Transfinite Sequences of Continuous and Baire Class 1 Functions

Any set F of real valued functions defined on an arbitrary set X is partially ordered by the pointwise order; that is, f ≤ g iff f (x) ≤ g(x) for all x ∈ X. Then, f < g iff f ≤ g and g ≤ f ; equivalently, f (x) ≤ g(x) for all x ∈ X and f (x) < g(x) for at least one x ∈ X. Our aim will be to investigate the possible lengths of the increasing or decreasing well-ordered sequences of functions in F with respect to this order. A classical theorem (see Kuratowski [7], §24.III, Theorem 2 ′ ) asserts that if F is the set of Baire class 1 functions (that is, pointwise limits of continuous functions) defined on a Polish space X (that is, a complete separable metric space), then there exists a monotone sequence of length ξ in F iff ξ < ω 1 . P. Komjáth [5] proved that the corresponding question concerning Baire class α functions for 2 ≤ α < ω 1 is independent of ZFC .

In the present paper we investigate what happens if we replace the Polish space X by an arbitrary metric space.

Section 1 considers chains of continuous functions. We show that for any metric space X, there exists a chain in C(X, R) of order type ξ iff |ξ| ≤ d(X).

Here, |A| denotes the cardinality of the set A, while d(X) denotes the density of the space X, that is

In particular, for separable X, every well-ordered chain has countable length, just as for Polish spaces.

Section 2 considers chains of Baire class 1 functions on separable metric spaces. Here, the situation is entirely different from the case of Polish spaces, since on some separable metric spaces, there are well-ordered chains of every order type less than ω 2 . Furthermore, the existence of chains of type ω 2 and longer is independent of ZFC + ¬CH . Under MA, there are chains of all types less than c + , whereas in the Cohen model, all chains have type less than ω 2 .

We note here that instead of examining well-ordered sequences, which is a classical problem, we could try to characterize all the possible order types of linearly ordered subsets of the partially ordered set F . This problem was posed by M. Laczkovich, and is considered in detail in [3].

1 Sequences of Continuous Functions Lemma 1.1 For any topological space X: If there is a well-ordered sequence of length ξ in C(X, R), then ξ < d(X) + .

Proof. Let {f α : α < ξ} be an increasing sequence in C(X, R), and let D ⊆ X be a dense subset of X such that d(X) = max(|D|, ω). By continuity, the f α ↾D are all distinct; so, for each α < ξ, choose a

The converse implication is not true in general. For example, if X has the countable chain condition (ccc), then every well-ordered chain in C(X, R) is countable (because X × R is also ccc). However, the converse is true for metric spaces: Lemma 1.2 If (X, ̺) is any non-empty metric space and ≺ is any total order of the cardinal d(X), then there is a chain in C(X, R) which is isomorphic to ≺.

Proof. First, note that every countable total order is embeddable in R, so if d(X) = ω, then the result follows trivially using constant functions. In particular, we may assume that X is infinite, and then fix D ⊆ X which is dense and of size d(X). For each n ∈ ω, let D n be a subset of D which is maximal with respect to the property ∀d, e

Then n D n is also dense, so we may assume that n D n = D. We may also assume that ≺ is a total order of the set D. Now, we shall produce

Since every x ∈ X has a neighborhood on which all but at most one of the ϕ n c vanish, we have

, so actually f d < f e whenever d ≺ e. Putting these lemmas together, we have: Theorem 1.3 Let (X, ̺) be a metric space. Then there exists a well-ordered sequence of length ξ in C(X, R) iff ξ < d(X) + . Corollary 1.4 A metric space (X, ̺) is separable iff every well-ordered sequence in C(X, R) is countable.

If we replace continuous functions by Baire class 1 functions, then Corollary 1.4 becomes false, since on some separable metric spaces, we can get well-ordered sequences of every type less than ω 2 . To prove this, we shall apply some basic facts about ⊂ * on P(ω). As usual, for x, y ⊆ ω, we say that x ⊆ * y iff x\y is finite. Then x ⊂ * y iff x\y is finite and y\x is infinite. This ⊂ * partially orders P(ω). Lemma 2.1 If X ⊂ P(ω) is a chain in the order ⊂ * , then on X (viewed as a subset of the Cantor set 2 ω ∼ = P(ω)), there is a chain of Baire class 1 functions which is isomorphic to (X, ⊂ * ).

Proof. Note that for each x ∈ X,

which is an F σ set in X. Likewise, the sets {y ∈ X : y ⊇ * x}, {y ∈ X : y ⊂ * x}, and {y ∈ X : y ⊃ * x}, are all F σ sets in X, and hence also G δ sets. It follows that if f x : X → {0, 1} is the characteristic function of {y ∈ X : y ⊂ * x}, then f x : X → R is a Baire class 1 function. Then, {f x : x ∈ X} is the required chain.

Lemma 2.2 For any infinite cardinal κ, suppose that (P(ω), ⊂ * ) contains a chain {x α : α < κ} (i.e., α < β → x α ⊂ * x β }). Then (P(ω), ⊂ * ) contains a chain X of size κ such that every ordinal ξ < κ + is embeddable into X.

Proof. Let S = 1≤n<ω κ n . For s = (α 1 ,

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