Two pieces, in every dimension

The crossing rule from the plane transfers to space unchanged, and nothing about it needs restating. Fire a ray from a point, count how many times it meets the closed surface, and the parity says which side of it the point is on.

What changes is what has to be crossed. In the plane the separator is a curve; in space it is a surface, and a curve in space separates nothing whatever. That difference is not a detail. It is the general theorem, and the general theorem’s corollary is the statement that gives the word “dimension” its meaning.

The rule survives the extra dimension intact #

The tube above is a genuine closed surface — 1,728 triangles glued edge to edge with no boundary — and the count is taken against those triangles rather than against the drawing. Every claim in the caption is a claim about the mesh.

Two hundred random directions give the same parity and different counts. That is exactly what happened in the plane, and for the same reason: moving the ray continuously changes the count only when the ray becomes tangent to the surface, and at such a moment two crossings appear or vanish together. Parity survives what counting does not.

The choice of a knotted tube is deliberate. A reader can settle inside from outside for a sphere by looking, and for a torus with some effort; on a trefoil tube nobody can. So the figure is in the same position as the spiral corridor in the plane — the eye is out of work, and what remains is the rule.

A closed curve in space separates nothing #

Take a loop of string in space, knotted or not, and ask what its complement looks like.

Any two points not on the curve can be joined by a path missing it. The reason is that a path can go round, and in three dimensions there is always room: the curve is one-dimensional, a generic path is one-dimensional, and two one-dimensional things in three-space can be pushed apart. In the plane the same argument fails because a path and a curve are both one-dimensional in a two-dimensional space, and there is no room to slide. The same accounting explains why a knot laid flat has to be drawn with breaks: a curve in space projects to a curve in the plane with crossings, and the crossings are the information the extra dimension was carrying.

So the complement of a knot in space is connected. It is not simply connected — a loop around the string carries a number saying how many times it goes round, and that number cannot change under any deformation avoiding the string. Connected but not simply connected is the whole content of knot theory’s first invariants, and it is a different statement from separation.

The counting to keep straight is a subtraction. A separator in $n$-space has dimension $n-1$: a point separates a line, a curve separates a plane, a surface separates space. Anything of lower dimension leaves the complement in one piece. The Jordan curve theorem is the case $n = 2$ of that pattern, and the reason it looks like a special fact is that it is the smallest case in which the separator is interesting to draw.

The general statement, and what proves it #

Jordan–Brouwer separation. If a subset of $\mathbb{R}^n$ is homeomorphic to the sphere $S^{n-1}$, its complement has exactly two connected components — one bounded, one not — and the subset is the boundary of each.

There is no elementary proof. The parity argument works on a triangulated surface and gives the theorem for polyhedral spheres, exactly as the ray argument gave it for polygons; the general statement covers embeddings with no triangulation available, and it needs an invariant that is defined without counting anything.

Two standard machines do it. Degree theory assigns to a map of spheres an integer counting how many times it wraps, computed without reference to any particular point’s preimage, and the separation follows from the degree of the map that sends a point of the complement to the direction of the nearest point of the sphere. Alexander duality relates the homology of a compact subset of the sphere to that of its complement, and reading off the zeroth homology of the complement gives the number of pieces directly: two.

Both are the same move made twice — replace count the crossings with evaluate an invariant — and it is the move the winding number makes in the plane. A ray’s crossings are a property of the ray; a degree is a property of the map, and only the second survives a curve with no tangents.

The degree route is worth a sentence more, because it is the one that shows why exactly two pieces appear rather than some other number. For a point $p$ off the sphere, the map sending each point $x$ of the sphere to the unit vector from $p$ toward $x$ is a map from $S^{n-1}$ to $S^{n-1}$, so it has a degree. That degree is unchanged when $p$ moves without crossing the sphere, so it is constant on each component of the complement. It is zero when $p$ is far away, because the whole sphere is then seen inside a small cone and the map misses most directions. And the theorem is that it takes exactly one other value, once, so there are exactly two components. The parity count above is that degree taken modulo two, on a surface where the preimages can be listed.

Why dimension is a property and not a description #

Now the corollary, which is the reason the theorem is worth having rather than merely believing.

Invariance of domain. An injective continuous map from an open subset of $\mathbb{R}^n$ into $\mathbb{R}^n$ has open image, and is a homeomorphism onto it.

Its consequence is that $\mathbb{R}^m$ and $\mathbb{R}^n$ are homeomorphic only when $m = n$. That is a statement nobody doubts and one that resisted proof for years, and the reason for both is that the obvious argument is wrong.

The history is short and instructive. Cantor’s bijection between a line and a square landed in 1878 and was received as a scandal precisely because it seemed to make dimension meaningless; Cantor himself wrote that he did not believe it. Dedekind’s reply named the missing condition at once — the bijection is not continuous — and conjectured that no continuous one with a continuous inverse exists. Peano’s space-filling curve of 1890 removed the other half of the easy repair by exhibiting a continuous surjection from the interval onto the square. What remained was exactly the conjunction, and it took Brouwer until 1912.

The obvious argument is about cardinality, and it fails: there is a bijection between a line and a square, so counting points cannot distinguish dimensions. The next argument is about continuous surjections, and it fails too: there is a continuous map of the interval onto the whole square. What has to be ruled out is a map that is both continuous and injective, with a continuous inverse, and that is precisely what invariance of domain rules out.

The two smallest cases can be settled by hand, and the figure settles them. A homeomorphism sends a removed point to a removed point and preserves the number of connected pieces. Remove a point from a line and there are two; remove a point from a plane and there is one. So no line is homeomorphic to a plane. Remove a line from a plane and there are two; remove a line from space and there is one. So no plane is homeomorphic to space.

That is the whole argument in those two cases, and it is honest. What it is not is general: distinguishing $\mathbb{R}^4$ from $\mathbb{R}^5$ by this route needs a notion of “how connected” that goes beyond counting pieces, and the notion is homotopy or homology. The pattern the two panels show — remove something of dimension $k$ from $\mathbb{R}^n$ and the complement’s connectivity depends on $n - k$ — is the shadow of the general fact.

What the counting rule costs, and where it stops being an algorithm #

There is a practical remark hiding in the hero figure, and it is the reason the ray rule is the one every graphics system uses in three dimensions as well as two.

Deciding whether a point is inside a closed surface of $m$ triangles costs $m$ ray–triangle tests and nothing else. No preprocessing, no data structure, no orientation to compute, no assumption that the surface is convex or connected or nicely shaped. The whole implementation is the intersection formula and a parity bit, and it behaves identically on a sphere and on a knotted tube.

What it consults, and this is the whole of its fragility, is one tolerance: whether the ray is parallel to a triangle’s plane. Every other decision is a sign test. A ray that grazes an edge is the degenerate case, and it is handled the way the plane version handles a ray through a vertex — by a half-open convention that counts the boundary of a triangle on exactly one side. Getting that convention wrong produces an error of exactly one crossing, which is an error of the whole answer, and it is the classic bug in this classic routine.

The theorem is what says the routine is answering a question at all. Without separation there is no “inside” for the parity to report, and on a surface with a boundary — a sheet rather than a closed surface — the count changes with the ray and the routine returns noise.

What the theorem does not say #

It says two pieces. It does not say what they look like, and the previous rung is where the difference bites.

In the plane the bounded piece is a disc, by Schoenflies. In space the bounded piece of a topological sphere’s complement need not be a ball. On a surface that is not a sphere the two-piece statement fails outright rather than weakening: a curve down the middle of a Möbius band leaves one piece rather than two, and the reason is the same one — separation is a statement about the ambient space as much as about the thing embedded in it. The separation statement generalises to every dimension and the straightening statement does not survive the first one it is offered.

That failure is not a technicality of the general case, either. It has a name and a construction, and both are the rung above: Alexander’s horned sphere separates space into two pieces exactly as promised, its inside is an ordinary ball, and its outside is not. The horned sphere is a counterexample to a statement this theorem never made, and one of the reasons the two statements are usually confused is that in the plane they are both true and one proof covers both.

The extra hypothesis that repairs it is local flatness: if the embedded sphere looks locally like a flat plane sitting inside space, the straightening does hold in every dimension, a result of Brown and of Mazur and Morse from around 1960. The horned sphere fails local flatness at exactly the points its horns converge to — a set of measure zero, and enough.

What the pictures cannot show #

The count is against a mesh, and the mesh is not drawn. Rendering 1,728 triangles would be ninety kilobytes of markup for a picture of the same tube; the tube is drawn as its own core stroked to its own width, and the crossings reported were computed against the triangles.

Two hundred directions are not every direction. The parity being the same for all of them is evidence, and the theorem is what makes it a fact. A direction exactly tangent to the surface would give a count that is neither, and the sampling avoids that case rather than resolving it.

The flood fills are on grids. Four panels report piece counts at a finite resolution, and a fine enough obstruction could in principle be missed by a coarse grid. Nothing here rules that out; what rules it out is the argument in the text.

The tube is knotted and the theorem does not care. Nothing in the separation statement is affected by how the surface sits in space; the knotting is there to defeat the reader’s eye rather than to be a hypothesis, and a round sphere would give the same parities with less trouble and less conviction.

And no figure here reaches four dimensions, which is where the theorem starts saying something a reader has no intuition to check it against. The two panels that settle low cases are the whole of what a picture contributes.

Where the ladder goes next #

The rung above is the horned sphere itself: what a surface in space can do that a curve in the plane cannot, why the outside of a topological ball can fail to be simply connected, and how a tree of clasped horns arranges it. It is the counterexample this rung has been pointing at twice.

Sideways: the linking number is the invariant that detects a loop’s failure to be pullable off, and it is what will make the horned sphere’s outside measurable rather than merely alarming. And surfaces in space are classified by two numbers, which is the statement that says what the separating objects in three dimensions can be.

What is worth carrying away #

A rule and the theorem it belongs to are different things, and the crossing rule is a good place to see the gap.

The rule is one line, works in every dimension, and proves nothing beyond the case it can be run on. Even the polygon it was written for turns out to be a special case of the objects the theorem is stated about. The theorem covers surfaces the rule cannot be run on at all, and its proof replaces the count with an invariant that is defined without any particular ray. Generalising an argument usually means finding the thing it was counting and computing that instead — and the payment for making the count go away is a corollary the count could never have reached, which here is the fact that dimension is a property of a space rather than a feature of how it was written down.