One of the really neat things you can do in topology is play games with dimensions. Topology can give you ways of measuring dimensions, and projecting structures with many dimensions into lower-dimensional spaces. One of the keys to doing this is understanding how to combine different topologies to create new structures. This is done using the *topological product*.
So what’s a topological product? It’s almost the same thing as a cartesian product of sets – except, of course, that it needs to preserve the topological structure of open sets (and therefore neighborhoods). One way of saying what you end up with is that it’s a particular topology whose parts are the cartesian product of the point-sets of the topologies being multiplied. But there are many possible ways of defining the open-sets on that product topology. The one which we’ll use is the topology on that set of points which has *the fewest* open sets, while always preserving the structure of its component topologies.
That’s definitely not an easy concept to understand. So let’s dive a bit deeper. Suppose we have a set of topological spaces. For convenience, we’ll number them: (T1,τ1), (T2,τ2), …, (Tn,τn). What does the product of that *set* of spaces look like?
Let’s name it, to make it easier to talk about:
(P,π) = (T1,τ1) × (T2,τ2) × … × (Tn,τn)
As I said, the set of points is the cartesian product of the sets of points in each of the topological spaces. So P = T1× … ×Tn. In addition to the set P, cartesian product also gives us a *projection function* for each Ti: so for each of the topologies (Ti,τi), there’s a function pi : P → Ti. The projection functions are the key to how we can generate that structure of the *fewest* open sets needed to preserve the topological structure of the elements of the product.
Let’s take one of the elements of the product, (Ti,τ1), and look at one of its open sets, O. What does it mean to preserve the structure of the open-set O in the product space (P,π)? What that means is: if we take the *inverse* pi-1 of the projection function pi, then pi-1(O) must be an open set in P.
If we take the union of the inverse projections of all of the open sets of all of the elements of the product, and result will be the open sets of the product space P.
To get an intuition of what this means, let’s think about two topologies that are also metric spaces: a circle on a plane, and a line.
For the circle, the primary open balls (that is, the open balls that really define the structure) are the circles of increasing radius centered on the same point. For the line, the open balls are the open segments of the line. The figure below illustrates these two spaces, and a sampling of the open sets on those spaces.
What’s the product of the two? It’s a cylindrical space. What do its open sets look like? Let’s look at it from the point of view of the circular space. Open sets on the cylindrical product space have to include anything whose projection onto the circle produces one of the circular open sets. What kind of shape in a cylinder will produce a circular projection? A cylinder. The open sets are the cylinders: the shapes formed by “stretching” the open sets of the circle using the other element of the product. Similarly, the open sets of the line space are line segments: what projection from the cylinder onto the line will produce an open segment? A finite length cylindrical section whose diameter is the size of the full circle.
Since open sets are closed over intersection, we can easily show that any cylidrical space centered on the same point as the circle is an open set. So, for example, the following figure illustrates our two original metrizable topological spaces, and the product space, with one of the open sets formed by intersecting the infinite cylinder that comes from a circular open set with the finite length cylinder that comes from a open segment on the line.
Just to reassure you that your normal intuitions still work: For the topological space consisting of a line metrizable as the metric space ℜ, the products of ℜ with itself *n* times is exactly the euclidean n-space, ℜn.