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.