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.
I was just wondering, what level of math should one be at to understand the math topics on this site? I want to learn it but it is hard to find good books, are there some you suggest? I’m in college level math, my major is programming. But it seems all they want me to learn is how to do math in different bases, and graphing. They move a little too slow for me, so I want to just start teaching myself, but don’t know where to start.
My intention is that the stuff on the site should be at least reasonably comprehensible (ie, follow the basics if not every detail) to anyone whose taken college level math.
Different topics of course vary. Things like topology, you wouldn’t expect to get all of the details before advanced undergrad/early grad level math classes. But I hope I make the basics understandable enough that you can see what the point of it is, if not exactly how it all works.
Since you have already spent so much time on category theory (kudos!), it seems very appropriate to tackle your exploration of topology with the language of functors! For example, in this post, stating/showing the product topology is uniquely determined by the universal mapping property of the product gives an easily comprehensible and wonderfully illuminating example of the Yoneda lemma in action. Moreover, it even helps understand better where this seemingly ad hoc construction comes from.
Ah, the joy of only vaguely comprehending something…
My favorite way of making the product topology less mysterious is using an indirect characterization:: the product topology on X x Y is the coarsest topology (i.e. the one with the least amount of open sets) for which the projection functions pi_X : X x Y -> X and pi_Y : X x Y -> Y are continuous. The idea is that these guys are the “god-given” functions involving the set-wise product of X and Y, so they damn well better be continuous in whatever topology we put on X x Y! And we shouldn’t add any needless open sets since pi_X and pi_Y are the _only_ god-given functions involving X x Y.
Have you ever noticed that cylindrical spaces are common to biological entities?
[A speculation of treating biology as a bio-physical-chemistry mathematical object – topology or as a game].
1 – For example, the yearly growth rings of a tree are very much like your diagram above.
In fact, tensor calculus appears to be able to correlate very well with the volume of a yearly tree growth ring from the upper most root to the lower most limb.
Tensor calculus can only approximate the ball-like volume of a planet circling the sun just as Archimedes polygons could only approximate the area of a circle. A more accurate calculus for solar system physics would appear to require multiple rotating 3-balls along helical geodesics with relative cylindrical planetary orbits as the sun moves about the galactic core. One might interpret this as a cylindrical integration along a string-like geodesic of planetary orbital loop differentials.
2 – Another biological example is the sequential gastrointestinal tract of most chordates which is a cylinder that might be mathematically interpreted as a genus-1 torus.
One might be able to make the case that the GI tract preserves Porifera [sponge] ancestorial evolution. The holdfast has been transformed into a sphincter. The colon most closely functions as a sponge with the intestine functioning in reverse. The stomach digests, the esophagus transports from the oral cavity which initiates digestion with chewing. Abstractly the GI tract functions something like a biological black hole, preserving information [Bekenstein] through transforming matter into smaller matter or energy. [This energy or smaller matter can be reconstiuted to matter within body organs.] Biological markers could be used to track individual atoms in this process.
If biological objects and relativity objects were interpreted as mathematical objects as
are quantum objects, then it might be easier to find a GUT or TOE for QM and GR.
I have not yet seen any game theory examples using tensor calculus or topologies but since vectors and matrices are common in game theory, such methods should be possible.
I had differential equations, and some linear algebra, but not topology. This article was reasonably understandable.
Now, as we do in physics all the time, if we extrapolate this single data point to a trend line…