Whence topology: the swamp
One is lured into the field of topology by the beauty of the previous problems. But serious attention to the subject reveals a swamp of difficulty with definitions.
For instance, Euler proved that \(V - E + F = 2\) holds for polyhedra. Then over the next few decades, various mathematicians found “exceptions” (i.e. counterexamples). From one point of view, the problem comes down to what shapes count as polyhedra and which do not. The long process of working out a definition of such a seemingly basic concept as polyhedron illustrates how careful one must be.1
Say we allow that polyhedra are piecewise-linear surfaces. But then this raises the question of what a surface is. A geometric object with two local degrees of freedom? What then about degenerate objects like a nodal surface? Do we just throw these out of consideration? Who doesn’t love conic sections as sections of a conic?
These sorts of problems obviously get worse the more dimensions you have to deal with. So you would hope that things would be simpler down in lower dimensions. They don’t improve by much. Zero-dimensional objects, or just sets, are really not so bad.2 But, down a dimension from point sets, the empty set has to be ruled out in your definitions, or else you will be constantly hounded by the possibility of an empty object. This is a minor irritation. And up a dimension from points, things are almost as bad as with surfaces. For instance, what is a curve? You might think the answer obvious, but Whyburn authored an entire essay on the subject. Those examples are but few of the difficulties of definition in topology. This could leave a novice unsure that the best definitions were made. This unsurety is why I cast this difficulty as a swamp. I would call it a jungle, but swamps are closer by to me, and in topology the difficulties are right at your feet.
Another swampy aspect of topology is how difficult the going can be even to achieve some very trivial-seeming results. For instance, after having made a decent working definition of “curve,” you would like to prove the following obvious result:
Theorem (Jordan)
Every simple closed curve in the plane separates the plane into two components, one bounded and one unbounded.
This should be easy. But the simplest proofs appeal to heavy machinery like complex analysis or algebraic topology. The shortest elementary proofs run several pages long. Why should something so innocuous be so hard to prove?
But these are minor concerns compared with topology’s monsters, about which more next time.