# 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

editThe empty set is denoted by symbol . A finite set consisting of elements is denoted . Set theorists commonly, albeit sloppily, do not distinguish strictly between a singleton set and its single element .

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 , is the set of all*x*that belong to either*A*or*B*(or both). - The
*intersection*of*A*and*B*, denoted , is the set of all*x*that belong to both*A*and*B*. - The
*difference*of*A*and*B*, denoted or , is the set of all such that .- In contexts where there is a set containing "everything," usually denoted
*U,*the*complement*of*A*, denoted , is .

- In contexts where there is a set containing "everything," usually denoted
- The
*symmetric difference*of*A*and*B*, denoted , is defined by . *A*is a*subset*of*B*, denoted , if and only if every element in also belongs to . In other words, when . A key property of these sets is that if and only if and .*A*is a*proper subset*of*B*, denoted , if and only if and . (We do not use the notation , as the meaning is not always consistent.)- The
*cardinality*of*A,*denoted , is the number of elements in*A*.- Examples

- The
*power set*of*A,*denoted , is the set of all subsets of*A.*- Examples

Note that .

Ordered *n*-tuples are denoted .
For two ordered sets and , we have if and only if .

*N*-tuples can be defined in terms of sets. For example, the ordered pair was defined by Kazimierz Kuratowski as .
Now *n*-tuples are defined as

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 , is the set of all possible ordered pairs where the first element comes from *A* and the second from *B*; that is,

- .

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 . If , 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 dom*R*.
The set of all *y* such that, for some *x*, *x* is in relation *R* with *y* is called the *range* of *R*, denoted ran*R*.
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 instead of *xFy*.

There are a few special types of functions we should discuss.
A function is said to be *onto* a set *B*, or a *surjective* function from *A* to *B*, if ran .
A function *F* is said to be *one-to-one* or *injective* if implies .
A function that is both injective and surjective is called *bijective*.

## Exercises

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

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