E^{\prime}$ denote the set of all limit points of $E$, we define the closure of $E$: $\bar{E}=E\cup E^{\prime}$. The theorem we have is:

Theorem 1 sup$E$ is a limit point of $E$
Let $E$ be a nonempty set of real numbers which is bounded above, then sup$E\in \bar{E}$. Hence sup$E\in E$ if $E$ 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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