some definitions related to open sets 5 (compact)

In preceding posts, a definition of the completeness of real numbers was given.
Here we shall give the similar definition of the completeness of a metric space.

In a metric space if any Cauchy sequences is always convergent, the metric space is complete.

Therefore, closed intervals on the real number line is complete. That is to say,
a closed subset in a complete metric space is complete.

The compactness is more stronger than the completeness. However the definition is most
difficult and abstract. A open cover must be prepared at first.

Given a set $X\subset S$ , $S=\cup A_i$  and all $A_i$  is open, $S$  is called a open cover of $X$ .

$[0,1)\subset \cup_{i=1}^{\infty}\left( -\frac{1}{i},1-\frac{1}{i} \right)$
Hence, $\cup_{i=1}^{\infty}\left( -\frac{1}{i},1-\frac{1}{i} \right)$  is a open cover of $[0,1)$

In general, there are many open covers of $X$ .
If $S'$  which is made by joining some selected $A_i$  also becomes a open cover of $X$ ,
$S'$ is called a sub cover. If $i$  is finite, $S$  is called a finite open cover.

If a open cover of $X$  always contains a finite open sub cover, the set $X$  is compact.

The famous Heine Borel theorem is as follows.

A set $X\subset \mathbb{R}$  is compact if and only if $X$  is closed and bounded.

It is not easy to understand the essence which the theorem insists.