ON A THEOREM OF E. HELLY

by

Sakaé Fuchino (Berlin) Szymon Plewik (Katowice)

to appear in Proc. of the A.M.S

Abstract

E. Helly’s theorem ([3]) asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

1991 Mathematics Subject Classification: 26A03, 06A05, 03E10, 03E35 Key-words: Helly’s theorem, splitting number, Saks’ problem

0. Introduction. E. Helly’s theorem ([3]) asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. In the present paper, we prove the following generalization of the theorem: for linearly ordered sets X and Y, if Y is sequentially compact with density less than the splitting number s, then any sequence of monotone functions from X to Y contains a pointwise convergent subsequence (Theorem 7). We also show that this theorem characterizes the splitting number (Theorem 9).

We begin with reviewing some definitions and elementary facts needed for our results:

1. Preliminaries: linearly ordered sets. A linearly ordered set X is said to be dense linear order, if, for any x, y ∈ X, x < y implies that there exists z ∈ X such that x < z < y. A subset D of a linearly ordered set X is said to be dense in X, if, for any x, y ∈ X, x < y implies that there exists z ∈ D such that x ≤ z ≤ y. The density of X is defined by

d(X) = min{ |D| : D ⊆ X and D is dense in X },

where |D| denotes the cardinality of the set D.

Let X and Y be linearly ordered sets. A function f : X → Y is said to be increasing, if, for any x, y ∈ X, x < y implies f(x) ≤ f(y); decreasing, if x < y implies f(y) ≤ f(x). A function is monotone if it is either increasing or decreasing. A sequence (x_{n})_{n∈IN} of elements of X is called increasing, decreasing or monotone respectively, if it is increasing, decreasing or monotone respectively as a function from the set of all natural numbers IN into X.

For a linearly ordered set X, the notion of convergence can be introduced in a canonical way: an increasing sequence (x_{n})_{n∈IN} in X converges to a point x, if x is the supremum of the set of all elements of this sequence; a decreasing sequence converges to x, if x is the infimum of the set of all elements of this sequence; in general, a sequence (x_{n})_{n∈IN} converges to x, if every monotone subsequence of this sequence converges to x. We say also that a sequence is convergent if it converges to some x. A linearly ordered set is said to be sequentially compact if each monotone sequence of its elements converges to some point in it. If a sequence (x_{n})_{n∈IN} converges to x, we denote this as usual by lim_{n→∞} x_{n} = x. For a sequence (x_{n})_{n∈I} indexed by an infinite subset I of IN, its convergence to a point x is defined similarly and denoted by lim_{n∈I} x_{n} = x. For an infinite I ⊆ IN and a sequence (f_{n})_{n∈I} of functions from a set X to a linearly ordered set Y, we say that (f_{n})_{n∈I} converges pointwise to f : X → Y, if lim_{n∈I} f_{n}(x) = f(x) holds for every x ∈ X. We shall also say that a sequence (f_{n})_{n∈I} is pointwise convergent if there is some function f to which the sequence converges pointwise.

Since any sequence in a linearly ordered set has a monotone subsequence, every sequence in a sequentially compact linearly ordered set has a convergent subsequence. Using this fact, we can see easily the following.

Lemma 1. If (x_{n})_{n∈IN} is a non-convergent sequence of elements of a sequentially compact linearly ordered set X, then there exist infinite subsets of natural numbers I and J such that subsequences (x_{n})_{n∈I} and (x_{n})_{n∈J} converge to different points of X. □

Lemma 2. Any infinite linearly ordered set X can be embedded into a dense linear order X̃ such that d(X̃) = d(X). If X is sequentially compact, then X̃ can be also chosen to be so. Also, convergent sequences in X remain convergent in X̃ with the same limit.

Proof. Let D be a dense subset of X of cardinality d(X). For points x, y ∈ X, let us call (x, y) a jump in X if x < y and there is no z ∈ X such that x < z < y. By definition of dense subsets, for each jump (x, y) one of the points x or y must be in D. Hence there are at most d(X) jumps. Let X̃ be the linearly ordered set constructed from X by inserting a copy of the reals into each of the jumps in X. Noting that the density of the reals IR, with respect to the canonical ordering, is countable and IR ∪ {−∞, +∞} is sequentially compact, it is easy to see that X̃ is as desired. □

Lemma 3. Suppose that X and Y are linearly ordered sets and (f_{n})_{n∈I} is a sequence of increasing functions from X to Y. If (f_{n})_{n∈I} converges pointwise to a function f : X → Y, then f is also increasing.

Proof. By Lemma 2, we may assume that Y is a dense linear order. The rest of the proof can be done just like the usual proof of the corresponding assertion on increasing real functions. □

Any linearly ordered set X can be densely embedded into a sequentially compact linearly ordered set \overline{X}. E.g. we can take the Dedekind completion of X as \overline{X}. Note that we have d(X) = d(\overline{X}) since X is dense in \overline{X}. In general, the Dedekind completion of X is not the minimal sequentially compact linearly ordered set containing dense copy of X since there can be an unfilled Dedekind cut (D, E) of X such that D has uncountable cofinality and E uncountable coinitiality. Let us call a sequence (x_{n})_{n∈IN} in a linearly ordered set X potentially convergent if it converges to some point in some \overline{X} as above. By virtue of Lemma 1, this is equivalent to say that there are no x, y ∈ X and no infinite I, J ⊆ IN such that x_{n} ≤ x < y ≤ x_{m} for every n ∈ I and m ∈ J.

2. The splitting number. A family S of infinite subsets of IN is said to be splitting if, for every infinite subset I ⊆ IN, there exists a set J ∈ S such that I ∩ J and I \ J are both infinite. The splitting number s is defined by

s = min{ |S| : S is a splitting family }.

In particular, if S is a family of infinite subsets of IN of cardinality less than s, then there exists an infinite subset I ⊆ IN such that I is almost included either in J or in IN \J for every J ∈ S. It is readily seen that s is uncountable and less than or equal to the cardinality of the reals. On the other hand, it is known that the value of s cannot be decided from the axioms of set theory alone. A splitting family was considered first by Sierpiński in [5]. He showed that under the Continuum Hypothesis there is a splitting family S with the property that every uncountable subfamily of S is still splitting. For more about the cardinal s and its relation to other cardinal invariants of reals the reader may consult [1], [2] or [7]. The role of splitting number in connection with convergence was also studied in [8].

The following lemma is the set-theoretic core of the generalization of Helly’s theorem.

Lemma 4. If X is a set of cardinality less than s and Y is a sequentially compact linearly ordered set of density less than s, then for any sequence (f_{n})_{n∈IN} of functions from X to Y there exists an infinite subset I ⊆ IN such that the sequence of functions (f_{n})_{n∈I} converges pointwise.

Proof. By Lemma 2 we may assume that Y is a dense linear order. Let D be a dense subset of Y of cardinality less than s. For x ∈ X and y ∈ D, let

C_{x}^{y} = { n ∈ IN : f_{n}(x) < y }.

Since |X × D| < s, there exists an infinite I ⊆ IN such that I is almost included either in C_{x}^{y} or in IN \ C_{x}^{y} for any x ∈ X and y ∈ D.

We shall show that the set I is as desired. Otherwise there would be some point a ∈ X such that the sequence (f_{n}(a))_{n∈I} of points in Y is not convergent. Then, by Lemma 1, there are infinite subsets J and K of I and a point d ∈ D such that sequences (f_{n}(a))_{n∈J} and (f_{n}(a))_{n∈K} of points in Y are convergent and we have

lim_{n∈J}f_{n}(a) < d < lim_{n∈K}f_{n}(a).

Hence we have f_{n}(a) < d for all but finitely many n ∈ J and d < f_{n}(a) for all but finitely many n ∈ K. It follows that the sets I ∩ C_{a}^{d} and I \ C_{a}^{d} are both infinite; but this contradicts the choice of I. □

Lemma 4 gives the consistency of a positive answer to the following question of S. Saks studied in [5]:

For arbitrary sequence (f_{n})_{n∈IN} of real functions, do there exist an infinite I ⊆ IN and an uncountable X ⊂ IR such that, for each x ∈ X the sequence of real numbers (f_{n}(x))_{n∈I} has a finite or infinite limit?

Under the Continuum Hypothesis, Sierpiński gave a negative answer to the question in [5]. By applying Lemma 4 for the sequentially compact linearly ordered set IR ∪ {−∞, +∞}, we see that, under s > ℵ_{1}, a positive answer to the question is obtained.

Since every linearly ordered set can be embedded densely into a sequentially compact linearly ordered set, the next lemma follows immediately from Lemma 4.

Lemma 5. If X is a set of cardinality less than s and Y is a linearly ordered set of density less than s, then for any sequence (f_{n})_{n∈IN} of functions from X to Y there exists an infinite subset I ⊆ IN such that the sequence (f_{n}(x))_{n∈I} is potentially convergent for every x ∈ X. □

Lemma 5 can be yet slightly improved. For any infinite I ⊆ IN, let us call a sequence (x_{n})_{n∈I} in a linearly ordered set X semi-monotone if there is a bijection φ : IN → I such that (x_{φ(n)})_{n∈IN} is eventually monotone, i.e. monotone from some m ∈ IN on. It is clear that a semi-monotone sequence is potentially convergent. If x = lim_{n∈I} x_{n} exists, then (x_{n})_{n∈I} is semi-monotone if and only if (x_{n})_{n∈I} approaches to x eventually from one side — i.e. for some m ∈ IN either x_{n} ≤ x for every n ≥ m or x ≤ x_{n} for every n ≥ m.

Lemma 6. If X is a set of cardinality less than s and Y is a linearly ordered set of density less than s, then for any sequence (f_{n})_{n∈IN} of functions from X to Y there exists an infinite subset I ⊆ IN such that the sequence (f_{n}(x))_{n∈I} is semi-monotone for every x ∈ X.

Proof. Without loss of generality, we may assume that Y is sequentially compact. By Lemma 4, there is an infinite I ⊆ IN such that (f_{n})_{n∈I} is pointwise convergent. For each x ∈ X let y_{x} = lim_{n∈I} f_{n}(x). Let Ỹ be the linearly ordered set obtained from Y by inserting a new point y′_{x} between y_{x} and { y ∈ Y : y_{x} < y } for each x ∈ X. Ỹ is still sequentially compact and d(Ỹ) < s since only fewer than s new points are added. Hence we can apply Lemma 4 again to (f_{n})_{n∈I} as a sequence of functions from X to Ỹ to obtain an infinite J ⊆ I such that (f_{n})_{n∈J} is pointwise convergent as a sequence of functions from X to Ỹ. For each x ∈ X, as (f_{n}(x))_{n∈J} should converge to y_{x} or y′_{x}, it follows that, for each x ∈ X, (f_{n}(x))_{n∈J} as a sequence of points in Y approaches to y_{x} eventually from one side. Hence by the remark before this Lemma, (f_{n}(x))_{n∈J} is a quasi-monotone sequence in Y. □

3. Generalized Helly’s theorem. Since the density of the reals is countable, Helly’s theorem ([3]) as cited in Introduction is just a special case of the following theorem.

Theorem 7. (Generalized Helly’s Theorem) Let X and Y be linearly ordered sets. If Y is sequentially compact with density less than s, then any sequence of monotone functions from X to Y contains a pointwise convergent subsequence.

Proof. Without loss of generality, we may assume that the sequence (f_{n})_{n∈IN} consists of increasing functions. By Lemma 2 we may also assume that Y is a dense linear order. Let D ⊆ Y be a dense subset of cardinality less than s. For b, d ∈ D and n ∈ IN, let x_{b,d}^{n} be an element of X such that b ≤ f_{n}(x_{b,d}^{n}) ≤ d if such an element exists; otherwise let x_{b,d}^{n} be an arbitrary element of X. Let

Z = { x_{b,d}^{n} : b, d ∈ D, n ∈ IN }.

Then we have |Z| < s. Hence, by Lemma 4, there exists an infinite subset L ⊆ IN such that the sequence of functions (f_{n} |` Z)_{n∈L} converges pointwise where f_{n} |` Z denotes the restriction of the function f_{n} to the set Z. Let

T = { x ∈ X : (f_{n}(x))_{n∈L} is a convergent sequence of points in Y }.

We have Z ⊆ T. Let h : T → Y be the function such that the sequence of functions (f_{n} |` T)_{n∈L} converges pointwise to h. By Lemma 3, h is an increasing function. Now let

U = { U : U is a maximal interval in X such that U ⊆ X \ T }.

For each interval U ∈ U we choose x_{U} ∈ U. By definition of Z, f_{n} is constant on each U ∈ U for every n ∈ IN. Hence, for any subset M ⊆ L, we have:

(∗) (f_{n}(x_{U}))_{n∈M} converges if and only if (f_{n} |` U)_{n∈M} converges pointwise.

Letting W = { x_{U} : U ∈ U }, we claim that |W | < s. To see this, let x ∈ W. The sequence (f_{n}(x))_{n∈L} is not convergent. Since Y is a dense linear order and sequentially compact, by Lemma 1, there are infinite subsets J, K ⊆ L and points b_{x}, c_{x}, d_{x} ∈ D such that sequences of points (f_{n}(x))_{n∈J} and (f_{n}(x))_{n∈K} are convergent and

lim_{n∈J}f_{n}(x) < b_{x} < d_{x} < c_{x} < lim_{n∈K}f_{n}(x).

If y ∈ T and y < x, then h(y) ≤ b_{x}, since lim_{n∈L}f_{n}(y) = h(y) and f_{n}(y) < b_{x} for infinitely many n ∈ L. Likewise, for any z ∈ T with x < z we have h(z) ≥ c_{x}. For x_{1}, x_{2} ∈ W with x_{1} < x_{2}, there is y ∈ T such that x_{1} < y < x_{2}. Hence the mapping from W to D defined by x ↦ d_{x} is injective. As |D| < s it follows that |W | < s.

Again by Lemma 4 we can find an infinite I ⊆ L such that (f_{n} |` W)_{n∈I} is pointwise convergent. By definitions and (∗) above, we have that the sequence of functions (f_{n})_{n∈I} converges pointwise. □

4. The splitting number is optimal. For an infinite subset V ⊆ IN, let

D(V) = ∑_{n∈V} \frac{1}{2^{n+1}} .

Note that D is a bijective mapping from infinite subsets of IN to the real numbers in the half-open interval (0, 1]. For a family S of subsets of IN, let us denote by D(S) the set { D(V) : V ∈ S }. Thus D(S) is a subset of the unit interval of cardinality |S|.

Assume now that S is a splitting family of cardinality s. Let

H = (D(S) × [0, 1]) ∪ ([0, 1] × {0})

be the linearly ordered set equipped with the lexicographical ordering, i.e. we let (x, y) < (p, q), whenever x < p, or x = p and y < q. Here, x < p and y < q denote the canonical ordering on the reals.

Lemma 8. The linearly ordered set H is sequentially compact and its density is equal to s.

Proof. Suppose that S = ((x_{n}, y_{n}))_{n∈IN} is a monotone sequence of points in H. Then (x_{n})_{n∈IN} is monotone as well. If (x_{n})_{n∈IN} is eventually constant, say x_{n} = x for all n > m, then (y_{n})_{n>m} is a monotone sequence. Hence lim_{n→∞} y_{n} exists and S converges to (x, lim_{n→∞} y_{n}). Otherwise there are infinitely many distinct x_{n}’s. If S is increasing then S converges to (lim_{n→∞} x_{n}, 0). If S is decreasing, then S converges to (lim_{n→∞} x_{n}, 1) provided that lim_{n→∞} x_{n} ∈ D(S); otherwise it converges to (lim_{n→∞} x_{n}, 0).

Let Q be the set of rational numbers in the unit interval [0, 1]. Then

H_{0} = (D(S) × Q) ∪ (Q × {0})

is dense in H and of cardinality s. This shows that d(H) ≤ s. If H′ ⊆ H is of cardinality less than s then there is some s ∈ D(S) such that {s} × [0, 1] is disjoint from H′. Hence H′ is not dense in H. Thus we also have d(H) ≥ s.

□

The following theorem is a variation of an example in [6].

Theorem 9. There exists a sequence (f_{n})_{n∈IN} of increasing functions from the subset D(S) of the unit interval to the linearly ordered set H such that (f_{n})_{n∈IN} does not have any pointwise convergent subsequence.

Proof. For each n ∈ IN and V ∈ S, let

f_{n}(D(V)) = \{ \begin{matrix}(D(V), 1), & if n ∈ V; \\ (D(V), 0), & otherwise. \end{matrix}

Each function f_{n} is obviously increasing. For any infinite subsequence (f_{n})_{n∈I}, let V ∈ S be such that the sets I ∩ V and I \ V are both infinite. Then we have f_{n}(D(V)) = (D(V), 1) for every n ∈ I ∩ V, and f_{n}(D(V)) = (D(V), 0) for every n ∈ I \ V. In particular, the sequence of points (f_{n}(D(V)))_{n∈I} is not convergent. □

The theorem above shows that the condition d(Y) < s in Theorem 7 is optimal. Using this fact, we obtain the following characterization of the splitting number.

Let τ_{1} be the supremum of the cardinals κ with the property that for every set of X of cardinality less than κ and for every sequentially compact linearly ordered set Y of density less than κ, any sequence of functions from X to Y has a pointwise convergent subsequence. Likewise, let τ_{2} be the least cardinal κ such that, for some set X of cardinality κ, it is not the case that any sequence of functions from X to {0, 1} has a pointwise convergent subsequence, where we consider {0, 1} as a linearly ordered set with 0 < 1. Finally, let μ be the supremum of the cardinals κ with the property that, for any linearly ordered set X and any sequentially compact linearly ordered set Y with d(Y) < κ, any sequence of monotone functions from X to Y has a pointwise convergent subsequence.

Theorem 10. s = τ_{1} = τ_{2} = μ.

Proof. By definition we have τ_{1} ≤ τ_{2}. Lemma 4 implies s ≤ τ_{1}; s ≤ μ follows from Theorem 7. Theorem 9 implies s ≥ μ.

To see τ_{2} ≤ s, we use a variant of Rademacher’s functions (see [4]): for each n ∈ IN, the function φ_{n} on the family of all infinite subsets of IN to {0, 1} is defined by

φ_{n}(V) = \{ \begin{matrix}1, & if n ∈ V; \\ 0, & otherwise. \end{matrix}

Let S be a splitting family of cardinality s. For any infinite subset I ⊆ IN, let V ∈ S be such that the sets I ∩ V and I \ V are both infinite. Then the 0-1 sequence (φ_{n}(V))_{n∈I} is not convergent as 0 and 1 both appear infinitely many times in this sequence. This shows that no subsequence of the sequence (φ_{n} |` S)_{n∈IN} of functions from S to {0, 1} can be pointwise convergent. □

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Authors’ addresses:

Sakaé Fuchino Institut für Mathematik II, Freie Universität Berlin Arnimallee 3, 14195 Berlin, Germany

e-mail: fuchino@math.fu-berlin.de

Present address: Department of computer sciences Kitami Institute of Technology, Kitami, Hokkaido 090 Japan.

  fuchino@info.kitami-it.ac.jp

Szymon Plewik Instytut Matematyki Uniwersytetu Śla̧skiego ul. Bankowa 14, 40 007 Katowice, Poland

e-mail: plewik@ux2.math.us.edu.pl