Journal of Applied Analysis Vol. 11, No. 1 (2005), pp. 49–62

SEPARATELY NOWHERE CONSTANT FUNCTIONS; n-CUBE AND α-PRISM

DENSITIES

K. CIESIELSKI and A. MILLÁN

Received April 8, 2003 and, in revised form, November 5, 2003

Abstract. A function f from a countable product X = ∏_{i} X_{i} of Polish spaces into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of X_{i}’s. This result is used to distinguish between n-cube density notions for different n ≤ ω, where ω-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of α < ω_{1}, between the notions of α-prism densities — the more refined density notions used also in CPA.

2000 Mathematics Subject Classification. Primary 03E40; Secondary 26B99, 03E35. Key words and phrases. Separately nowhere constant functions, cube density, prism

density.

The work of the first author was partially supported by NATO Grant PST. CLG. 977652 and 2002/03 West Virginia University Senate Research Grant.

This work was completed when the second author was a Ph.D. student at West Virginia University. Some part of this work is likely to be included in his dissertation, written under the supervision of K. Ciesielski.

ISSN 1425-6908 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag.

1. Introduction

The set theoretical notation used in the paper is standard and follows [1]. For a topological space X a function f: X → Y is nowhere constant if f is not constant on any non-empty open subset of X. For a subset G of a product space X = ∏_{i∈I} X_{i} we say that a function f: G → Y is separately nowhere constant if for every t ∈ G and k ∈ I function f restricted to the section G_{k}^{t} = {x ∈ G: x ↾ I \ {k} = t ↾ I \ {k}} is nowhere constant. This notion is the most natural when G = X. In this case it is related in a natural way to the notion of a separately continuous function f: X → Y, that is, such that f restricted to every section X_{k}^{t} is continuous.

Note that every separately nowhere constant function is nowhere constant. However, the converse implication is false, as shown by the polynomial functions from ℝ^{2} into ℝ defined by w_{0}(x, y) = xy and w_{1}(x, y) = x. This implications pattern stays in contrast with the implications for separate continuity: continuity implies separate continuity, but the converse implication is false.

We will consider the notion of being separately nowhere constant only for the product of Polish spaces which we define here as a complete separable metric spaces without isolated points. (No function is nowhere constant if it is defined on a space containing isolated points.) Let ℭ stand for the Cantor set 2^{ω} and for a Polish space X let Perf(X) be the family of all subsets of X homeomorphic to ℭ. Our main theorem on separately nowhere constant functions is the following result, where a subset P of ∏_{i∈I} X_{i} is a perfect cube provided it is of the form P = ∏_{i∈I} P_{i} with P_{i} ∈ Perf(X_{i}) for all i ∈ I.

Theorem 1. Let G be a dense G_{δ} subset of a product ∏_{i∈I} X_{i} of Polish spaces and let f be a continuous function from G into a Polish space Y. If f is separately nowhere constant then there is a perfect cube P in ∏_{i∈I} X_{i} such that P ⊂ G and f restricted to P is one-to-one.

It is not difficult to see that the conclusion of the theorem remains true for the function w_{0}(x, y) = xy, despite the fact that w_{0} is not separately nowhere constant. On the other hand, the theorem’s conclusion is false for the nowhere constant function w_{1}(x, y) = x.

The theorem will be used to provide the examples distinguishing between the notions of n-cube densities defined as follows. For 0 < n ≤ ω we say that a family F ⊂ Perf(X) is n-cube dense provided that for every continuous injection f: ℭ^{n} → X there is a perfect cube C ⊂ ℭ^{n} such that f[C] ∈ F.

To put these notions in a better perspective notice that 1-cube density is just the standard perfect set density, that is, F ⊂ Perf(X) is 1-cube dense provided every perfect subset of X contains a set from F. To see that in general n-cube density is a stronger notion recall the following example, which is extracted from Miller’s construction [6, Theorem 5.10] of a Marczewski s_{0}-set of cardinality continuum.

Example 1 ([2, 4]). Let X = ℭ × ℭ and let E be the family of all P ∈ Perf(X) such that either all vertical sections of P are countable, or all horizontal sections of P are countable. Then E is dense in Perf(X) but it is not 2-cube dense.

The interest in cube densities comes from the work of Ciesielski and Pawlikowski [4], in which authors formulate the Covering Property Axiom CPA, notice that it holds in the iterated perfect set model, and discuss in depth its consequences. The simplest form of the axiom is that of CPA_{cube}:

CPA_{cube}: 𝔠 = ω_{2} and for every Polish space X and every ω-cube dense family E ⊂ Perf(X) there exists an E_{0} ⊂ E such that |E_{0}| < 𝔠 and |X \ ⋃ E_{0}| < 𝔠.

Another version of CPA axiom, stronger than CPA_{cube}, is axiom CPA_{prism} which is obtained from CPA_{cube} by replacing the notion of ω-cube density with the weaker notion of density known as prism density: a family F ⊂ Perf(X) is prism dense if it is α-prism dense for every 0 < α < ω_{1}. Now, αprism density is defined in a way similar to n-cube density. First one needs to identify the family ℙ_{α} of subsets of ℭ^{α}, termed iterated perfect sets. This family, which includes all perfect cubes in ℭ^{α}, will be defined in Section 3. Then, we say that a family F ⊂ Perf(X) is α-prism dense provided that for every continuous injection f: ℭ^{α} → X there is a P ∈ ℙ_{α} such that f[P] ∈ F.

The next two theorems discuss these notions of density. In particular, the first of them shows that essentially all these notions are different. The second theorem shows that any strengthening of the axiom CPA_{prism} obtained by replacing the prism density with a proper subclass of the densities we consider leads to the statement which is false in ZFC.

Theorem 2. For a Polish space X, a family F ⊂ Perf(X), and 1 < α < ω_{1} consider the following sentences:

C_{α}: family F is β-cube dense for every 0 < β < α;

P_{α}: family F is β-prism dense for every 0 < β < α.

Then, for 2 < m < n < ω and ω + 1 < α < γ < ω_{1}, they are related by the following implications.

[figure: C_{2}; P_{2}; C_{m}; P_{m}; C_{n}; P_{n}; C_{ω}; P_{ω}; C_{ω+1}; P_{ω+1}; C_{α}; P_{α}; C_{γ}; P_{γ}]

Moreover, none of these implications can be reversed.

For α < ω_{1} we say that a family F ⊂ Perf(X) is α-prism^{∗} dense provided F is β-prism dense for every 0 < β < α and it is n-cube dense for every 0 < n < ω.

Theorem 3. For every α < ω_{1} and for every Polish space X there is an α-prism^{∗} dense family F ⊂ Perf(X) for which |X \ ⋃ F| = 𝔠.

These theorems will be proved in Section 3.

2. Separately nowhere constant functions

As usual ℝ and ℚ represent the sets of real and rational numbers, respectively. The space ℝ^{ω} is the product space of countably many copies of ℝ with its usual topology. This is a Polish space and a vector space over ℝ with the operations defined pointwise from the usual operations in ℝ. In this context we consider for every k < ω the canonical unit vectors e⃗_{k} ∈ ℝ^{ω}: e⃗_{k}(k) = 1 and e⃗_{k}(i) = 0 for all other i < ω. If S ⊂ ℝ^{ω} and δ ∈ ℝ then δ · e⃗_{k} + S = {δ · e⃗_{k} + s : s ∈ S}. If ε > 0 and x ∈ ℝ^{ω} then B(x, ε) denotes the open ball with center x and radius ε and \overline{B}(x, ε) is the corresponding closed ball. If X is a Polish space and A ⊂ X, the closure of A is denoted by \overline{A}. If m < ω then we identify ℝ^{ω} with ℝ × ℝ^{ω\{m}}; if y ∈ ℝ^{ω\{m}} and G ⊂ ℝ^{ω} then the section of G along y is the set (G)^{y} = {x ∈ ℝ: ⟨x, y⟩ ∈ G}.

The following variant of Kuratowski-Ulam theorem will be useful in what follows. Although it looks like it should be well known, we could not locate it in the literature. Thus, we present it here with a proof.

Lemma 2. Let X be a Polish space and consider X^{T} with the product topology, where T ≠ ∅ is an arbitrary set. Fix at most countable family K of sets K ⊊ T. Then for every comeager set H ⊂ X^{T} there exists a comeager set G ⊂ H such that for every x ∈ G and K ∈ K the set

G_{x↾K} = {y ∈ X^{T \K}: (x ↾ K) ∪ y ∈ G}

is comeager in X^{T \K}.

Proof. Let {K_{i}: i < ω} be an enumeration of K with infinite repetitions. We construct, by induction on i < ω, a decreasing sequence ⟨G_{i}: i < ω⟩ of comeager subsets of H such that for every i < ω

(i) the set (G_{i})_{x↾K_{i}} is comeager in X^{T \K_{i}} for every x ∈ G_{i}.

Put G_{−1} = H and assume that for some i < ω the comeager set G_{i−1} is already constructed. To define G_{i} identify X^{T} with X^{K_{i}} × X^{T \K_{i}}. Then, by Kuratowski-Ulam theorem, the set

A = {y ∈ X^{K_{i}}: (G_{i−1})_{y} is comeager in X^{T \K_{i}}}

is comeager in X^{K_{i}}. Put G_{i} = G_{i−1} ∩ (A × X^{T \K_{i}}).

Clearly G_{i} ⊂ G_{i−1} is a comeager subset of X^{T}. If x ∈ G_{i} then x ↾ K_{i} ∈ A so (G_{i})_{x↾K_{i}} = (G_{i−1})_{x↾K_{i}} is comeager in X^{T \K_{i}}. So, (i) holds. This completes the definition of the sequence ⟨G_{i}: i < ω⟩.

Let G = ⋂_{i<ω} G_{i}. Clearly G ⊂ H is comeager in X^{T}. To see the additional part, take a K ∈ K. Since G = ⋂{G_{i}: i < ω & K_{i} = K}, for every x ∈ G the set

G_{x↾K} = ⋂ {(G_{i})_{x↾K_{i}}: i < ω & K_{i} = K}

is comeager in X^{T \K}.

The next lemma is an immediate consequence of Lemma 2 applied to X = ℝ, T = ω, and K = {ω \ {n}: n < ω}.

Lemma 3. For every comeager set G ⊂ ℝ^{ω} there exists a comeager set H ⊂ G such that for every x ∈ H and n < ω the set H ∩ (x + ℝ · e⃗_{n}) is comeager in x + ℝ · e⃗_{n}.

The following lemma will facilitate the inductive step in the next theorem.

Lemma 4. Let G be a comeager subset of ℝ^{ω} such that (•) G ∩ (x + ℝ · e⃗_{k}) is comeager in x + ℝ · e⃗_{k} for every x ∈ G and k < ω. Let f be a continuous separately nowhere constant function from G into a Polish space Y. If S ∈ [G]^{<ω} is such that f is one-to-one on S, then for every k < ω and ε > 0 there exists a δ ∈ (0, ε) such that (S + δ · e⃗_{k}) ⊂ G and f is one-to-one on S ∪ (S + δ · e⃗_{k}).

Proof. Let S = {x_{i}: i < n} ⊂ G be such that f ↾ S is one-to-one. Since f is continuous, decreasing ε if necessary, we can assume that (∗) if S^{∗} = {x_{i}^{∗}: i < n} is such that x_{i}^{∗} ∈ G ∩ B(x_{i}, ε) for every i < n, then

f is also one-to-one on S^{∗}.

For each x ∈ S consider the sets M_{x} = {δ ∈ ℝ: x + δ · e⃗_{k} ∈ G}, which by (•) are comeager, and N_{x} = {δ ∈ M_{x}: f(x + δ · e⃗_{k}) ∈ f[S]}. Since f ↾ G ∩ (x + ℝ · e⃗_{k}) is nowhere constant, the set N_{x} is meager in ℝ. So, B = ⋂_{x∈S} M_{x} \ ⋃_{x∈S} N_{x} is comeager in ℝ.

Pick a δ ∈ (0, ε) ∩ B. Then S + δ · e⃗_{k} ⊂ G as δ ∈ ⋂_{x∈S} M_{x}. To see that f is one-to-one on S ∪ (S + δ · e⃗_{k}) take x ≠ y in this set. We need to show that f(x) ≠ f(y). This follows from the assumption when x, y ∈ S, from (∗) when x, y ∈ S + δ · e⃗_{k}, and from δ ∉ ⋃_{x∈S} N_{x} otherwise.

Theorem 4. Let G be a comeager subset of ℝ^{ω} and let f be a continuous separately nowhere constant function from G into a Polish space Y. Then there is a perfect cube P in ℝ^{ω} such that P ⊂ G and f is one-to-one on P.

Proof. Let {m_{k}: k < ω} be an enumeration of ω where every natural number appears infinitely often. By Lemma 3, shrinking G if necessary, we can assume that G satisfies the condition (•) from Lemma 4. Since G is a dense G_{δ} subset of ℝ^{ω} we have G = ⋂_{n<ω} G_{n}, where each G_{n} is open and dense subset of ℝ^{ω}.

We will construct by induction on k < ω the sequences ⟨S_{k} ∈[G]^{2^{k}}: k <ω⟩, ⟨ε_{k}: k < ω⟩, and ⟨δ_{k}: k < ω⟩ such that for every k < ω:

(1) 0 < δ_{k} < ε_{k} ≤ 2^{−k},

(2) S_{k+1} = S_{k} ∪ (δ_{k} · e⃗_{m_{k}} + S_{k}) ⊂ ⋃{B(x, ε_{k}): x ∈ S_{k}},

(3) \overline{B}(x, ε_{k}) ⊂ G_{k} for every x ∈ S_{k},

(4) f[\overline{B}(x, ε_{k})] ∩ f[\overline{B}(x^{∗}, ε_{k})] = ∅ for every distinct x, x^{∗} ∈ S_{k}.

We start the construction with an arbitrary S_{0} = {s} ⊂ G, and ε_{0} ≤ 1 ensuring (3). If for some k < ω the set S_{k} and ε_{k} are already constructed we choose δ_{k} using Lemma 4 with k = m_{k} and ε ≤ ε_{k} small enough that it insures (2) and |S_{k+1}| = 2^{k+1}. Then f is one-to-one on S_{k+1} ⊂ G and, using continuity of f, we can choose ε_{k+1} satisfying (1), (3), and (4). This finishes the construction.

If for n, k < ω we put A_{k,n} = {x(n): x ∈ S_{k}} then it is easy to see that:

(a) S_{k} = ∏_{n<ω} A_{k,n},

(b) A_{k+1,n} = A_{k,n} for every n ≠ m_{k+1},

(c) A_{k+1,m_{k+1}} = A_{k,m_{k+1}} ∪ (δ_{k} + A_{k,m_{k+1}}).

We define P_{n} = \overline{⋃_{k<ω} A_{k,n}} and put P = ∏_{n<ω} P_{n}. We will show that each P_{n} is a perfect subset of ℝ, P ⊂ G, and f is one-to-one on P. Notice that this will finish the proof, because as a final adjustment (necessary, when P_{n} has a non-empty interior in ℝ) we can shrink each P_{n} to a subset from Perf(ℝ).

Clearly each P_{n} is closed and, by (1) and (2), it has no isolated points. We need to show that

\overline{⋃_{k<ω} S_{k}} = ∏_{n<ω} P_{n}.

The inclusion \overline{⋃_{k<ω} S_{k}} ⊂ ∏_{n<ω} P_{n} follows from (a). In order to prove the other inclusion pick an x ∈ ∏_{n<ω} P_{n}. Then for every n < ω there exists a sequence {a_{i}^{n}: i < ω} ⊂ ⋃_{k<ω} A_{k,n} with distinct terms such that lim_{i→∞} a_{i}^{n} = x(n). For every m < ω let x_{m} ∈ ℝ^{ω} be defined as x_{m}(i) = a_{i}^{m} if i ≤ m and x_{m}(i) = s(i) if i > m. Then {x_{m}: m < ω} ⊂ ⋃_{k<ω} S_{k} and lim_{m→∞} x_{m} = x. This proves that ∏_{n<ω} P_{n} ⊂ \overline{⋃_{k<ω} S_{k}}.

Next notice that if for k < ω we put T_{k} = ⋃\{\overline{B}(x, ε_{k}): x ∈ S_{k}\} then condition (2) gives us \overline{⋃_{k<ω} S_{k}} ⊂ ⋂_{k<ω} T_{k} while the other inclusion is obvious. In particular we have

∏_{n<ω} P_{n} = \overline{⋃_{k<ω} S_{k}} = ⋂_{k<ω} T_{k}.

In order to prove that f is one-to-one on ∏_{n<ω} P_{n} pick distinct x and y from ∏_{n<ω} P_{n}. Then there are sequences {x_{m}} and {y_{m}} such that for every m < ω we have x_{m}, y_{m} ∈ S_{m}, x ∈ \overline{B}(x_{m}, ε_{m+1}), and y ∈ \overline{B}(y_{m}, ε_{m+1}). Since x ≠ y, there is an m < ω such that x_{m} ≠ y_{m}. So, by (5), we have f[\overline{B}(x_{m}, ε_{m+1})] ∩ f[\overline{B}(y_{m}, ε_{m+1})] = ∅. Hence f(x) ≠ f(y). This shows that f is one-to-one on ∏_{n<ω} P_{n}.

Finally note that by (3) we have ∏_{n<ω} P_{n} = ⋂_{k<ω} T_{k} ⊂ G.

Corollary 5. Let {X_{n}: n < ω} be a family of Polish spaces, G be a dense G_{δ} subset of ∏_{n<ω} X_{n}, and let f be a continuous separately nowhere constant function from G into a Polish space Y. Then there exist perfect sets P_{n} ∈ Perf(X_{n}), n < ω, such that f is one-to-one on ∏_{n<ω} P_{n}.

Proof. For every n < ω let G_{n} be a dense G_{δ} subset of X_{n} homeomorphic to the Baire space ω^{ω}.^{1} Since ω^{ω} is homeomorphic to ℝ \ ℚ, there is a homeomorphism h_{n}: G_{n} → ℝ \ ℚ. Then, h: ∏_{n<ω} G_{n} → (ℝ \ ℚ)^{ω} defined by h = ⟨h_{n}: n < ω⟩ is a cube-preserving homeomorphism. We can apply Theorem 4 to the function f ∘h^{−1} on a dense G_{δ} subset h [G ∩ ∏_{n<ω} G_{n}] of ℝ^{ω} to obtain a perfect cube ∏_{n<ω} Q_{n} on which f ∘ h^{−1} is one-to-one. Then, h^{−1} [∏_{n<ω} Q_{n}] is a perfect cube in ∏_{n<ω} X_{n} on which f is one-to-one.

1Every Polish space X (without isolated points) has a dense subspace G homeomorphic to ω^{ω} constructed as follows. Let {B_{n}: n < ω} be a basis for X. Then Y = X \ ⋃_{n<ω} bd(B_{n}) is a zero-dimensional dense G_{δ} subspace of X, where bd(B_{n}) denotes the boundary of B_{n}. Take a countable dense subset D of Y and put G = Y \ D. Then G is a dense G_{δ} subspace of X. Also, G is Polish, zero-dimensional, and every compact subset of G has an empty interior. So, by Alexandrov-Urysohn theorem [5, Theorem 7.7], it is homeomorphic to ω^{ω}.

Proof of Theorem 1. We can assume that the index set I is a cardinal number κ. Let X = ∏_{i∈κ} X_{i}.

The case κ = ω is true by Corollary 5.

If κ = n < ω and f: G → Y is continuous and separately nowhere constant consider F: G×ℭ^{ω\n} →Y×ℭ^{ω\n} defined by F (x)=(f(x↾n), x↾ω\n). Then, F is continuous and separately nowhere constant function defined on a dense G_{δ} subset of ∏_{i∈ω} X_{i} where X_{i} = ℭ for every i ∈ ω \ n. Thus, by case κ = ω, there are {P_{i} ∈ Perf(X_{i}): i < ω} such that F is one-to-one on ∏_{i<ω} P_{i} ⊂ G × ℭ^{ω\n}. This implies that f is one-to-one on ∏_{i<n} P_{i} ⊂ G.

If κ > ω then the result is trivial because in this case f cannot be simultaneously continuous and separately nowhere constant on a dense G_{δ} subset G of X. To see this first notice that G contains a subset of the form H × ∏_{i∈κ\A} X_{i}, where A is a countable subset of κ and H is a dense G_{δ} subset of ∏_{i∈A} X_{i}. This is the case, since every dense open subset U of X contains a dense open subset in similar form: a union of a maximal pairwise disjoint family of basic open subsets of U. So, we can assume that G is in this form. Pick an x_{0} ∈ G. By the continuity of f at x_{0}, for every n < ω there exists an open subset U_{n} in X containing x_{0} such that the diameter of f [G ∩ U_{n}] is less than 2^{−n}. By the definition of the product topology each U_{n} contains a set of the form ∏_{i∈F_{n}}{x_{0}(i)} × ∏_{i∈κ\F_{n}} X_{i}, where each F_{n} ⊂ κ is finite. Put F = A ∪ ⋃_{n<ω} F_{n} and notice that Z = ∏_{i∈F} {x_{0}(i)} × _{∏i∈κ\F} X_{i} ⊂ G ∩ ⋂_{n<ω} U_{n}. So, f[Z] has the diameter equal to 0, that is, f is constant on Z. But this contradicts the fact that f is separately nowhere constant on G, since for ξ ∈ κ \ F set Z contains the section {x ∈ X: x ↾ κ \ {ξ} = x_{0} ↾ κ \ {ξ}}.

3. Cube and prism densities

We start with recalling the definition of the family ℙ_{α} of iterated perfect sets in ℭ^{α}, where 0 < α < ω_{1}. So, let Φ_{α} be the family of all continuous injections f: ℭ^{α} → ℭ^{α} such that for every β < α

f ↾↾ β =^{def} {⟨x ↾ β, y ↾ β⟩: ⟨x, y⟩ ∈ f}

is a one-to-one function from ℭ^{β} into ℭ^{β}. For example, if α = 3 = {0, 1, 2} then f ∈ Φ_{α} provided there exist continuous functions f_{0} : ℭ → ℭ, f_{1} : ℭ^{2} → ℭ, and f_{2} : ℭ^{3} → ℭ such that f(x_{0}, x_{1}, x_{2}) = ⟨f_{0}(x_{0}), f_{1}(x_{0}, x_{1}), f_{2}(x_{0}, x_{1}, x_{2})⟩ for all x_{0}, x_{1}, x_{2} ∈ ℭ and maps f_{0}, ⟨f_{0}, f_{1}⟩, and f are one-to-one. Note that Φ_{α} is closed under compositions and that for every 0 < β < α if f ∈ Φ_{α} then f ↾↾ β ∈ Φ_{β}. We define

ℙ_{α} = {range(f): f ∈ Φ_{α}}.

The following properties can be easily deduced from these definitions. (For (D) see [4, (3.13)].) Here π_{β} is the projection from ℭ^{α}, for some α ≥ β, onto the first β coordinates, that is, π_{β}(x) = x ↾ β. (A) Every perfect cube in ℭ^{α} belongs to ℙ_{α}. (B) If P ∈ ℙ_{α+1} and x ∈ π_{α}[P] then |({x} × ℭ) ∩ P | = 𝔠. (C) If 0 < β < α and P ∈ ℙ_{α} then π_{β}[P] ∈ ℙ_{β}. (D) If 0 < β < α then Q = {x ∈ P: π_{β}(x) ∈ R} ∈ ℙ_{α} for every P ∈ ℙ_{α}

and R ∈ ℙ_{β} with R ⊂ π_{β}[P].

Lemma 6. For 0 < n < ω and any continuous f: ℭ^{n} → Y there exist a basic clopen subset U = ∏_{i<n} U_{i} of ℭ^{n}, an A ⊂ n, and, if A ≠ n, a dense G_{δ} subset G of W = ∏_{i∈n\A} U_{i} such that

• f ↾ U does not depend on the variables x_{j} for j ∈ A;

• if A ≠ n then f ↾ U, considered as a function of the variables x_{i} with i ∈ n \ A, is separately nowhere constant on G.

Proof. We proceed by induction on n. If n = 1 the lemma is true by the definition of nowhere constant function. Suppose the lemma is true for n and let f: ℭ^{n+1} → Y be continuous. Denote by B_{0} a countable basis for the topology on ℭ consisting of non-empty clopen sets. For each i ≤ n and V ∈ B_{0} consider the closed set S_{i}(V) = {x⃗ ∈ ℭ^{n+1\{i}}: f ↾ {x⃗} × V is constant}.

First assume that for every i ≤ n and V ∈ B_{0} the set S_{i}(V) has empty interior. Then each set H_{i} = ℭ^{{i}} × (ℭ^{n+1\{i}} \ ⋃{S_{i}(V): B ∈ B_{0}}) is comeager in ℭ^{n+1}. So, we can apply Lemma 2 to X = ℭ, T = n + 1, K = {n + 1 \ {i}: i ≤ n}, and H = ⋂_{i≤n} H_{i} to find a comeager set G ⊂ H such that for every x ∈ G and i ≤ n the set G_{x↾n+1\{i}} = {y ∈ ℭ^{{i}}: (x ↾ n + 1 \ {i}) ∪ y ∈ G} is comeager in ℭ^{{i}}. Note also that this last property implies that f ↾ G is separately nowhere-constant, since for every x ∈ G its restriction x ↾ n+1\{i} does not belong to ⋃{S_{i}(V): B ∈ B_{0}}. Thus, in this case the lemma is satisfied with U = ℭ^{n+1}, A = ∅, and the above chosen G.

So, assume that there exist i ≤ n and V_{i} ∈ B_{0} such that the set S_{i}(V_{i}) has non-empty interior. Let V^{∗} ⊂ S_{i}(V_{i}) be a non-empty basic clopen subset of ℭ^{n+1\{i}}. Then V^{∗} = ∏_{j≠i} V_{j}, where V_{j} ⊂ ℭ is a basic clopen set for every j ≠ i. If V = ∏_{j≤n} V_{j} then V is homeomorphic to ℭ^{n+1}, f ↾ V does not depend on the variable x_{i}, and we can consider f ↾ V as a function g from V^{∗} to Y. By applying our inductive hypothesis to g we can find a basic clopen subset U^{∗} = ∏_{j≠i} U_{j} of V^{∗}, a set A^{∗} ⊂ n + 1 \ {i}, and, if A^{∗} ≠ n + 1 \ {i}, a dense G_{δ} subset G of W = ∏_{n+1\A} U_{j}, where A = A^{∗} ∪ {i}, satisfying the lemma for g. But then U = ∏_{j≤n} U_{j}, where U_{i} = V_{i}, and the sets A and G are as desired.

Here is the main example of the paper.

Example 7. For every 0 < α < ω_{1} there is a family G_{α} ⊂ Perf(ℭ^{α}) such that

(a) G_{α} does not contain any iterated perfect set, that is, G_{α} ∩ ℙ_{α} = ∅;

(b) G_{α} is γ-prism dense for every 0 < γ < α;

(c) G_{α} is n-cube dense for every 0 < n < min{α, ω};

(d) if G^{∗} ∈ [G_{α}]^{<𝔠} then |ℭ^{α} \ ⋃ G^{∗}| = 𝔠.

Proof. For ξ < α let K_{ξ} = {P ∈ Perf(ℭ^{α}): π_{ξ} ↾ π_{ξ+1}[P] is one-to-one}, where in case of ξ = 0 we understand the definition of K_{0} as {P: π_{1}[P] is a singleton}. It is worth to note that {P ∈Perf(ℭ^{α}): π_{ξ} ↾P is one-to-one} ⊂ K_{ξ}. We define G_{α} = ⋃_{ξ<α} K_{ξ}.

To see (a) take P ∈ ℙ_{α} and ξ < α. We need to show that P ∉ K_{ξ}. But by (C) we have π_{ξ+1}[P] ∈ ℙ_{ξ+1} and then (B) shows that P ∉ K_{ξ}.

We prove (b) by induction on α. Clearly it holds for α = 1. So, assume that for some 1 < α < ω_{1} condition (b) holds for every non-zero α′ < α. To see that (b) holds for α fix 0 < γ < α and a continuous injection f: ℭ^{γ} → ℭ^{α}. We need to find a Q ∈ ℙ_{γ} for which f[Q] ∈ G_{α}.

Let g = π_{γ} ∘ f. By [4, Lemma 3.2.2] there exist P ∈ ℙ_{γ} and 0 < β ≤ γ such that h = g ∘ π_{β}^{−1} is a function on π_{β}[P] (i.e., g ↾ P does not depend on coordinates δ ≥ β) and this function is either one-to-one or constant. If h is constant then π_{1}[f[P]] is a singleton and f[P] ∈ K_{0} ⊂ G_{α}. So, assume that h is one-to-one.

If β = γ then g = π_{γ} ∘ f is one-to-one on P and so π_{γ} is one-to-one on f[P]. Then f[P] ∈ K_{γ} ⊂ G_{α}. So, assume that β < γ. Then h is an injection from π_{β}[P] ∈ ℙ_{β} into ℭ^{γ}. Let φ ∈ Φ_{β} witness π_{β}[P] ∈ ℙ_{β}. Then φ maps ℭ^{β} onto π_{β}[P]. Since h ∘ φ: ℭ^{β} → ℭ^{γ} is a continuous injection, by the inductive hypothesis used for α′ = γ there exists an E ∈ ℙ_{β} such that Z = h ∘ φ[E] ∈ G_{γ}, that is, there exists a ξ < γ for which π_{ξ} ↾ π_{ξ+1}[Z] is one-to-one.

Next notice that R = φ[E] ∈ ℙ_{β}, since Φ_{β} is closed under the composition, and R ⊂ π_{β}[P]. So, by (D), Q = {x ∈ P: π_{β}(x) ∈ R} ∈ ℙ_{α}. Moreover,

Z = h ∘ φ[E] = h[R] = (g ∘ π_{β}^{−1})[π_{β}[Q]] = g[P] = π_{γ}[f[Q]]

and so π_{ξ+1}[Z] = π_{ξ+1}[π_{γ}[f[Q]]] = π_{ξ+1}[f[Q]]. Thus, π_{ξ} ↾ π_{ξ+1}[f[Q]] is one-to-one and so f[Q] ∈ K_{ξ} ⊂ G_{α}.

To show (c) we will prove by induction on 0 < n < ω the statement

for every 0 < α < ω_{1} if n < α then G_{α} is n-cube dense.

So, take 0 < n < ω and assume that the statement holds for all non-zero k < n. Take an α > n. To prove that G_{α} is n-cube dense fix a continuous injection f: ℭ^{n} → ℭ^{α}. Then π_{n}∘f: ℭ^{n} → ℭ^{n} is continuous. Apply Lemma 6 to π_{n} ∘ f to find U = ∏_{i<n} U_{i} ⊂ ℭ^{n}, A ⊂ n, and G satisfying the lemma.

If A = n then π_{n}[f[U]] = (π_{n} ∘ f)[U] is a singleton and f[U] ∈ K_{0} ⊂ G_{α}. If A = ∅ then π_{n} ∘ f ↾ G is continuous separately nowhere constant.

So, by Theorem 1, there exist perfect sets {P_{i} ⊂ U_{i}: i < n} such that (π_{n} ∘ f) ↾ ∏_{i<n} P_{i} is one-to-one. Then, π_{n} ↾ f [∏_{i<n} P_{i}] is one-to-one and so f [∏_{i<n} P_{i}] ∈ K_{n} ⊂ G_{α}.

So, assume that ∅ ≠ A ≠ n and let k = |n \ A|. Then 0 < k < n. Since (π_{n} ∘ f) ↾ U does not depend on the variables x_{j} for j ∈ A, it can be considered as a function g on W = ∏_{i∈n\A} U_{i}. Moreover, g ↾ G is separately nowhere constant. Thus, by Theorem 1, we can find a perfect cube P = ∏_{i∈n\A} P_{i} ⊂ G ⊂ ∏_{i∈n\A} U_{i} on which g is one-to-one. Thus, g is a continuous injection from P, which can be identified with ℭ^{k}, into ℭ^{n}. Since, by the inductive assumption, G_{n} is k-cube dense, there exists a perfect cube C = ∏_{i∈n\A} C_{i} ⊂ ∏_{i∈n\A} P_{i} such that g[C] ∈ G_{n}. Let C_{i} = U_{i} for i ∈ A. Then Q = ∏_{i<n} C_{i} ⊂ ∏_{i<n} U_{i} is a perfect cube and π_{n}[f[Q]] = (π_{n} ∘ f)[Q] = g[C] ∈ G_{n}. So, there exists a ξ < n such that π_{ξ} is one-to-one on π_{ξ+1}[π_{n}[f[Q]]] = π_{ξ+1}[f[Q]], So, f[Q] ∈ K_{ξ} ⊂ G_{α}.

Now, to argue for (d) fix a G^{∗} ∈ [G_{α}]^{<𝔠}. We need to show that |ℭ^{α}\⋃ G^{∗}| = 𝔠. For ξ < α let G_{ξ}^{∗} = G^{∗} ∩ K_{ξ}. By induction on ξ < α choose

x(ξ) ∈ ℭ \ {z(ξ): z ∈ G_{ξ}^{∗} & z(η) = x(η) for every η < ξ}.

Note that at each step we have less than continuum many restricted points since for every z ∈ K_{ξ} the set {z(ξ): z(η) = x(η) for every η < ξ} may have at most one element. It is easy to see that x = ⟨x(ξ): ξ < α⟩ ∈ ℭ^{α}\⋃_{ξ<α} G_{ξ}^{∗} =ℭ^{α}\G^{∗}. To finish the proof it is enough to notice that the value of x(0) can be chosen in continuum many ways, so indeed |ℭ^{α}\⋃ G^{∗}|=𝔠.

To transport the above example into an arbitrary Polish space we will use the following simple fact.

Fact 8. Let h be a homeomorphic embedding of a Polish space Y into a Polish space X, let F ⊂ Perf(Y), and put F^{∗} = {h[F]: F ∈ F} ∪ Perf(X \ h[Y [). Then for every 1 ≤ α ≤ ω_{1} the following conditions are equivalent.

(a) F is α-cube (α-prism) dense in Y.

(b) F^{∗} is α-cube (α-prism) dense in X.

Proof. “(a)⟹(b)” Let f: ℭ^{α} → X be injective and continuous. Since h[Y] is a G_{δ}-set in X we can apply [2, Claim 3.2] (see also [4, Claim 1.21.5]) to find a perfect cube C ⊂ ℭ^{α} such that either f[C] ⊂ h[Y] or f[C] ∩ h[Y] = ∅.

If f[C] ∩ h[Y] = ∅ then f[C] ∈ F^{∗} and we are done. If f[C] ⊂ h[Y] then h^{−1} ∘ f: C → Y is a continuous injection. Identifying C with ℭ^{α} and using to h^{−1} ∘ f the α-cube (α-prism) density of F in Y we can find a C′ ⊂ C such that C′ is a perfect cube (belongs to ℙ_{α}) and F = (h^{−1} ∘ f)[C′] ∈ F. So f[C′] = h[F] ∈ F^{∗}. The family F^{∗} is as desired.

The other implication is easy.

Corollary 9. For every 1 < α < ω_{1} and every Polish space X there exists a family F_{α} ⊂ Perf(X) such that: F_{α} is not α-prism dense; F_{α} is β-prism dense for every 0 < β < α; F_{α} is n-cube dense for every 0 < n < min{α, ω}; |X \ ⋃ F_{α}| = 𝔠.

Proof. First note that it is enough to find such an F_{α} for X = ℭ^{α}. Indeed, if F is such a family and h, X is an arbitrary Polish space, and h is an embedding from ℭ^{α} into X, then the family F^{∗} from Fact 8 is as desired.

Thus, it is enough to notice that the family G_{α} from Example 7 is not α-prism dense. But this is the case since for the identity function f on ℭ^{α} there is no P ∈ ℙ for which f[P] = P ∈ G_{α}.

Proof of Theorem 3. Use Corollary 9 with α ≥ ω.

Proof of Theorem 2. The vertical implications, that α-cube density implies α-prism density, follows from the fact (A), that every perfect cube in ℭ^{α} is also in ℙ_{α}. For 0 < β < α < ω_{1} the implications “C_{α} ⟹ C_{β}” and “P_{α} ⟹ P_{β}” are obvious. P_{2} implies C_{2} since 1-prism density is just perfect set density (ℙ_{1} = Perf(ℭ^{1}), as Φ_{1} consists just of autohomeomorphisms of ℭ^{1}) and so it implies 1-cube density.

To see that for ω < α < ω_{1} we have “C_{ω+1} ⟹ C_{α}” it is enough to notice that any ω-cube dense family is also β-cube dense for any ω ≤ β < ω_{1}. This is the case since the coordinatewise homeomorphism between ℭ^{ω} and ℭ^{β} preserves perfect cubes.

The fact that no other horizontal implication can be reversed is justified by the family F_{α} from Corollary 9 for different values of α. Indeed, F_{α} clearly justifies “P_{α} ≠⇒ P_{γ}” for any 1 < α < γ since it satisfies P_{α} but not P_{γ} as it is not α-prism dense. If 1 < m < n ≤ ω then F_{m} also witness “C_{m} ≠⇒ C_{n}” since it satisfies C_{m} but not C_{n}, since it cannot be m-cube dense without being m-prism dense.

The fact that none of the vertical implications “C_{α} ⟹ P_{α}”, for 2 < α < ω_{1}, can be reversed is justified by any family which is α-prism dense for every α but is not 2-cube dense. There are many such families. For example, this is the case for the family F of all linearly independent (over ℚ) subsets of ℝ. It is shown in [3] (see also [4, Corollary 5.1.2]) that this F is α-prism dense for every 0 < α < ω_{1}. On the other hand it is not 2-cube dense, as shown by the following function f. (See [3, Remark 5.2] or [4, Remark 5.1.4].) Let P_{1} and P_{2} be disjoint perfect subsets of ℝ such that P_{1} ∪ P_{1} is linearly independent over ℚ. Let f: P_{1} × P_{2} → ℝ be defined by f(x_{1}, x_{2}) = x_{1} + x_{2}. Identifying P_{1} and P_{2} with ℭ we think about f as defined on ℭ^{2}. It is easy to see that if each of the sets Q_{1} ⊂ P_{1} and Q_{2} ⊂ P_{2} has at least two elements then f[Q_{1} × Q_{2}] is linearly dependent.

Another such example is a family F of all P ∈ Perf(ℭ^{2}) such that the projection on one of the coordinates is one-to-one. It follows quite easily from [4, Lemma 3.2.2] that F is α-prism dense for every α. (See e.g. [4, Proposition 4.1.3].) It is not 2-cube dense since for the identity function f: ℭ^{2} → ℭ^{2} there is no perfect cube C for which f[P] ∈ F.

4. Final remarks

It is also worth to notice that we have the following implications.

Proposition 10. If β + 1 ≤ α < ω_{1} then every α-prism (α-cube) dense family is also (β + 1)-prism ((β + 1)-cube) dense. In particular, if 0 < m < n ≤ ω then every n-cube dense family is also m-cube dense.

Proof. Let g: ℭ^{α\β} → ℭ be a homeomorphism, and let h: ℭ^{α} → ℭ^{β+1} be defined by h(x)(ξ) = x(ξ) for every ξ < β and h(x)(β) = g(x ↾ α \ β). It is easy to see that h is a homeomorphism and that if P ⊂ ℭ^{α} is a perfect cube (belongs to ℙ_{α}) then h[P] is a perfect cube (belongs to ℙ_{β+1}).

Now, let F ⊂ Perf(X) be α-prism α-cube) dense in X. To see that F is (β + 1)-prism ((β+1)-cube) dense take a continuous injection f: ℭ^{β+1} → X. Then f ∘ h: ℭ^{α} → X is also a continuous injection. Since F is α-prism (αcube) dense, there exists a P ⊂ ℭ^{α} such that P belongs to ℙ_{α} (is a perfect cube) and f[h[P]] = (f ∘ h)[P] ∈ F. But h[P] belongs to ℙ_{β+1} (is a perfect cube), so F is (β + 1)-prism ((β + 1)-cube) dense.

We do not know if, in general, for a limit ordinal λ < ω_{1} the (λ+1)-prism density implies λ-prism density.

The next example shows that Lemma 6 fails, in a strong way, for functions defined on infinite product.

Example 11. There exists a continuous function f: ℭ^{ω} → ℭ^{ω} such that for every perfect cube P there is an n < ω such that f ↾ P is one-to-one on some section of n-th variable, and is constant on some other sections of the same variable.

Proof. For n < ω let f_{n}: ℭ^{2} → ℭ be defined by f_{n}(x, y)(i) = y(n) · x(i). Clearly f_{n} is continuous. Moreover, if y(n) = 1 then f_{n}(·, y) is the identity function, so it is one-to-one; if y(n) = 0 then f_{n}(·, y) is constant equal to 0.

For ⟨x_{n}: n < ω⟩ ∈ ℭ^{ω} define f(⟨x_{n}: n < ω⟩) = ⟨f_{n}(x_{n+1}, x_{0}): n < ω⟩. Then f is clearly continuous. Consider f restricted to a perfect cube P = ∏_{n<ω} P_{n}. Let a, b ∈ P_{0} be distinct and let n < ω be such that a(n) ≠ b(n). Assume that a(n) = 0 and let z ∈ ℭ^{ω\{n+1}}. Look at f ↾ P on a section given by z and note that: if z(0) = a then f ↾ P is constant on this section; if z(0) = b then f ↾ P is one-to-one on this section.

References

[1] Ciesielski, K., Set Theory for the Working Mathematician, London Math. Soc. Stud. Texts 39, Cambridge Univ. Press, Cambridge, 1997.

[2] Ciesielski, K., Pawlikowski, J., Covering Property Axiom CPA_{cube} and its consequences, Fund. Math. 176(1) (2003), 63–75. (Preprint^{⋆} available.^{2})

[3] Ciesielski, K., Pawlikowski, J., Nice Hamel bases under the Covering Property Axiom, Acta Math. Hungar. 105(3) (2004), 197–213. (Preprint^{⋆} available.)

[4] Ciesielski, K., Pawlikowski, J., Covering Property Axiom CPA. A Combinatorial Core of the Iterated Perfect Set Model, Cambridge Tracts in Math.164, Cambridge Univ. Press, Cambrigde, 2004.

[5] Kechris, A., Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer Verlag, New York, 1994.

[6] Miller, A. W., Special subsets of the real line, in “Handbook of Set-Theoretic Topology” (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, 201–233.

Krzysztof Ciesielski Andrés Millán Department of Mathematics Department of Mathematics West Virginia University West Virginia University Morgantown, WV 26506-6310 Morgantown, WV 26506-6310 USA USA e-mail: K Cies@math.wvu.edu e-mail: amillan@math.wvu.edu

web page: http://www.math.wvu.edu/~kcies

2Preprints marked by ^{⋆} are available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html