# Topology/Basic Concepts Set Theory

This chapter concisely describes the basic set theory concepts used throughout this book—not as a comprehensive guide, but as a list of material the reader should be familiar with and the related notation. Readers desiring a more in-depth understanding of set theory should read the Set Theory Wikibook.

## Basic Definitions

The empty set is denoted by symbol ${\displaystyle \varnothing }$ . A finite set consisting of elements ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$  is denoted ${\displaystyle \{x_{1},x_{2},\ldots ,x_{n}\}}$ . Set theorists commonly, albeit sloppily, do not distinguish strictly between a singleton set ${\displaystyle \{x\}}$  and its single element ${\displaystyle x}$ .

For a more in depth understanding of how elements of sets relate to each other, we must first define a few terms. Let A and B denote two sets.

• The union of A and B, denoted ${\displaystyle A\bigcup {B}}$ , is the set of all x that belong to either A or B (or both).
• The intersection of A and B, denoted ${\displaystyle A\bigcap {B}}$ , is the set of all x that belong to both A and B.
• The difference of A and B, denoted ${\displaystyle A\backslash B}$  or ${\displaystyle A-B}$ , is the set of all ${\displaystyle x\in A}$  such that ${\displaystyle x\notin B}$ .
• In contexts where there is a set containing "everything," usually denoted U, the complement of A, denoted ${\displaystyle A^{c}}$ , is ${\displaystyle U\backslash A}$ .
• The symmetric difference of A and B, denoted ${\displaystyle A\Delta B}$ , is defined by ${\displaystyle A\Delta B=(A\backslash B)\bigcup {(B\backslash A)}}$ .
• A is a subset of B, denoted ${\displaystyle A\subseteq B}$ , if and only if every element in ${\displaystyle A}$  also belongs to ${\displaystyle B}$ . In other words, when ${\displaystyle \forall x\in A:x\in B}$ . A key property of these sets is that ${\displaystyle A=B}$  if and only if ${\displaystyle A\subseteq B}$  and ${\displaystyle B\subseteq A}$ .
• A is a proper subset of B, denoted ${\displaystyle A\subsetneq B}$ , if and only if ${\displaystyle A\subseteq B}$  and ${\displaystyle A\neq B}$ . (We do not use the notation ${\displaystyle A\subset B}$ , as the meaning is not always consistent.)
• The cardinality of A, denoted ${\displaystyle \left|A\right|}$ , is the number of elements in A.
Examples
• ${\displaystyle \left|\left\{1,2,3,4,5\right\}\right|=5}$
• ${\displaystyle \left|\varnothing \right|=0}$
• ${\displaystyle \left|\left\{\varnothing \right\}\right|=1}$
• The power set of A, denoted ${\displaystyle P(A)}$ , is the set of all subsets of A.
Examples
• ${\displaystyle P(\varnothing )=\left\{\varnothing \right\}}$
• ${\displaystyle P(\left\{x\right\})=\left\{\varnothing ,\left\{x\right\}\right\}}$
• ${\displaystyle P(\left\{x,y\right\})=\left\{\varnothing ,\left\{x\right\},\left\{y\right\},\left\{x,y\right\}\right\}}$

Note that ${\displaystyle \left|P(A)\right|=2^{\left|A\right|}}$ .

Ordered n-tuples are denoted ${\displaystyle (x_{1},x_{2},\ldots ,x_{n})}$ . For two ordered sets ${\displaystyle X=(x_{1},x_{2},\ldots ,x_{n})}$  and ${\displaystyle Y=(y_{1},y_{2},\ldots ,y_{n})}$ , we have ${\displaystyle X=Y}$  if and only if ${\displaystyle \forall i\in \mathbb {N} ,1\leq i\leq n:x_{i}=y_{i}}$ .

N-tuples can be defined in terms of sets. For example, the ordered pair ${\displaystyle \langle x,y\rangle }$   was defined by Kazimierz Kuratowski as ${\displaystyle \left(x,y\right):=\left\{\{x\},\{x,y\}\right\}}$ . Now n-tuples are defined as

${\displaystyle (x_{1},x_{2},\ldots ,x_{n})\ :=\ \{\langle 1,x_{1}\rangle ,\langle 2,x_{2}\rangle ,\ldots ,\langle n,x_{n}\rangle \}.}$

We now can use this notion of ordered pairs to discuss the Cartesian Product of two sets. The Cartesian Product of A and B, denoted ${\displaystyle A\otimes B}$ , is the set of all possible ordered pairs where the first element comes from A and the second from B; that is,

${\displaystyle A\otimes B=\left\{(a,b)~\left|~a\in A,~b\in B\right.\right\}}$ .

Now that we have defined Cartesian Products, we can turn to the notions of binary relations and functions. We say a set R is a binary relation from A to B if ${\displaystyle R\subseteq A\otimes B}$ . If ${\displaystyle (x,y)\in R}$ , it is customary to write xRy. If R is a relation, then the set of all x which are in relation R with some y is called the domain of R, denoted domR. The set of all y such that, for some x, x is in relation R with y is called the range of R, denoted ranR. A binary relation F is called a function if every element x in its domain has exactly one element y in its range such that xFy. Also, if F is a function, the typical notation is ${\displaystyle F(x)=y}$  instead of xFy.

There are a few special types of functions we should discuss. A function ${\displaystyle F:A\to B}$  is said to be onto a set B, or a surjective function from A to B, if ran${\displaystyle F=B}$ . A function F is said to be one-to-one or injective if ${\displaystyle a_{1}\in {\text{dom }}F,~a_{2}\in {\text{dom }}F,{\text{ and}}~a_{1}\neq a_{2}}$  implies ${\displaystyle F(a_{1})\neq F(a_{2})}$ . A function that is both injective and surjective is called bijective.

## Exercises

If you can successfully answer the following problems, you are ready to study topology! Please take the time to solve these problems.

1. Prove that the empty set is a subset of every set.
2. Consider the set ${\displaystyle A_{n}=(-n,n)}$  for each n in the set of natural numbers. Does the union over all ${\displaystyle A_{n}}$  (for n in the set of natural numbers) equal ${\displaystyle \mathbb {R} }$  (the set of all real numbers)? Justify your answer.
3. Using ${\displaystyle A_{n}}$  from above, prove that no finite subset of ${\displaystyle A_{n}}$  has the property that the union of this finite subset equals ${\displaystyle \mathbb {R} }$ . Once you study topology, you will see that this constitutes a proof that ${\displaystyle \mathbb {R} }$  is not compact.