# 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 $A \ne B$. (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 $\forall i \in \mathbb{N}, 1 \le i \le n:x_i = y_i$.

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 $a_{1}\in \text{dom }F,~a_{2}\in \text{dom }F,\text{ and}~a_{1}\ne a_{2}$ implies $F(a_{1})\ne F(a_{2})$. 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.