Background

Here is a collection of background results and conventions that we will follow throughout the course. You have hopefully seem before all of the material below.

Sets and Functions

We will not worry too much about set theory in the strict sense, dealing usually with familiar sets of numbers, of sequences, and of functions, their subsets and their power sets. In particular we use the following standard notation.

$\emptyset$ for the empty set
$N$ for the natural numbers
$Q$ for the set of rational numbers
$R$ for the set of real numbers
$C$ for the set of complex numbers

The power set of a set is the set consisting of all its subsets. Write $P (X)$ for the power set of a set $X$ . Thus $E \in P (X)$ is the same as writing $E \subset X$ . For example $\begin{aligned} P ({1, 2}) & = {\emptyset, {1}, {2}, {1, 2}} \\ P (\emptyset) & = {\emptyset} \\ P ({\emptyset}) & = {\emptyset, {\emptyset}} \end{aligned}$

We use the standard notation for intervals in $R$ . Thus $\begin{array}{r} (a, b) = {t \in R : a < t < b} \\ [a, b) = {t \in R : a \leq t < b} \\ (a, b] = {t \in R : a < t \leq b} \\ [a, b] = {t \in R : a \leq t \leq b} \end{array}$ for all real numbers $a, b$ . If $a > b$ then all sets above are empty, and if $a = b$ then all sets above except $[a, b]$ are empty.

We will make special use of the notation $[0, \infty] = [0, \infty) \cup {\infty}$ which is the set of non-negative real numbers united with an additional symbol $\infty$ called infinity. We impose addition, multiplication and order on $[0, \infty]$ as in $R$ together with $\begin{matrix} t + \infty = \infty = \infty + t for all t > 0 \\ \infty + \infty = \infty \\ t * \infty = \infty for all t > 0 \\ t < \infty for all t \geq 0 \end{matrix}$ all of which are reasonable. We also take $\infty * 0 = 0$ .

Given a function $f : X \to Y$ we can pull back subsets of $Y$ to get subsets of $X$ . In other words, to $f$ we can associate a new function $\begin{aligned} f^{- 1} : P (Y) & \to P (X) \\ A & \mapsto {x \in X : f (x) \in A} \end{aligned}$ for all $A \in P (Y)$ . This new function interacts nicely with set-theoretic operations. For example $\begin{aligned} f^{- 1} (A \cup B) & = f^{- 1} (A) \cup f^{- 1} (B) \\ f^{- 1} (A \cap B) & = f^{- 1} (A) \cap f^{- 1} (B) \\ f^{- 1} (Y ∖ B) & = X ∖ f^{- 1} (B) \end{aligned}$ whenever $A, B \subset Y$ .

Topology

A topology on a set $X$ is any collection $T$ of subsets of $X$ satisfying the following properties.

$\emptyset \in T$
$X \in T$
For any two sets $U, V \in T$ the intersection $U \cap V$ belongs to $T$
For any collection $α \mapsto U_{α}$ in $T$ the set $⋃_{α} U_{α}$ belongs to $T$

By a topological space we mean a pair $(X, T)$ where $X$ is a set and $T$ is a topology on $X$ .

Given a topology $T$ on $X$ the members of $T$ are called the open subsets of $X$ . A subset of $X$ is then closed if its complement is open. Note that some subsets of $X$ might be open and closed, and that some subsets of $X$ may be neither open nor closed.

If $(X, d)$ is a metric space then collection of metrically open subsets of $X$ form a topology on $X$ . Write $B (x, r) = {y \in X : d (x, y) < r}$ for the open ball centered at $x$ with radius $r$ .

In the special case $X = R$ and $d$ the Euclidean distance $d (x, y) = | x - y |$ the open balls are just the open intervals $(a, b)$ with $a < b$ . We have an explicit description of open subsets of $R$ . Every open set $U \subset R$ is of the form $U = ⋃_{n = 1}^{\infty} (a_{n}, b_{n})$ where the intervals $(a_{n}, b_{n})$ are pairwise disjoint, at most one $a_{i}$ is $- \infty$ and at most one $b_{i}$ is $\infty$ .

Let $(X, T)$ and $(Y, S)$ be topological spaces. A map $f : X \to Y$ is continuous if $f^{- 1} (U) \in T$ for every $U \in S$ . When $T$ comes from a metric $d_{X}$ on $X$ and $S$ comes from a metric $d_{Y}$ on $Y$ we can rephrase continuity of $f : X \to Y$ as follows: for every $x \in X$ and every $ϵ > 0$ there is $δ > 0$ such that $d_{X} (x, y) < δ$ implies $d_{X} (f (x), f (y)) < ϵ$ .