FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

CHAPTER 1
Preliminaries
In order to ground our later discourse, a brief review of the fundamentals is in order. The
purpose of the present discussion is twofold: first, it will establish the linguistic and symbolic
conventions which we will follow throughout the balance of this exposition. Secondly, and
perhaps more importantly, it will provide (or perhaps refresh) for the reader a set of concepts
and results, the comprehension of which is sufficient to understand the motivation, proof,
and consequences of the five theorems which are the main focus of this work.
The reader is assumed to be familiar with the rudiments of set theory, algebra, and
analysis. However, some of this material will be introduced in this chapter for the reasons
outlined above, as well as material from topology and analysis in metric spaces.
Set Theory
A set is a collection of objects. We will often refer to the set which has no elements.
This set is called the null, void, or empty set, and is represented by the symbol ∅.
We will typically use upper-case letters to denote sets, and lower-case letters to denote
their elements. The statement ‘object x belongs to the set X’ or ‘x is a member of X’ will be
expressed symbolically as x ∈ X. Sometimes, it is more convenient to define a set in terms
of a condition which its members must meet. Let C(x) be such a proposition. We define the
set X of objects x which satisfy C(x) as
X = {x| C(x) } .
Table 1 defines a number of standard sets used throughout this paper.
If X and Y are sets, we say that X is a subset of Y if x ∈ X guarantees that x ∈ Y ,
and in this case we write X ⊂ Y . Two sets X and Y are equal if, and only if, X ⊂ Y and
Y ⊂ X.
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