Why isn't the Jordan Curve Theorem axiomatic?

Question

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a plane simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. (Wikipedia)

That a continuous closed non-self-intersecting loop would divide a normal euclidean plane into two regions which can be joined only by a path crossing the loop would have been axiomatic, or so I would have thought. It is inherent in the idea of a closed non-intersecting curve that it defines an inside and an outside.

Are the difficulties in proving the theorem present in a simple euclidean plane, or only when it is applied to non-euclidean geometries or higher dimensions?

Answer 1

Some comments in no particular order:

• You can have closed non-intersecting curves on the sphere. This will divide the sphere into two halves, but it's not clear which one is the "inside" and which is the "outside." Also, you can have a closed non-intersecting curve in the circle. This doesn't divide the circle in any meaningful way; indeed, such curves will always cover the entire circle. And of course, you can have a closed non-intersecting curve in real $3$-space; this won't disconnect the space.
• Usually we base math on the principles of set theory (or perhaps $\infty$-groupoid theory) or something like that. I don't know of any genuinely successful attempts to found mathematics on geometry. So from this point of view, we can't really choose our axioms of geometry. They're meant to flow from the definitions, and certain basic principles about the universe of sets ($\infty$-groupoids, whatever.)
• It might be possible to characterize the plane as the unique topological manifold satisfying the Jordan curve theorem. (And this is a good question btw, +1).

Answer 2

You write "it is inherent in the idea of a closed non-intersecting curve that it defines an inside and an outside".

Mathematics makes progress by questioning what seems to be inherent. For example, although Euclid included in his axioms the famous parallel postulate, which perhaps seemed inherent at the time, from very early in the history of mathematics people were not always willing to accept the parallel postulate as an axiom. There were many (failed) attempts to prove the parallel postulate from the other axioms of Euclid. Eventually, after about 2 millennia it was discovered that the parallel postulate was not inherent at all, in fact it is independent of the other axioms, because one can construct the hyperbolic plane which satisfies all of Euclid's axioms except the parallel postulate. From this we can conclude that it is good to include the parallel postulate as an axiom of Euclidean geometry.

The problem with simply deciding that an "inherent" property should be an axiom is that it is not good to accept unwarranted assumptions. What if the JCT turned out to be false? What if it turned out that there existed a Jordan curve in the plane whose complement had 3 or more connected components, contrary to the conclusion of the JCT that there are exactly 2 connected components? It's hard to know ahead of time whether that's how things will turn out. If one thought, back in the 19th century, that the JCT is inherent in our understanding of the Euclidean plane, to make progress one should still question that thought.

One may raise a valid question about any seemingly "inherent" property: Is it really always true? Or is there a counterexample?

In the case of the parallel postulate, there exists a counterexample, i.e. a mathematical object satisfying all axioms of Euclidean geometry except for the parallel postulate, namely the hyperbolic plane.

In the case of the JCT, it is always true: the complement of every Jordan curve in the Euclidean plane has 2 connected components (one bounded, one unbounded). Nowadays this theorem is taught in a first year algebraic topology course, requiring a lot of carefully developed topological machinery. From this we can conclude that it is bad to include the JCT theorem as an axiom of Euclidean geometry.

Appendix 1: I'm beginning to think that you might find the Kline Sphere Characterization Theorem to be interesting and relevant to your question. What that theorem says is this:

• If $X$ is a compact, connected, locally connected metric space with more than one point, if no pair of points separates $X$, but if every simple closed curve separates $X$, then $X$ is a homeomorphic to the 2-sphere.

Here is an immediate corollary:

• If $X$ is a noncompact, connected, locally connected, nonempty metric space, if no single point separates $X$ and no pair of points separates $X$, if every simple closed curve separates $X$, and if every closed subset homeomorphic to $\mathbb{R}$ separates $X$, then $X$ is homeomorphic to the plane.

So one can conclude that the JCT is indeed inherent in the topological concept of "being homeomorphic to the plane", rather than to the Euclidean concept of "being a Euclidean plane".

Appendix 2: Regarding axiom systems, you can think of almost any definition as an axiom system, and sometimes that is enforced by the language. For instance, sometimes you will hear one speak about the definition of a group as the "axioms for a group": a set, with a binary operation satisfying the associative law, and with an identity element satisfying the inverse law. When you study group theory, every property of a group should be viewed through the lens of whether it is "inherent" or "not inherent".

As a simple example: is the commutative law an inherent property of groups? Translation: Is every group commutative, or on the other hand do there exist noncommutative groups?

This is just one example, but this frame of mind is a great way to study and learn new mathematics.

Answer 3

You are correct in a very real sense. If you look at Hilbert's "Foundation of Geometry" you will discover there is a betweenness axiom characterized by a line passing across a triangle. The triangle acts to select a line segment all of whose points lie between the two points of intersection with the triangle.

That this does not translate to foundational views reflects the emphasis upon arithmetization of mathematics that had been popular at the time foundational studies arose. Frege, however, had retracted his logicism and proposed that all mathematics is based upon geometry. In this regard, it is of interest that the sixteen basic Boolean functions relate to one another as a finite affine plane. You can see the significant pattern leading to this recognition by studying negations and de Morgan conjugations as involutions over the truth tables. Such facts, however, are irrelevant to those who promote the received view based upon arithmetization and the import of Goedel's theorems.