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

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

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

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

$(x_{1},x_{2},\ldots ,x_{n})\ :=\ \{\langle 1,x_{1}\rangle ,\langle 2,x_{2}\rangle ,\ldots ,\langle n,x_{n}\rangle \}.$ 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 $A\otimes B$ , is the set of all possible ordered pairs where the first element comes from A and the second from B; that is,

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

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 $R\subseteq A\otimes B$ . If $(x,y)\in R$ , 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 $F(x)=y$ instead of xFy.

There are a few special types of functions we should discuss. A function $F:A\to B$ is said to be onto a set B, or a surjective function from A to B, if ran$F=B$ . A function F is said to be one-to-one or injective if $a_{1}\in {\text{dom }}F,~a_{2}\in {\text{dom }}F,{\text{ and}}~a_{1}\neq a_{2}$ implies $F(a_{1})\neq F(a_{2})$ . 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 $A_{n}=(-n,n)$ for each n in the set of natural numbers. Does the union over all $A_{n}$ (for n in the set of natural numbers) equal $\mathbb {R}$ (the set of all real numbers)? Justify your answer.
3. Using $A_{n}$ from above, prove that no finite subset of $A_{n}$ has the property that the union of this finite subset equals $\mathbb {R}$ . Once you study topology, you will see that this constitutes a proof that $\mathbb {R}$ is not compact.