Definition 1 Given $S \subset C$ is compact if $S$ is both closed and bounded.
The true story of the definition of compactness is quite a bit more complicated. In fact,
the definition stated above is a celebrated theorem of topology. The following fills in many of
the details of the full definition of compactness and the conditions under which our definition
is equivalent to actual definition.
We must first make some definitions about topological spaces in general. First and
foremost, what exactly is a topological space?
Definition 2 A topological space is a set $T$, together with a collection of subsets of $T$,
called the open sets of $T$, denoted $O$, that satisfy the following properties:
* the empty set and $T$ itself are both elements of $O$: $\emptyset, T \in O$
* given any finite collection of elements of $O$, their intersection must also be an element
of $O$:
$U_1, \dots, U_n \in O \Rightarrow \bigcap_{i=1}^n U_i \in O$
* given any collection of elements of $O$, their union must be an element of $O$:
$U_\alpha \in O, \alpha \in A \Rightarrow \bigcup_{\alpha \in A} U_\alpha \in O$
It is of vital importance that we understand $O$ is closed under finite intersections and
arbitrary unions. The set $A$ referenced in the above definition is called an index set; for
cases of finite unions or intersections we usually let the index set be $\{1, 2, \dots, n\}$. The point
of setting our index set to be $A$ in the last bullet point is that it may be any size, even the
size of $\mathbb{R}$ (or larger)! Of course, in these cases $A$ cannot be enumerated (Why?).
Examples: (1) Consider the set of real numbers, $T = \mathbb{R}$ and let the collection of open
sets $O$ be generated by all open intervals (i.e. sets of the form $(a, b)$ with $a, b \in \mathbb{R}$):
$O = \{\text{finite intersections of sets of form } (a, b), \text{ arbitrary unions of sets of the form } (a, b)\}$
Then the pair $(T, O)$ is the topological space of real numbers with the usual topology that
we use to develop calculus (secretly!) in calculus 1 and 2.
(2) Nearly the same construction as above can be repeated, letting $T = \mathbb{C}$ and replacing
in the definition of $O$ "open intervals" with open disks, sets of the form
$B_\rho(z_0) = \{z \in \mathbb{C} \mid |z - z_0| < \rho\} \subset \mathbb{C}$
Formally, our development of calculus will at some level follow the same lines as what you
learned in calculus 1 and 2, but the nature of the standard topology of $\mathbb{C}$ described here
manifest many important and fundamental differences in the development of calculus.
Exercise 1: Let $T$ be any set. Prove that (1) setting $O_1 = \{\emptyset, T\}$ makes a topological
space $(T, O_1)$ and (2) setting $O_2 = P(T)$, the powerset of $T$ makes $(T, O_2)$ a topological
space. (Note: the powerset of $T$, $P(T)$, is the set of all subsets of $T$).
