Topology/Basic Concepts Set Theory

This chapter is meant to be a short, concise introduction to the basic set concepts used throughout this book. It is not meant to be a comprehensive text book on set theory. Rather, it will list the material that the reader should be familiar with and showcase the notation used. Readers desirous of a more in-depth understanding of set theory should read the Set Theory Wikibook.

Basic Definitions

The empty set is denoted by symbol $\emptyset$.  A (finite) set consisting of elements $x_1, x_2, \ldots, x_n$ is denoted $\{x_1, x_2, \ldots, x_n\}$.
It is a bit sloppy but common practice not to distinguish very 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. Then we define the union of A and B, denoted $A\bigcup{B}$, as 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 $x\notin B$.
The symmetric difference of A and B, denoted AΔB, is defined by $A\Delta B=(A\backslash B)\bigcup{(B\backslash A)}$.

If every element in A also belongs to B, we say that A is a "subset" of B. In other words, $\forall x\in A:x\in B$ is equivalent to $A\subseteq B$. A key property of these sets is that A = B if and only if $A\subseteq B$ and $B\subseteq A$. If $A\subseteq B$ and $A \ne B$ then A is a proper subset of B, which we denote $A\subsetneq B$. We do not use the notation $A\subset B$, as the meaning varies between subset and proper subset in various sources.

We may want to focus on properties of elements that are not in a particular set. To do so we introduce the notion of the complement of a set. Let X and U be sets such that $X\subseteq U$. We say the complement of X, denoted X^c, is the set of all $y\in U$ such that $y\notin X.$.

Our next definition deals with "sizes" of sets. Let A be some set. Then we define the cardinality of A, denoted $\left| A \right|$, as the number of elements in A.
Examples:

1) $\left| \left\{ 1,2,3,4,5 \right\} \right|=5$

2) $\left| \varnothing \right|=0$

3) $\left| \left\{ \varnothing \right\} \right|=1$

Another important concept to note is that of the power set. We define the power set to be the set of all subsets of some set A, and denote it by P(A).

Examples:

1) $P(\varnothing )=\left\{ \varnothing \right\}$

2) $P(\left\{ x \right\})=\left\{ \varnothing ,\left\{ x \right\} \right\}$

3) $P(\left\{ x,y \right\})=\left\{ \varnothing ,\left\{ x \right\},\left\{ y \right\},\left\{ x,y \right\} \right\}$

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

Finite ordered sets (or 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 $\forall n:x_n = y_n$ (compare with the statement above regarding equality of unordered sets).

Ordered sets can be defined in terms of unordered sets.
For example, the ordered pair $\langle x,y\rangle$  was defined by Kazimierz Kuratowski as $\langle x,y\rangle := \left\{\{x\}, \{x,y\}\right\}$. It is tedious but not difficult to check that $\ \langle x_1,y_1\rangle =\langle x_2,y_2\rangle$  if and only if $\ x_1 = x_2$  and $\ y_1 = y_2$.
Now n-tuple is defined as follows:

$(x_1, x_2,\ldots, x_n)\ :=\ \{\langle 1,x_1\rangle ,\langle 2,x_2 \rangle ,\ldots,\langle n,x_n\rangle \}$

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 relationfrom 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 ranF = 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 two 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 a) 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 R (the set of all real numbers)? Justify your answer.

2 b) Prove that no finite subcollection of A_n has the property that the union of this finite subcollection equals R. Once you study topology, you will see that this constitutes a proof that R is not compact.