There is some preliminary information we need to cover before a discussion of product spaces makes sense.

Topological Spaces

First, we must define a topology and a topological space.

A topology defined on a set is a collection of subsets of that has the following properties:

  1. and are contained in
  2. Any arbitrary union of the elements of a subcollection of is contained in
  3. Any finite intersection of the elements of a finite subcollection of is contained in

Then we say that , along with a topology defined on is called a topological space. Further, we say that if a subset is an element of , then is an open set of .

Example

Let . Then there are many possible topologies that can be defined on . Two such topologies are the collections

The empty set and the whole space are clearly contained in both topologies. Now take any subcollection of either topologies, say or . Note that , which is contained in , and , which is also contained in . Similarly, note that is contained in , and is also contained in .

Bases for Topological Spaces

Second, we must define what a basis for a topology is. Usually listing out every member of a topology can get a bit… tiresome. So we define something easier to get our hands on, and build a topology from there.

If is a set, a basis for a topology on is a collection of subsets of (called basis elements) such that

  1. For each there is at least one basis element containing
  2. If belongs to the intersection of two basis elements and , then there exists a third basis element such that and

Then we define the topology generated by as follows: A subset is said to be open in (that is, an element of ) if for every , there exists of basis element such that and .

basis-element-intersection

If is the collection of all open intervals in , then the topology generated by is called the standard or usual topology on .

Cartesian Products

Third, we must discuss cartesian products in a more general sense than you might be familiar.

Let be a nonempty collection of sets. An indexing function for is a surjective function . The set is called the index set. The collection , along with the function , is called an indexed family of sets. For some , we typically denote as , and denote the whole family of sets by .

Note that no restrictions are given on the index set or the collection ! This means that both and could be finite, countable, or even uncountable.

We will use this notation to redefine arbitrary unions, intersections, and products. Given some indexed family of sets , we define the arbitrary union of elements of as

and the arbitrary intersection of elements of as

We will define cartesian products of indexed families of sets in stages.

Finite Products

Let . Given a set , we define an -tuple of elements of to be a function

We typically denote as , and often denote the entire function as

Now let be a finite family of sets indexed by the set , and let . We define the finite cartesian product of this family of sets, denoted by

or equivalently

as the collection of all -tuples of elements of , such that for every .

Countable Products

Given a set , we define an -tuple of elements of to be a function

We also call this function a sequence of elements of , and typically denote it by the symbols

or

Now let be a family of sets indexed by , and let . Then the countable cartesian product of this family of sets, denote by

or equivalently

is defined as the set of all -tuples of elements of such that for every .

If each , then the product is denoted by , which is the collection of all -tuples of elements of .

Arbitrary Products

Let be any index set. Given a set , we define a -tuple of elements of to be a function . If , we typically denote as , and we often denote the function itself by

and we denote the set of all -tuples of elements of by .

Now let be some indexed family of sets, and let . Then the arbitrary cartesian product of this indexed family of sets, denoted by

is defined to be the collection of all -tuples of elements of such that for every . That is, it is the set of all functions

such that for each .

If is understood, we often denote the product simply by and an element of the product as .

Product Spaces

As with the cartesian product, we will build our understanding of the product space in pieces.

The Product of Two Spaces

Let and be topological spaces. The product topology on is the topology with basis . We call a product space.

Now what if and are given in terms of their bases? If is a basis for the topology of , and is a basis for the topology of , then

is a basis for the topology of .

Now define be defined by and be defined by . and are called projections of onto the first and second factors respectively. But then if is open in , then the set is the set . Similarly, if is open in , then is the set .

Note that .

Example

Let us denote the space and an open set as

space-x

Then given an open set , we can draw , and as

x-times-y

However, we will soon see that this fails to generalize to higher dimensions, say for an uncountable product space.

The Product of Arbitrarily Many Spaces

Recall that for the product of two spaces discussed above, we defined that basis of the product space as the collection where each and are basis elements of the bases for and respectively. However, this was a bold-faced lie. For reasons beyond the scope of this discussion, extending this definition to arbitrarily many topological spaces really sucks.

Box Topology

Let be an indexed family of topological spaces. We define a topology on the product space using the basis

This basis generates the box topology on .

Product Topology

Let be an indexed family of topological spaces. We define a topology on the product space using the basis

where each except for finitely many values of . This basis generates the product topology on . Note that the box and product topologies are identical when the family of topological spaces is finite!

Pictures that do Generalize

As we remarked earlier, the visualization of a cartesian product below does not generalize to higher dimensions.

x-times-y

Let us now introduce a new visualization method. Let us begin by visualizing the ordered pairs and in . We lay the spaces and vertically, and visualize points in their product as a sequence of dots joining the two spaces.

visualizing-a-point

Since is a product of finitely many spaces, the box topology and the product topology are one and the same. So a basis element of one is a basis element of the other. In particular, if and are basis elements of and respectively, then is a basis element in the product space. Well, what does look like?

Recall from the definition of the cartesian product, that is defined as the collection of all points of such that the first coordinate is an element of and the second an element of .

xy-basis

The point of laying each space vertically is intended to provide intuition for higher dimensional product spaces, even to the point of uncountably many dimensions. However it does have some limitations.

higher-dimensional-product-space

For example, drawing distinct lines, and indexing them with the positive integers makes the assumption that the spaces are countable. It may make more sense to draw the product space as a plane, where each infinitely thin vertical slice is a particular . However, this too has a shortcoming. It assumes that there is some order on – that is, we can somehow claim that some should be drawn before .

However, with these (and possibly other) shortcomings, this kind of visualization does prove useful for building intuition regarding product spaces.

An Example of the Difference Between the Box and Product Topologies

Let be given by

where for each . Suppose has the product topology. Then the function is continuous if and only if each function is continuous.

This is not true if is given the box topology. To see why, consider , the countably infinite product of with itself.

Now for the sheer hell of it, define a function given by

sequence

Note that each of the coordinate functions by are continuous. Therefore, if is given the product topology, is continuous by the preceding theorem. However, what happens if we give to box topology?

Consider .

basis-element

is a basis element of the box topology, and is thus open. However, is not open, so then cannot be continuous. To see this, suppose by way of contradiction that was open.

Note that , and thus . Now since is open by assumption, for every point in there is some neighborhood of contained in . Thus there is some open interval around contained in .

sequence-interval

However, this implies that , but this implies that for every , but this is absurd and a contradiction.

Therefore cannot be open, and therefore cannot be continuous.