Chapter V

SEPARATION THEOREMS

§ 1. CHAINS ON A GRATING

1. It is convenient in the following chapters to use X^{2} to denote either R^{2} (the open plane) or Z^{2} (the closed plane). In any one argument X^{2} denotes, of course, one definite space.

A rectangular grating, G, in the open or closed plane, is formed by drawing a finite number of segments across a square, parallel to its sides. The lines go right across the square: divisions like that in Fig. 22(b) are not allowed. The grating in which the square is left undivided is not excluded.

Fig. 22

For convenience we suppose that the original square (the frame of the grating) has its centre at the origin and its sides parallel to the coordinate axes, so that the words “horizontal”, “above”, “below”, etc. may be used in an obvious sense.

The 2-cells of G are (1) the closures of the rectangular domains into which the interior of the frame is divided, and (2) the closure of the exterior of the frame (including the point at infinity if there is one). The edges, or 1-ceIls, are the sides of the 2-cells of finite diameter, and the vertices, or 0-cells, are their corners.

Thus all cells are bounded closed sets of points, except the 2-cell outside the frame, which is closed but not bounded.†

Every edge of the grating evidently belongs to just two 2-cells, which lie on opposite sides of it.

Theorem 1·1. If F_{1} and F_{2} are non-intersecting closed sets in Z^{2}, there exists a grating, G, no cell of which meets both F_{1} and F_{2}.

† To avoid confusion with another sense of “bounded”, used in this chapter, this will be called the infinite 2-cell; and all others finite cells.

At least one of the closed sets, say F_{1}, does not contain w, and therefore lies, for some α, in the interior of the set

E_{α} : [|ξ_{1}| ≤ α, |ξ_{2}| ≤ α].

The non-intersecting compact sets F_{2}E_{α} and F_{1} in R^{2} are at a positive distance δ apart. Let m be an integer exceeding 4α/δ. If the frontier of E_{α} is taken as frame, and m equidistant segments are drawn between, and parallel to, each pair of sides the result is evidently a grating with the required property.

In R^{2} the additional condition that em at least one of the closed sets is compact is required. The proof is then as before.

Fig. 23 Fig. 24

2. A k-chain, C^{k} (k = 0, 1, 2), on a grating G, is any set of k-cells of G. The sum (modulo 2) of two k-chains, C_{1}^{k} and C_{2}^{k}, is the set of k-cells that belong to one, but not both, of C_{1}^{k} and C_{2}^{k}. It is denoted by C_{1}^{k} + C_{2}^{k}. The complement CC^{2} of a 2-chain C^{2} is the set of 2-cells of G not belonging to C^{2}. Thus if Ω^{2} denotes the 2-chain containing all the 2-cells of G, CC^{2} = C^{2} + Ω^{2}.

Example. 1. In Fig. 23 the set of three shaded cells is a 2-chain, and the six thick edges form a 1-chain. In Fig. 24, if C_{1}^{2} and C_{2}^{2} are the 2-chains whose cells are those inside the rectangles abcd and pqrs, C_{1}^{2} + C_{2}^{2} is the set of shaded 2-cells. CC_{1}^{2} + CC_{2}^{2} is also the set of shaded 2-cells.

Addition of k-chains is commutative and associative, and

∑_{1}^{q} C_{i}^{k} = C_{1}^{k} + C_{2}^{k} + … + C_{q}^{k}

is the set of k-cells that belong to an odd number of the chains C_{i}^{k}. Clearly for any C^{k}, C^{k} + C^{k} = 0 (the “zero chain”), and hence the equation C_{1}^{k} + X^{k} = C_{2}^{k} is satisfied by X^{k} = C_{1}^{k} + C_{2}^{k} and no other k-chain. Thus the k-chains on G form a commutative group under the operation of addition modulo 2. Some of the consequences of this fact are considered in the final section of Chapter VI, but otherwise no use is made of the language or theorems of grouptheory.

The boundary, Ċ^{k}, of the k-chain C^{k} on G is (for k = 1, 2) the set of (k − 1)-cells of G that are contained in an odd number of k-cells of C^{k}. (The boundary of a 0-chain is not defined.)

Example. 2. The boundary of an edge is its pair of end-points. The boundary of the 1-chain in Fig. 23 is the set of vertices a, b, c, d. The boundary of the shaded 2-chain in Fig. 24 is the set of the thickened edges.

Theorem 2·1. (C_{1}^{k} + C_{2}^{k})˙ = Ċ_{1}^{k} + Ċ_{2}^{k} (k = 1, 2).

A (k − 1)-cell A^{k−1} belonging to n_{1} k-cells of C_{1}^{k} and n_{2} of C_{2}^{k} belongs to Ċ_{i}^{k} if n_{i} is odd, and therefore to

Ċ_{1}^{k} + Ċ_{2}^{k}

if n_{1} +n_{2} is odd. This is the condition that it belongs to (C_{1}^{k} +C_{2}^{k})˙ . By induction on the number of chains it follows that

(C_{1}^{k} + C_{2}^{k} + … C_{q}^{k})˙ = Ċ_{1}^{k} + Ċ_{2}^{k} + … + Ċ_{q}^{k}.

In particular the boundary of any k-chain is the sum, mod 2, of the boundaries of its k-cells.

Since Ω̇^{2} = 0 it follows from 2·1 that, for any C^{2},

(CC^{2})˙ = (C^{2} + Ω^{2})˙ = Ċ^{2}.

A k-cycle is, for k = 1, 2, a k-chain whose boundary is zero; a 0-cycle is a 0-chain with an even number of 0-cells. The sum mod 2 of any set of k-cycles is a k-cycle (by 2·1, or directly for k = 0).

Theorem 2·2. The boundary of any k-chain is a (k − 1)-cycle (k = 1, 2).

The boundary of a 1-cell is two vertices—a 0-cycle. The boundary of a 2-cell is a 1-chain in which every vertex belongs to just two edges, i.e. it is a 1-cycle. From these special cases the theorem follows by addition.

3. The k-chain C^{k} (a finite set whose members are k-cells) is to be distinguished from the union of its k-cells, a set of points denoted by |C^{k}| and called the locus of C^{k}, or the set of points covered by C^{k}. The functions, F, K (or ¯ ) and J are applicable to the locus |C^{k}| but not, of course, to C^{k} itself. To avoid clumsy constructions, a chain (instead of, more correctly, the locus of a chain) will often be said to contain a point, to lie in a set of points, or to meet a set or another chain; but this licence will not extend to symbolic relations.†

Clearly |C_{1}^{k} + C_{2}^{k}| ⊆ |C_{1}^{k}| ∪ |C_{2}^{k}| in all cases. The two sets are identical if, and only if, C_{1}^{k} and C_{2}^{k} have no common k-cell. (Thus in Example 1 above, |C_{1}^{k}|∪|C_{2}^{k}| is the whole closed region bounded by the polygon abyqrspx, |C_{1}^{k} + C_{2}^{k}| is the shaded region.) Note that whereas C|C^{2}| is an open set, |CC^{2}| is closed, and is in fact \overline{C|C^{2}|}.

A k-chain C^{k} is, by definition, connected if its locus |C^{k}| is connected. The maximal connected k-chains contained in any k-chain C^{k} are called the components of C^{k}. They have as loci the components of |C^{k}|.

Theorem 3·1. If K^{k} is a component of C^{k}, K̇^{k} is the part of Ċ^{k} in K^{k} (k > 0).

For if a cell A^{k−1} is contained in a k-cell of K^{k}, all the cells of C^{k} containing A^{k−1} are in K^{k}, and therefore A^{k−1} has the same parity in C^{k} and K^{k}.

The following simple deduction from 3·1 is one of the main links between the combinatorial theory of this section and the point-set notions of Part I.

Theorem 3·2. If x and y form the boundary of a 1-chain C^{1}, they are connected in |C^{1}|.

If not, the component of C^{1} containing x has, by 3·1, the single odd vertex x as its boundary, contrary to 2·2.

Theorem 3·3. For any C^{2}, F|C^{2}| = |Ċ^{2}|.

If x ∈ |Ċ^{2}| then

x ∈ |C^{2}| ∩ |CC^{2}| = |C^{2}| ∩ \overline{C|C^{2}|} = F|C^{2}|.

If x ∈ F|C^{2}|, at least one 2-cell containing x is in C^{2}, and one is in CC^{2}. Therefore a 2-cell of C^{2} and one of CC^{2} have an edge in common that contains x. Thus x ∈ |Ċ^{2}|.

† Thus we shall write “x lies in C^{k}”, but “x ∈ |C^{k}|”.

It follows from 3·3 that Ω^{2} and 0 are the only 2-cycles. Hence if Ċ^{2} is connected so also is C^{2}, for either C^{2} = Ω^{2} or else every component of C^{2} contains a non-null component of Ċ^{2}.

4. It is sometimes necessary to pass from one grating, G, to another, G^{∗}, by introducing additional lines; or by enlarging the frame, i.e. by extending all the lines to meet a larger square containing the original frame; or by a combination of both processes. Such a new grating G^{∗} is called a refinement of the old. (It is convenient to agree that G is a refinement of itself.) A common refinement can be formed for any two gratings G_{1} and G_{2}, by taking as frame any square, with sides parallel to the axes, containing the frames of G_{1}, and G_{2} and as cross-lines all the lines of G_{1} and G_{2}, suitably extended.

To each k-chain, C^{k}, on G corresponds the subdivided k-chain C_{∗}^{k} on G^{∗}, which is the sum of the k-chains into which the k-cells of C^{k} are subdivided. (0-chains are unaltered by subdivision: C_{∗}^{0} = C^{0}.) A subdivided chain has the same locus as its original: |C_{∗}^{k}| = |C^{k}|.

Theorem 4·1. (C_{1}^{k} + C_{2}^{k})_{∗} = C_{1}^{k}_{∗} + C_{2}^{k}_{∗}, (Ċ^{k})_{∗} = (C_{∗}^{k})˙ .

The first relation is clear from the definitions.

The effect of subdivision on an edge is to introduce new intermediate vertices, but since they each belong to two new edges the boundary of the resulting 1-chain is still the original two endpoints. Hence, by addition, (C_{∗}^{1})˙ = Ċ^{1} = (Ċ^{1})_{∗}. If C^{2} is any 2-chain,

|(C_{∗}^{2})˙ | = F|C_{∗}^{2}| = F|C^{2}| = |Ċ^{2}| = |(Ċ^{2})_{∗}|.

Thus (C_{∗}^{2})˙ and (Ċ^{2})_{∗} are chains on G^{∗} with the same locus, and therefore identical.

Corollary. C_{∗}^{k} is a k-cycle if, and only if, C^{k} a k-cycle.

5. Theorem 5·1. Every 1-cycle, on a rectangular grating in Z^{2} or R^{2} is the boundary of just two 2-chains.

This theorem is called the Fundamental Lemma because, in spite of its simplicity, it is the geometrical kernel of the whole of the theory in this Chapter. It is a property of the plane and sphere, and distinguishes them topologically from all other surfaces. For example, if the surface of a solid ring is divided into squares, a meridian circle, though a “1-cycle”, is not the boundary of any 2-chain on the surface (Fig. 25).

The proof is by induction on the number of lines drawn across the original square to make the grating. On the grating consisting of the frame alone, the only 1-cycles are the null-set, which bounds two 2-chains (the zero-chain and Ω^{2}); and the square itself, which bounds the two 2-cells possessed by this grating.

Fig. 25 Fig. 26

Let the given grating, G_{1}, be formed from a grating G_{0}, for which the theorem may be assumed true, by the addition of a line λ across the square, which we may suppose to be parallel to the ξ_{1}-axis. Let Γ^{1} be the given 1-cycle on G_{1}. We denote by C^{2} the sum of the 2-cells of G_{1}, whose lower edges lie in the line λ and belong to Γ^{1}. The 1-cycle Γ^{1} + Ċ^{2} therefore contains no edge in λ. It follows that it is the subdivided form of a 1-cycle Γ_{0}^{1} on G_{0}, for Γ^{1} + Ċ^{2} contains no horizontal edge at a vertex, x, of λ, and therefore, since it is a cycle, contains both or neither of the vertical edges at x, which together make up an edge of G_{0}.

By hypothesis there is a 2-chain C_{0}^{2}. on G_{0} such that Γ_{0}^{1} = Ċ_{0}^{2}, and if C_{1}^{2} is the subdivided form of C_{0}^{2} on G_{1}, Ċ_{1}^{2} = Γ^{1}+Ċ^{2}. Hence

(C_{1}^{2} + C^{2})˙ = (Γ^{1} + Ċ^{2}) + Ċ^{2} = Γ^{1}.

The residual 2-chain C(C_{1}^{2} + C^{2}) also has boundary Γ^{1}.

It has thus been shewn that Γ^{1} bounds at least two 2-chains. No 1-cycle bounds more than two 2-chains. For if Ċ_{a}^{2} = Ċ_{b}^{2}, (C_{a}^{2} + C_{b}^{2})˙ = 0 and therefore C_{a}^{2} + C_{b}^{2} is 0 or Ω^{2}, i.e. C_{b}^{2} = C_{a}^{2} or CC_{a}^{2}.

6. The convention is now introduced that if C^{k} is a k-chain on a grating G, the corresponding subdivided chain on any refinement G^{∗} shall be denoted by the same symbol C^{k}, with the addition when necessary of the words “on G^{∗}”. Thus if C_{1}^{k} and C_{2}^{k} are chains on the gratings G_{1} and G_{2}, and G^{∗} is any common refinement of G_{1} and G_{2}, “C_{1}^{k} + C_{2}^{k} (on G^{∗})” will denote (C_{1}^{k})_{∗} + (C_{2}^{k})_{∗}. No ambiguity can arise from the various orders in which a combination of subdivision, addition, and taking the boundary can be carried out, since, by 2·1 and 4·1 the result is always the same.

Let G be an open set and Γ^{k} a k-cycle on G in G. The cycle Γ^{k} bounds in G (Γ^{k} ∼ 0 in G) if there is, on some refinement G^{∗} of G, a (k + 1)-chain C^{(}k + 1) such that |C^{k+1}| ⊆ G and Ċ^{k+1} = Γ^{k} on G^{∗}. “Γ^{k} is non-bounding in G” means that |Γ^{k}| ⊆ G but Γ^{k} does not bound in G.

Examples. 1. If A^{2} is a 2-cell of G, Ȧ^{2} ∼ 0 in X^{2} on G itself; but if a ∈ J |A^{2}| and a ∈ C|A^{2}| does not bound in X^{2} − (a ∪ b) because (as will be shewn below) every 2-chain bounded by Ȧ^{2} contains a or b.

2. If x and y are finite points of the plane, (x) + (y) ∼ 0 in X^{2}, for if G is a grating having x and y as vertices, (x) + (y) bounds either an edge or an ⌊-shaped 1-chain on G.

It is often unnecessary to make explicit reference to a grating in the enunciation of theorems. Thus the opening phrase of the following theorem (6·1) means “If Γ_{i}^{k} is a k-cycle in G on a grating G_{i}” or still more explicitly “If Γ_{i}^{k} is a k-cycle on G_{i} and |Γ_{i}^{k}| ⊆ G”. Similarly if x and y are any points, “(x) + (y) ∼ 0 in G” means “(x) + (y), regarded as a 0-cycle on some grating G having x and y as vertices, bounds in G”.

Theorem 6·1. If Γ_{i}^{k} is a cycle in G and Γ_{i}^{k} ∼ 0 in G, for

i = 1, 2, …, q,

then ∑_{1}^{q} Γ_{i}^{k} ∼ 0 in G.

If Γ_{i}^{k} = Ċ_{i}^{k+1} on G_{i}^{∗}, and if G_{0} is a common refinement of all the G_{i}^{∗}, then

∑_{1}^{q} Γ_{i}^{k} = ( ∑_{1}^{q} C_{i}^{k+1})˙

on G_{0}.

Corollary. If Γ^{k} is non-bounding in G, at least one of its components is non-bounding in G, for since the components do not meet each other their (mod 2) sum is Γ^{k}.

Theorem 6·2. If CG is connected every 1-cycle in G bounds in G. For if CG met both the 2-chains bounded by Γ^{1} it would meet

the common frontier, |Γ^{1}|, of their loci.

The relation Γ_{1}^{k} + Γ_{2}^{k} ∼ 0 in G is also written Γ_{1}^{k} ∼ Γ_{2}^{k} in G, and read “Γ_{1}^{k} is homologous to Γ_{2}^{k} in G”. Homology in G is an equivalence relation, for it is clearly symmetrical; it is reflexive since Γ^{k} + Γ^{k} = 0; and it is transitive, for if Γ_{1}^{k} ∼ Γ_{2}^{k} and Γ_{2}^{k} ∼ Γ_{3}^{k} in G, then

Γ_{1}^{k} + Γ_{3}^{k} = (Γ_{1}^{k} + Γ_{2}^{k}) + (Γ_{1}^{k} + Γ_{2}^{k}) ∼ 0 in G.

Exercise. State and prove 6·1 without making use of the convention introduced at the beginning of the paragraph.

Theorem 6·3. A necessary and sufficient condition that the finite points x and y be connected in an open set G is that (x) + (y) ∼ 0 in G.

Fig. 27

Necessary. By hypothesis x and y belong to the same component, D, of G. Let E_{x}, be the set of all points z such that (x) ∼ (z) in D. E_{x} is open, for if z ∈ E_{x} every point z′ in a sufficiently small neighbourhood of z is connected to z by either one or two segments in D. The sum, mod 2, of this chain and the one which (by hypothesis) is bounded by x and z is a 1-chain in D bounded by x and z′, i.e. z′ ∈ E_{x}. D − E_{x} is open, for, as before, if x and a point z′, in a small neighbourhood of z of D − E_{x}, were together the boundary of a 1-chain in D it would follow that x and z bound in D, contrary to the definition of D − E_{x}. Since D is connected one of the sets E_{x} and D − E_{x} must be null; and since x ∈ E_{x}, D − E_{x} = 0. Therefore y ∈ E_{x}.

Sufficient. This follows immediately from 3·2.

Theorem 6·4. If x and y do not lie on |Γ_{1}^{1}| or |Γ_{2}^{1}|, at least one of the 1-cycles Γ_{1}^{1}, Γ_{2}^{1}, Γ_{1}^{1} + Γ_{2}^{1} bounds in X^{2} − (x ∪ y).

Let C_{i}^{2} be the 2-chain, on G_{i} say, that contains x and is bounded by Γ_{i}^{1} (i = 1, 2). If, for i = 1 or 2, y is also in |C_{i}^{2}| then Γ_{i}^{1} = (CC_{i}^{2})˙ ∼ 0 in X^{2} − (x ∪ y). If y is in neither |C_{1}^{2}| nor |C_{2}^{2}| it is not in |C_{1}^{2} + C_{2}^{2}|. Since every cell containing x belongs to both C_{1}^{2} and C_{2}^{2}, x is not in |C_{1}^{2} + C_{2}^{2}|. Thus

Γ_{1}^{1} + Γ_{2}^{1} = (C_{1}^{2} + C_{2}^{2})˙ ∼ 0 in X^{2} − (x ∪ y).

7. Change of notation. It has so far been possible to discuss the properties of 0-, 1- and 2-chains to some extent together, but from now on their parts in the theory are different. To avoid a multiplicity of suffixes it is convenient to adopt new notations. We use

A for a 2-cell, K for a 2-chain;

α for a 1-cell, γ for a 1-cycle, κ for a general 1-chain;

italic letters x, y, a, b, etc. for vertices.

General 0-chains are not often referred to, and are then denoted, as before, by C^{0}, 0-cycles by Γ^{0}.

The 0-chain (x) is always to have its brackets, e.g. in (x) ∼ (y). This prevents the possibility of confusion between the 0-chain (x) + (y) and the vector sum of points x + y.

8. Two 1-chains, κ_{1} and κ_{2}, on a grating have general intersection if they cross at every common vertex, that is to say if (i) no common vertex lies on the outer frame of the grating, and (ii) at each common vertex the two edges parallel to ξ_{1} = 0 belong to one 1-chain, and the other two edges to the other.

Theorem 8·1. If a 1-cycle γ and a 1-chain κ with boundary (x) + (y) have general intersection, γ ∼ 0 in X^{2} − (x ∪ y) if, and only if, the number, n, of crossings is even.

Let κ_{1} and κ_{2} be the subchains of κ in K_{1} and K_{2}, the 2-chains bounded by γ. Each common vertex of κ and γ belongs to just one edge of κ_{1}, and therefore κ̇_{1} contains all these vertices. If n is odd, the 0-cycle κ̇_{1} cannot consist precisely of this odd set of vertices, and therefore κ_{1} contains one of the vertices x, y—the only other available odd vertices—and κ_{2} contains the other. If n is even, κ̇_{1}, contains both or neither of x and y, and κ_{2} neither or both, accordingly.

Corollary. Two 1-cycles, γ_{1} and γ_{2}, with general intersection have an even number of crossings.

Theorem 8·2. If, for i = 1, 2, κ_{i} is a 1-chain with boundary (x_{i}) + (y_{i}); if κ_{1} and κ_{2} have general intersection, with an odd number of crossings; and if the continuum F contains x_{1} and y_{1} but does not meet κ_{2}; then F ∪ |κ_{1}| separates x_{2} and y_{2}.

If not let κ_{3} bea 1-chain in C(F ∪|κ_{1}|) with boundary (x_{2})+(y_{2}). The 1-cycle κ_{2} + κ_{3} has general intersection with κ_{1}, the crossings being the same as those of κ_{1} and κ_{2} and therefore odd in number. Therefore x_{1} and y_{1} belong to different 2-chains bounded by κ_{2}+κ_{3}. But this is impossible, for they are joined by the continuum F, meeting neither κ_{2} nor κ_{3}.

These theorems can be made the basis of alternative proofs of the separation theorems that occupy the rest of this chapter. More general forms of 8·1 and 8·2 are proved in Chapter VII (8·5 Corollary).

§ 2. ALEXANDER’S LEMMA AND JORDAN’S THEOREM

9. A theorem (“Alexander’s Lemma”)^{(16)} will now be proved which gives an answer to a question that often arises in the course of this chapter. Suppose it is given that two points can be joined by a path not meeting a closed set F_{1}, and also by a path not meeting a closed set F_{2}; in what circumstances can they be joined by a path meeting neither F_{1} nor F_{2}? Or, in terms of open sets, given that (x) ∼ (y) in G_{1} and in G_{2}, in what circumstances does (x) ∼ (y) in G_{1}G_{2}?

The lemma can be stated in two ways, which are easily seen to be equivalent, by taking G_{1} and G_{2} in the first version to be the complements of F_{1} and F_{2} in the second.

Theorem 9·1·1. Let G_{1} and G_{2} be open sets in the closed plane Z^{2}. If the points x and y bound the 1-chains κ_{1} in G_{1} and κ_{2} in G_{2}, and if κ_{1} + κ_{2} in G_{1} ∪ G_{2}, then (x) ∼ (y) in G_{1}G_{2}.

Theorem 9·1·2. Let F_{1} and F_{2} be closed sets in the closed plane Z^{2}. If the points x and y bound the 1-chains κ_{i} not meeting F_{i} (for i = 1, 2) and if κ_{1} + κ_{2} ∼ 0 in Z^{2} − F_{1}F_{2} then x and y are not separated by F_{1} ∪ F_{2}.

We shall prove the second version, 9·1·2, which, in spite of its less direct form, is more readily applicable in most cases.

By hypothesis, one of the 2-chains bounded by κ_{1} + κ_{2} say K on a grating G, does not meet F_{1}F_{2}, and hence |K|F_{1} and |K|F_{2} do not meet. Let G^{∗} be a refinement of G such that no cell meets both |K|F_{1} and |K|F_{2}, and let K_{1} be the set of 2-cells of K^{∗} (= K as subdivided on G^{∗}) that meet F_{1}. Consider the 1-chain κ_{0} = κ_{2} +K̇_{1} on G^{∗}. (In Fig. 28 K_{1} is horizontally shaded, and κ_{0} is the thickened lines.) Since K̇_{1} is a cycle κ̇_{0} = κ̇_{2}, i.e. κ_{0} is bounded by x and y.

Fig. 28

κ_{0} does not meet F_{1} ∪ F_{2}. By the definition of κ_{2}, |κ_{2}| ∪ F_{2} = 0, and since all cells of K_{1} meet F_{1}, |K_{1}|F_{2} = 0. Therefore κ_{0} does not meet F_{2}. The set F_{1} does not meet K + K_{1}, nor a fortiori its boundary κ_{1} + κ_{2} + K̇_{1}. Since |κ_{1}|F_{1} = 0, and

|κ_{0}| = |κ_{2} + K̇_{1}| = |κ_{1} + (κ_{1} + κ_{2} + K̇_{1})| ⊆ |κ_{1}| ∪ |κ_{1} + κ_{2} + K̇_{1}|,

it follows that κ_{0} does not meet F_{1}.

This completes the proof.

Corollary. In 9·1·1 the vertex-pair (x) + (y) may be replaced by any 0-cycle Γ^{0} bounding κ_{1} in G_{1} and κ_{2} in G_{2}. The proof is unaffected.

In the open plane the condition “F_{1} or F_{2} is bounded” must be added, to ensure the existence of the grating G^{∗}; but this extra condition is not necessary in the case F_{1}F_{2} = 0, for then K can always be chosen to be finite.

Theorem 9-2. If the common part of the two closed sets F_{1} and F_{2} in Z^{2} is connected, two points which are connected in CF_{1} and CF_{2} are connected in C(F_{1} ∪ F_{2}). (Holds in R^{2} if F_{1}F_{2} = 0.)

By a preliminary topological mapping of the whole space, which does not disturb separation properties, the points may both be made finite. They then bound chains κ_{i} not meeting F_{i}, and since F_{1}F_{2} is connected κ_{1} + κ_{2} bounds in C(F_{1}F_{2}).

Corollary 1. If CF_{1}, CF_{2} and F_{1}F_{2} are connected, C(F_{1} ∪ F_{2}) is connected. The extra condition “F_{1} or F_{2} is bounded” is again required in R^{2}, for 9·2 and Corollary 1, except when F_{1}F_{2} = 0. (Example: F_{1} and F_{2} are the positive ξ_{1}- and ξ_{2}-axes, including the origin in both cases.)

Corollary 2. If D_{1} and D_{2} are domains in X^{2} (= Z^{2} or R^{2}) such that D_{1} ∪ D_{2} = X^{2}, D_{1}D_{2} is connected.

Theorem 9·3. If a closed set F is contained in a domain D in X^{2}, and D_{1}, D_{1}, … are the components of CF, the components of D − F are DD_{1}, DD_{2}, …, provided F ≠ X^{2}.

DD_{i} is not null. For CF − D_{i} is open, since it is the union of components of CF (IV. 6·2), and therefore if DD_{i} were null

D_{i} | (CF − D_{i}) ∪ D

would be a partition of CF ∪ D, = X^{2}, which is impossible.

DD_{i} is connected. If x and y are any two of its points they are connected in D and in CF, and F ∩ CD = 0. Therefore, by Theorem 9·2, x and y are connected in DCF, i.e. they are joined by a continuum in D not meeting F, and therefore lying in D_{i}. Thus x and y are connected in DD_{i}.

Since the points of DD_{i} and DD_{j} (i ≠ j), are clearly not connected in D − F the theorem is proved.

It follows that CF and D − F have the same number of components.

Fig. 29

Theorems 9·2 and 9·3 do not hold on surfaces in general. For example, on the ring-surface in Fig. 29 the circle γ determines one domain in the complete surface, but two in the white domain.

Exercise. If the closed set F in X^{2} has the components F_{1}, F_{2}, …, F_{p} (p finite), and if each F_{i} has a finite (positive) number, n_{i}, of residual domains, then CF has ∑_{i=1}^{p}(n_{i} − 1) + 1 components. [Induction on p.]

The following theorems form a partial converse of Alexander’s Lemma.

Theorem 9·4·1. Let G_{1} and G_{2} be open sets in X^{2} such that CG_{1} and CG_{2} are connected. Let the 0-cycle Γ^{0} in G_{1}G_{2} bound the 1chain κ_{i} in G_{i}, for i = 1, 2. Then if Γ^{0} ∼ 0 in G_{1}G_{2}, κ_{1} + κ_{2} ∼ 0 in G_{1} ∪ G_{2}.

Let Γ_{0} = κ̇_{0}, |κ_{0}| ⊆ G_{1}G_{2}. Then the 1-cycle κ_{0} + κ_{i} does not meet the connected set CG_{i}, and therefore bounds in G_{i}, and all the more in G_{1} ∪ G_{2}. Hence

(κ_{1} + κ_{2}) = (κ_{0} + κ_{1}) + (κ_{0} + κ_{2}) ∼ 0 in G_{1} ∪ G_{2}.

Theorem 9·4·2. Let F_{1} and F_{2} be connected closed sets in X^{2} and let the 0-cycle Γ_{0} in C(F_{1} ∪ F_{2}) bound the 1-chain κ_{i} not meeting F_{i} (i = 1, 2). Let the vertices of Γ^{0} be paired in any way as

(x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{q}, y_{q}).

Then if κ_{1} + κ_{2} is non-bounding in X^{2} − F_{1}F_{2}, at least one x_{r} is separated from y_{r} by F_{1} ∪ F_{2}.

If, for each vertex pair, (x_{r}) + (y_{r}) ∼ 0 in C(F_{1} ∪ F_{2}), their sum Γ^{0} ∼ 0 in C(F_{1} ∪ F_{2}), and therefore κ_{1} + κ_{2} ∼ 0 in X^{2} − F_{1}F_{2}, by 9·4·1.

10. A simple arc (or Jordan arc) is a set of points homeomorphic with the closed segment ⟨0, 1⟩. A simple closed curve (or Jordan curve) is a set of points homeomorphic with the circle ξ_{1}^{2} + ξ_{2}^{2} = 1 in R^{2}. The arc and curve are connected and compact, and are therefore closed sets in any containing space. It was shewn in IV. para. 2 that a simple arc or closed curve in R^{2} or Z^{2} is nowhere dense, and therefore if E is one or other of these curves,

FE = F(CE) = E.

It is convenient in dealing with simple arcs and curves to suppose a definite topological mapping chosen in either case, and we may then speak of “the point τ of the curve”, meaning in the case of the are the image of the point τ of ⟨0, 1⟩, and in the case of the closed curve the image of the point with angular coordinate 2πτ on the circle. In either caser ranges from 0 to 1, but on the closed curve the points 0 and 1 are identical. The points 0 and 1 of the simple are are called its end-points: it was proved on p. 62, Example 5, that they are distinguished topologically from other points of the curve.

Any point τ, other than 0 or 1, of the segment ⟨0, 1⟩ divides it into two segments with τ as their only common point. It follows that if x is any point, other than an end-point, of a simple arc, the arc is the union of two simple arcs having no common point except x, which is an end-point of each. From the corresponding property of the circle it follows that if x and y are any two points of a simple closed curve, the curve is the union of two simple arcs having no common points except x and y, which are the end-points of both. It was shewn in IV. § 5 that these properties distinguish the simple arc and simple closed curve from all other compact connected spaces, and they are in fact the only properties that are used in proving the following theorems.

Fig. 30 Fig. 31

To understand the difficulties that would attend any attempt to deal with simple arcs and curves by means of linear approximations, and to appreciate the power of the “direct” method of Veblen, Brouwer and Alexander used in the following proofs, it should be borne in mind that these curves, although topologically “simple” in the sense that they have no multiple points, may from the point of view of differential geometry present highly condensed singularities. The arcs of two spirals r = (θ + α_{1})^{−λ} and r = (θ + α_{2})^{−λ} (λ > 0, θ > −α_{i}), cut off by the circle r = β form a simple arc with its end-points on the circle. A simple closed curve can be constructed having a spiral cusp of this kind in every are, however small. The method of construction is the usual “condensation of singularities”: first spiral cusps of a certain diameter ϵ_{1} are introduced at regular intervals round a circle; then cusps of diameter ϵ_{2} at closer intervals round this curve; and so on. It is easy to see that if the numbers ϵ_{n} tend to zero sufficiently fast the curves tend to a simple closed curve as limit.

Theorem 10·1. A simple arc in the open or closed plane has a single complementary domain.

Let L be the arc. Its complementary set is certainly not null, since L is nowhere dense (IV. 2).

Suppose the points x and y are separated by L. Then if a is the point τ = \frac{1}{2} of L, dividing it, say, into the two arcs L_{1} and L_{2}, either L_{1} or L_{2} separates x from y. For if not, L_{1} and L_{2} satisfy the conditions of 9·2 (L_{1}L_{2} = a, and if the containing space is R^{2} the arc, being compact, is bounded). Therefore their union, L, does not separate x and y.

A bisection argument now leads us to a contradiction. Of the two halves L_{1}, L_{2} of L we select one that separates x and y, then a half of this half separating x and y, and so on, where a “half” of a simple arc means the image of half the pattern segment. If L^{(n)} is the nth arc so chosen, the segments of which L^{(1)}, L^{(2)}, … are images converge to a point of ⟨0, 1⟩ (since each is a half of its predecessor). Therefore, owing to the continuity of the mapping, the arcs L^{(1)}, L^{(2)}, … themselves converge to a point a_{0} of L. For some n there is a square neighbourhood† of a_{0} containing L^{(n)} but not x or y. The points x and y are connected by a continuum not meeting this square, and therefore not meeting L^{(n)}.

Thus the assumption that x and y are separated by L has led to a contradiction.

Theorem 10·2. (Jordan’s Theorem.)^{(17)} A simple closed curve in the open or closed plane has two complementary domains, of each of which it is the complete frontier.

Fig. 32

Let J be the curve. If the containing space is R^{2}, J is bounded. I. CJ has at most two components. Let a and b be any two points

of J, dividing it into the simple arcs L_{1} and L_{2}. Suppose that x, y and z are finite points of three different residual domains of J. By 10·1 there exist 1-chains κ_{1} and κ_{2} bounded by x and y, such that κ_{i} does not meet L_{i} (i = 1, 2). Since x and y are separated by L_{1} ∪ L_{2} it follows from Alexander’s Lemma that κ_{1} + κ_{2} is non-bounding in C(L_{1}L_{2}), i.e. in

X^{2} − (a ∪ b).

Similarly, (y)+(z) is the common boundary of two 1-chains κ_{3} and κ_{4} not meeting L_{1} and L_{2} respectively, and κ_{3}+κ_{4} is non-bounding in X^{2} − (a ∪ b). Therefore, by 6·4,

(1)  (κ_{1} + κ_{2}) + (κ_{3} + κ_{4}) ∼ 0 in X^{2} − (a ∪ b).

The 1-chain κ_{1} + κ_{3} does not meet L_{1} and it is bounded by (x + y) + (y + z), i.e. by x and z. Similarly, κ_{2} + κ_{4} does not meet L_{2} and is bounded by x and z. The sum

(κ_{1} + κ_{3}) + (κ_{2} + κ_{4})

† A square neighbourhood of the point at infinity is the exterior of any square. is identical with (1), and therefore bounds in C(L_{1}L_{2}). Hence x and z are not separated by L_{1} ∪ L_{2}, contrary to the hypothesis.

II. CJ has at least two components. Let a, b, L_{1}, L_{2} be as before and let Q be a square† with sides parallel to the coordinate axes separating a from b. Since the closed sets L_{1}Q and L_{2}Q do not meet, there is a grating G (which may be so chosen that Q is the locus of a 1-chain σ) no cell of which meets both L_{1}Q and L_{2}Q. Let κ_{0} be the set of 1-cells of σ that meet L_{1}.

The 0-cycle κ̇_{0} bounds κ_{0} in CL_{2} and κ_{0} + σ in CL_{1}, and

κ_{0} + (κ_{0} + σ) = σ

is non-bounding in X^{2} − (a ∪ b) = X^{2} − L_{1}L_{2}. Hence by 9·4·2 a pair of vertices of κ̇_{0} are separated by L_{1} ∪ L_{2}.

III. If D_{0} and D_{1} are the components of CJ, FD_{0} = FD_{1} = J. In the course of proving II it was shewn that there are points of both residual domains on Q. Now a is an arbitrary point of J and Q may be chosen so that it lies in an assigned neiohbourhood of a. Therefore every point of J belongs to both FD_{0} and FD_{1}. By IV. 3·1, both frontiers are contained in the frontier of CJ, i.e. in J.

This completes the proof of the whole theorem.

*

16 (p. 110). J. W. Alexander, Trans. Amer. Math. Soc. 23 (1922), 342.

17 (p. 115). C. Jordan was the first to state, and to attempt to prove, the theorem that bears his name. The first correct proof was given by Veblen in 1905. The text proof, which is that of Alexander, is based on Brouwer’s proof (Math. Annalen, 1909).

† I.e. the 1-dimensional frontier.