ON TWO CONSEQUENCES OF CH ESTABLISHED BY SIERPI ´NSKI

. We study the relations between two consequences of the Continuum Hypothesis discovered by Wac law Sierpi´nski, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, deﬁned on sub-sets of the real line of cardinality continuum.


Introduction
In [11] we studied the following two consequences of the Continuum Hypothesis (CH) distinguished by Wac law Sierpiński in his classical treaty Hypothèse du continu [14] (the notation is taken from [14]): C 8 There exists a continuous function f : E → R, E ⊆ R, |E| = c, not uniformly continuous on any uncountable subset of E. C 9 There is a sequence of functions f n : E → R, E ⊆ R, |E| = c, converging pointwise but not converging uniformly on any uncountable subset of E. Sierpiński [13] checked that C 8 implies C 9 .The status of the converse implication remains unclear.Let us notice, however, that in Topology I by Kuratowski [7], footnote (3) on page 533 suggests that the two statements are in fact equivalent.
In [11] we considered the following stratifications of statements C 8 and C 9 for uncountable cardinals κ ≤ λ ≤ c: C 8 (λ, κ) There exists a set E ⊆ R of cardinality λ and a continuous function f : E − → R, which is not uniformly continuous on any subset of E of cardinality κ.C 9 (λ, κ) There exists a set E ⊆ R of cardinality λ (equivalently: for any set E ⊆ R of cardinality λ) and there is a sequence of functions f n : E → R, converging on E pointwise but not converging uniformly on any subset of E of cardinality κ.In particular, we proved in [11] that: • C 8 (c, c) ⇔ C 9 (c, c), and each of these statements is equivalent to the assertion d = c, provided that the cardinal c is regular.
• C 8 (ℵ 1 , ℵ 1 ) ⇔ C 9 (ℵ 1 , ℵ 1 ), and each of these statements is equivalent to the assertion b = ℵ 1 .Here d and b denote, as usual, the smallest cardinality of a dominating and, respectively, an unbounded family in N N corresponding to the ordering of eventual domination ≤ * (cf.[4]).
An important role in our considerations was played by the notion of a K-Lusin set (cf. [1]) which we extended declaring that an uncountable subset E of a Polish space X is a κ-K-Lusin set in X, ℵ 1 ≤ κ ≤ c, if |E ∩ K| < κ for every compact set K ⊆ X.We proved in [11] that C 9 (λ, κ) is equivalent to the statement that there is a Polish space X and a κ-K-Lusin set of cardinality λ in X.
In this note we present additional results related to the subject of [11].Most of them were earlier announced in [11,Section 4].
In Sections 2, 3 and 4 we investigate C 8 -like phenomena in a more general setting.We are interested in two closely related problems (X and Y are fixed separable metric spaces).ß Problem 1.Is the existence of a set E ⊆ X of cardinality λ and a continuous function on E with values in Y , which is not uniformly continuous on any subset of E of cardinality κ, related to either C 8 (λ, κ) or C 9 (λ, κ)? ßProblem 2. Does the existence of a set E ⊆ X of cardinality λ and a continuous function on E with values in Y , which is not uniformly continuous on any subset of E of cardinality κ, imply that there also exists such a function on E with values in R?
ßConcerning Problem 1, we observe that the existence of a separable metric space X of cardinality λ, a metric space Y , and a continuous function on X with range in Y , which is not uniformly continuous on any subset of X of cardinality κ, already implies that there exists a κ-K-Lusin set of cardinality λ in some Polish space and, consequently, that C 9 (λ, κ) holds true (cf.Proposition 2.1).
Conversely, C 9 (λ, κ) implies that there exists a κ-K-Lusin set E of cardinality λ in P, the set of irrationals of the unit interval I = [0, 1], such that for every non-σ-compact Polish space Y there is a continuous function on E which is not uniformly continuous on any subset of E of cardinality κ (cf.Theorem 2.2).
On the other hand, if Y is compact, then the existence of a set E ⊆ I of cardinality λ, and a continuous function f : E → Y , which is not uniformly continuous on any subset of E of cardinality κ, implies C 8 (λ, κ) (cf.Theorem 3.2).
ß Concerning Problem 2, we show that if a set E in the Hilbert cube I N is zero-dimensional and there exists a continuous function on E with range in any uncountable compact metric space Y , not uniformly continuous on any subset of X of cardinality κ, then there is also such a function with range in any uncountable compact metric space Z, and in particular, in I (cf.Theorem 3.1).
On the other hand, assuming CH, we prove the existence of a set E ⊆ I N of cardinality c such that there is a continuous function f : E − → I N , which is not uniformly continuous on any subset of E of cardinality c but each continuous function g : E → R is constant on a subset of E of cardinality c.The construction of a witnessing pair E and f falls under a general scheme, described in Section 4, of constructions of C 8like examples based on a generalization of the notion of a κ-K-Lusin set.
Section 5 is a slight departure from the topic but it is closely related to a reasoning of Sierpiński concerning C 9 .We shall show that a Hausdorff space X is Čech-complete and Lindelöf if and only if there is a sequence f 0 ≥ f 1 ≥ . . . of continuous functions f n : X → I converging pointwise to zero but not converging uniformly on any closed non-compact set in X (cf.Theorem 5.2).The existence of such a sequence for any Polish space X was a key step in proving that statement C 9 (λ, κ) is equivalent to the existence of a κ-K-Lusin set E of cardinality λ in P (cf.[11,Theorem 2.3]).
In this note P always denotes the set of irrationals of the unit interval I = [0, 1].It is homeomorphic to the Baire space N N , the countable product of the set of natural numbers N = {0, 1, 2, . ..} with the discrete topology (cf.[6]).

Mappings into non-compact spaces and C 9
We start with a general observation.Proposition 2.1.If there exist a separable metric space (X, d X ) of cardinality λ, a metric space (Y, d Y ), and a continuous function on X with range in Y , which is not uniformly continuous on any subset of X of cardinality κ, then there exists a κ-K-Lusin set of cardinality λ in some Polish space and, consequently, C 9 (λ, κ) holds true.
Proof.Let f : X → Y be a continuous function which is not uniformly continuous on any subset of X of cardinality κ.The function f extends (cf.[6,Theorem 3.8]) to a continuous function f : By the choice of X, we have that |K ∩ X| < κ, which shows that X is a κ-K-Lusin set of cardinality λ in G.By [11,Theorem 2.3], this proves C 9 (λ, κ).
On the other hand, statement C 9 (λ, κ) already implies (and in view of Proposition 2.1, is equivalent to) the existence of C 9 -like example for functions with values in arbitrary non-σ-compact Polish spaces.Theorem 2.2.For any uncountable cardinals κ ≤ λ ≤ c the following are equivalent: (1) C 9 (λ, κ), (2) there exist a set E ⊆ R of cardinality λ, a non-σ-compact Polish space Y , and a continuous function f : E → Y which is not uniformly continuous (with respect to arbitrary complete metric on Y ) on any subset of E of cardinality κ.
Moreover, any κ-K-Lusin set E of cardinality λ in P has the property expressed in (2) with respect to any non-σ-compact Polish space Y .
Let Y be an arbitrary non-σ-compact Polish space.Let h : P − → Y be a homeomorphic embedding of P onto a closed subspace h(P) of Y (cf.[6,Theorem 7.10]).We will show that E together with f = h|E have the required properties.
To that end, let us fix a set A ⊆ E with |A| = κ.Then Ā, the closure of A in I, is not contained in P since otherwise Ā would be a compact set in P, intersecting E on a set of cardinality κ.So let us pick a k ∈ A, k ∈ N, such that lim k→∞ a k = a and a ∈ I \ P. Now, if f were uniformly continuous on A with respect to a complete metric d on Y , f would take Cauchy sequences in A to Cauchy sequences in Y .In particular, the sequence (f (a n )) n∈N would be Cauchy in Y , hence lim k→∞ f (a k ) = z for some z ∈ Y .This, however, is not the case: since the set {a 0 , a 1 , . ..} has no accumulation point in P, the set {f (a 0 ), f (a 1 ), . ..} has no accumulation point in h(P) and hence also in Y , as h(P) is closed in Y .ß The implication (2) ⇒ (1) follows immediately from Proposition 2.1.

Mappings into compact spaces and C 8
The results of the previous section show that C 8 -like statements for functions with values in non-σ-compact Polish spaces are actually equivalent to statement C 9 .Apparently, the situation changes when we consider functions with values in compact spaces.As the following result shows, if a set E ⊆ I N is zero-dimensional and there exists a continuous function on E with range in any uncountable compact metric space Y , not uniformly continuous on any subset of X of cardinality κ, then there is also such a function with values in R, witnessing that C 8 (λ, κ) holds true.Theorem 3.1.Let E be a zero-dimensional subset of a compact metric space X.If there exists a continuous function on E with range in a compact metric space Y , not uniformly continuous on any subset of E of cardinality κ, then there is also such a function with range in the Cantor ternary set C in I. Consequently, for any uncountable compact metric space Z, there is also such a function with values in Z.
Proof.Let h : E → Y be a continuous function not uniformly continuous on any set of cardinality κ.
Using the compactness of Y , let as fix a sequence (K n , L n ) n∈N of pairs of disjoint compact sets in Y such that for any pair (K, L) of disjoint compact sets in Y , there is n with K ⊆ K n and L ⊆ L n .
For each n ∈ N, we let , and using the fact that E is zero-dimensional and separable, we choose a continuous function u n : E → {0, 2} taking on C n value 0 and on We shall show that the function f : E → I defined by the formula which takes values in the Cantor ternary set C, is not uniformly continuous on any subset of E of cardinality κ.
To that end, let us fix a set A ⊆ E with |A| = κ.We shall first make the following observation.Let a ∈ Ā, the closure of A in X.Then for any n, since u n takes on A values 0 or 2 only, the oscillation Let us note that h|A : A → Y cannot be extended to a continuous (hence, by the compactness of Ā, uniformly continuous) function h : Ā → Y , since otherwise the function h|A = h|A would itself be uniformly continuous, contrary to the the fact that |A| = κ.Consequently, there must be closed disjoint sets K, L in Y such that the closures of (h|A) −1 (K) and (h|A It follows that u n |A has the oscillation at a equal to 2 and let us assume that n is the smallest index with this property.The oscillation of each of the functions u 0 |A, . . ., u n−1 |A is then equal to 0, hence we can find a neighbourhood V of a in X such that all these functions have constant values on A ∩ V .

Let us pick x
It follows that the oscillation of f |A at a is at least 3 −n−1 .In effect, f |A has no continuous extension over Ā, which means that f is not uniformly continuous on A.
ß Finally, if (Z, d) is an arbitrary compact metric space and e : C → Z is a homeomorphic embedding of C into Z, then since e −1 is uniformly continuous, the function e • f : E → Z has desired properties.
As an immediate corollary we obtain the following equivalent form of statement C 8 (λ, κ).Theorem 3.2.For any uncountable cardinals κ ≤ λ ≤ c if the cofinality of λ is uncountable, then the following are equivalent: (1) C 8 (λ, κ), (2) there exists a set E ⊆ R of cardinality λ, a compact metric space Y and a continuous function f : E → Y , which is not uniformly continuous on any subset of E of cardinality κ.Moreover, any set E ⊆ I that witnesses C 8 (λ, κ) for some continuous function from E to I has the property expressed in (2) with respect to any uncountable compact metric space.
Proof.(1) ⇒ (2).If f : E − → R is a continuous function on a set E ⊆ R of cardinality λ, which is not uniformly continuous on any subset of E of cardinality κ, then since the cofinality of λ is uncountable, by shrinking E, if necessary, we may assume that the range of f is contained in a closed interval Y of length 1.
If additionally E ⊆ I and f : E → I, then since E contains no non-trivial interval, it is zero-dimensional, and Theorem 3.1 applies.ß (2) ⇒ (1).Now let f : E → Y be a continuous function with values in a compact metric space Y , which is not uniformly continuous on any subset of E of cardinality κ.We may again assume that E is contained in a closed interval X.Then, E being zero-dimensional, it is enough to apply the final part of the assertion of Theorem 3.1 to Z = I.
By the results of [11] (cf. Section 1), Theorems 2.2 and 3.2 lead to the following corollary.Throughout this section we assume that G is an uncountable G δ -set in a compact metric space X.
Given a collection K of compact sets in X containing all singletons, we say that an uncountable subset The C 8 -like examples, presented later in this section by means of K -Lusin sets, are based on the following observation, applied also in the proofs of [11, Theorems 3.8 and 3.9].Proposition 4.1.Let ϕ : X − → Y be a continuous map onto a compact metric space Y such that ϕ|G is a homeomorphism onto ϕ(G).Let K be a collection of compact sets in X such that whenever A ⊆ G and ϕ|A extends to a homeomorphism over Ā, the closure of A in X, then we obtain a continuous function on a set of cardinality λ, which is not uniformly continuous on any subset of E of cardinality κ.
Proof.Aiming at a contradiction, assume that f |B is uniformly continuous (with respect to any metric compatible with the topology of Y ) on a set B ⊆ E of cardinality κ and let A = f (B) = ϕ −1 (B).Then, since ϕ|A : A → B is also uniformly continuous, the function ϕ|A extends to a homeomorphism over Ā (cf.[2, Theorem 4.3.17]).Consequently, Ā ∈ K .This, however, is impossible, since on one hand we have |T ∩ Ā| < κ, T being a κ-K -Lusin set in X, but on the other hand A ⊆ T ∩ Ā has cardinality κ.

4.1.
Zero-dimensional spaces.Throughout this subsection let us additionally assume that the (compact metric) space X is zero-dimensional.ß A proof of the following fact is given in [9, Lemma 4.2] (it is based on an idea similar to that in [3, proof of Lemma 5.3]).Lemma 4.2.For any G δ -set G in X there is a continuous map ϕ : X → Y onto a compact metric space Y , such that ϕ|G is a homeomorphism onto ϕ(G), ϕ(X \ G) ∩ ϕ(G) = ∅ and the set Y \ ϕ(G) is countable.
Various C 8 -like examples could be constructed with the help of Lemma 4.2 and the following observation.By a σ-ideal on X we mean a collection I of Borel sets in X, closed under taking Borel subsets and countable unions of elements of I, and containing all singletons.Proposition 4.3.Let I be a σ-ideal on X and let K be the collection of all compact sets from I.
If G is a G δ set in X such that G ∈ I and ϕ : X − → Y is a continuous map described in Lemma 4.2, then Ā ∈ K for any A ⊆ G such that ϕ|A extends to a homeomorphism onto Ā.
we obtain a continuous function on a set of cardinality λ, which is not uniformly continuous on any subset of E of cardinality κ.

Proof. Let us fix A ⊆ G such that ϕ|A extends to a homeomorphism φ between Ā and ϕ(A). Then the set Ā
The final assertion follows directly from Proposition 4.1.
As the following proposition shows, the above observation could be applied to various natural σ-ideals.Let us recall that un uncountable set T in a Polish space Y is a Lusin set in Y , if |T ∩ D| < ℵ 1 for every closed nowhere-dense subset of Y .Proposition 4.4.Let I be a σ-ideal on X, let K be the collection of all compact sets from I and let us assume that I is not generated by K (i.e., there is a set from I which is not covered by any F σ -set from I).Then, there exists a G δ -set in X such that G ∈ I but no non-empty, relatively open set in G is covered by an F σ -set from I. Consequently, every Lusin set T in G is ℵ 1 -K -Lusin in G hence, it gives rise to a C 8 -example (as described in Proposition 4.3).
Proof.Let B ∈ I be (a Borel) set not covered by any F σ -set from I. Then the existence of a G δ -set in X such that G ⊆ B but G is not covered by any F σ -set from I follows from a theorem of Solecki (see [15]).By shrinking G, if necessary, we may assume that G has the desired properties.
It follows that if K ∈ K , then G ∩ K is meager in G, and, consequently, T ∩ K is countable for any Lusin set T in G.
Remark 4.5.A typical example of the situation described in Propositions 4.3 and 4.4 is when X is a copy of the Cantor in R of positive Lebesgue measure, I is the σ-ideal of Lebesgue measure zero Borel sets in X (then K is the family of closed Lebesgue measure zero sets in X) and G is a dense copy of irrationals in X of Lebesgue measure zero.
Then the function f = ϕ −1 |H : H → R, where H = ϕ(G), is a homeomorphic embedding of H, a copy of irrationals, to R, with the property that for every Lusin set L of cardinality c in H, the function f |L is not uniformly continuous on any uncountable subset of L. This provides an alternative proof of the theorem of Sierpiński that the existence of a Lusin set of cardinality c in P implies C 8 (cf.[14, proof of Théorème 6 on page 45]).

Infinite-dimensional spaces.
The assumption that the space X is zero-dimensional in Theorem 3.1 is essential, as demonstrated by the following result (CH in this theorem can be weakened to the assumption that no family of less than c meager sets covers R, cf.Remark 4.8).
Theorem 4.6.Assuming CH, there exists a set E ⊆ I N of cardinality c such that (1) there is a continuous function f : E → I N , which is not uniformly continuous on any subset of E of cardinality c, (2) each continuous function g : E → R is constant on a subset of E of cardinality c.
A key element of the proof of this theorem is a Henderson compactum -a compact metrizable infinite-dimensional space whose all compact subsets of finite dimension are zero-dimensional, cf.[10].
More specifically, we shall need the following fact, where punctiform sets are the sets without non-trivial subcontinua, cf.[3, 1.4.3].Lemma 4.7.There exists a non-empty punctiform G δ -set M in a Henderson compactum H ⊆ I N such that for each M ′ ⊆ M with dim (M \ M ′ ) ≤ 0, every continuous function u : M ′ → R is constant on a set of positive dimension.
A justification of this lemma is rather standard, but since we did not find convenient references, we shall give a proof to this effect at the end of this subsection.For now, taking this fact for granted, we shall proceed with a proof of our main result.
Proof of Theorem 4.6.Let us adopt the notation of Lemma 4.7, and let K be the collection of all compact zero-dimensional sets in the Henderson compactum H.
Using CH, we inductively construct a c-K -Lusin set T in M such that |T ∩ L| = c for every Borel set L in M with dim L > 0. To that end, we list all elements of K as K α : α < c and all Borel sets L in M with dim L > 0 as L α : α < c , repeating each such set L continuum many times.Then, we subsequently pick for ξ < c using the fact that, under CH, the set in brackets is zero-dimensional, by the sum theorem for dimension zero (cf.[3, 1.3.1]).Finally, we let In particular, T meets each Borel set in M of positive dimension in continuum many points.
Indeed, w can be extended to a continuous function w : Let us now apply a counterpart of Lemma 4.2 for infinite-dimensional spaces to the following effect (cf. the proof of [3, Lemma 5.3.1]):there exists a continuous surjection ϕ : Claim 2. The set E = ϕ(T ) satisfies the assertion of Theorem 4.6.
Since E is homeomorphic to T , Claim 1 shows that (2) in Theorem 4.6 is satisfied.
To check also assertion (1) in this theorem, we shall make sure that the function f = ϕ −1 |E : E → H is not uniformly continuous on any set of cardinality c.
Since T is a c-K -Lusin set in M, by Proposition 4.1 it is enough to verify that for any non-empty A ⊆ M, whenever ϕ|A is an embedding (the closure of A is in H), then Ā ∈ K , i.e., dim Ā = 0.
In the assumptions of Theorem 4.6, CH can be weakened to the assertion (usually denoted by cov(M ) = c, cf.[5]) that no family of less than c meager sets covers R. The only change in the proof requires checking that the inductive definition of the sequence p ξ : ξ < c is correct.More precisely, the following is true.
Then U K is open and dense in the function space C(I N , R) and this space being perfect Polish, the assumption cov(M ) = c guarantees that the set [6, 8.32]).
Let us recall (cf.[3]) that a closed set L in a topological space Z separates the space Z between sets A 1 , A 2 ⊆ Z, if Z \ L = U 0 ∪ U 1 , where U 0 , U 1 are open, disjoint and A i ⊆ U i , for i = 0, 1.If A 1 and A 2 are singletons, then we say that L separates Z between the respective points.
Let F = K .To prove that dim F = 0, it is enough to show that for any pair of disjoint compact sets A, B in I N , I N can be separated between A and B by a closed set disjoint from F .Indeed, let x ∈ F and let U be a relatively open subset of F with x ∈ U and F \ U = ∅.Then U = V ∩ F for an open V in I N with B = I N \ V = ∅, and separating I N between A = {x} and B by a closed set in So let A, B be disjoint compact sets in I N .We can pick f ∈ G such that f (A) ⊆ (−∞, 0), f (B) ⊆ (0, +∞), as such functions form an open, non-empty set in C(I N , R).Then L = f −1 (0) is a closed set in I N separating I N between A and B with the property that L ∩ K = ∅ for all K ∈ K , cf. ( 1) and (2).
Proof of Lemma 4.7.Let us fix a Henderson continuum K in I N .We shall consider on I N the metric assigning to points (s 0 , s 1 , . ..), (t 0 , t 1 , . ..)This theorem goes back to Mazurkiewicz, and can be justified by the reasoning in the proof of [10,Theorem 3.9.3]We adopt the notation from Claim 1.Since K is a compact, connected set in I N , we can find open connected neighbourhoods Let D be a countable set dense in S, and let (c n , d n ) : n ∈ N be an enumeration of all ordered pairs of distinct points from D, such that each such pair appears in the sequence infinitely many times.
Let us fix n > 0. We shall define an embedding h n : I N → U n such that the distance from h n (a) to c n and the distance from h n (b) to d n is less than 1 n .To that end, let us notice that U n is arcwise connected, being an open, connected set in I N (cf.[16,Proposition 12.25]), and hence there is a continuous f : Let m ≥ n be large enough to ensure that for any x, y ∈ I N , whenever the first m coordinates of x and f (y) coincide, then x ∈ U n .Now, denoting by p j : I N → I the projection onto j'th coordinate, we define the embedding h n by p i (h n (x)) = p i (f (x)) for i ≤ m and p m+i (h n (x)) = p i (x) for i = 1, 2, . ...
Having defined the embeddings h n , we shall check that the set M in Claim 2 satisfies the assertion of Lemma 4.7.
Proof.In view of [11,Theorem 2.1] we only have to prove the "if" part of the above equivalence.So let (X, d) be a separable metric space.Let f n : X − → I, n ∈ N, be a sequence of continuous functions with f 0 ≥ f 1 ≥ . .., which converges to zero pointwise but does not converge uniformly on any set with non-compact closure in X.
Let ( X, d) be the completion of (X, d); clearly, the space X is Polish.There exists a G δ -set X in X, containing X such that each f n extends to a continuous function fn : X → I. Since X is dense in X, we have f0 ≥ f1 ≥ . . .and in particular, the set Since X ⊆ G, it remains to make sure that G ⊆ X, to conclude that X = G is completely metrizable as a G δ -subset of the (Polish) space X.
So let c ∈ G and let us pick x k ∈ X, k ∈ N, such that lim k→∞ x k = c.Let K = {c} ∪ {x k : k ∈ N}.Then K is compact, so the sequence ( fn ) n∈N converges uniformly on K (this is a special instance of the Dini theorem [2, Lemma 3.2.18]).It follows that the sequence (f n ) n∈N converges uniformly on K ∩X, a closed set in X which therefore, by the assumed property of (f n ) n∈N , is compact.Since the sequence (x n ) n∈N converges to c and all x n are elements of K ∩ X, so is c.In particular, c ∈ X.
With the help of Theorem 5.1 we shall establish the following more general result (for terminology see [2]).
Theorem 5.2.Let X be a Hausdorff space.Then X is a Čechcomplete Lindelöf space if and only if there is a sequence f 0 ≥ f 1 ≥ . . . of continuous functions f n : X → I converging pointwise to zero but not converging uniformly on any closed non-compact set in X.
Proof.First, let X be a Čech-complete Lindelöf space.By a theorem of Frolík, it follows that there is a perfect map p : X → Y onto a Polish space Y , cf. [2, 5.5.9].Let us recall that for a Hausdorff space X and a metrizable space Y this means, cf.[2, Theorems 3.7.2 and 3.7.18],that p is a continuous, closed mapping and the inverse image of every compact subset of Y is compact.It is straightforward to check that if functions f n : Y → I, n ∈ N, with f 0 ≥ f 1 ≥ . .., satisfy the assertion of Theorem 5.1, the functions f n • p : X → I, n ∈ N, have the required properties.
ß Next, given a sequence f 0 ≥ f 1 ≥ . . . of functions f n : X → I, described in the theorem, let us consider the diagonal map (1) F = (f 0 , f 1 , . ..) : X → I N and let Y = F (X).

Corollary 3 . 3 . 4 .
If either κ = λ = c or κ = λ = ℵ 1 , then the following are equivalent: (1) There exist a set E ⊆ R of cardinality λ, a non-σ-compact Polish space Y , and a continuous function f : E → Y which is not uniformly continuous (with respect to any complete metric on Y ) on any subset of E of cardinality κ. (2) There exist a set E ⊆ R of cardinality λ, a compact metric space Y and a continuous function f : E → Y , which is not uniformly continuous on any subset of E of cardinality κ.Constructing C 8 -like examples from K -Lusin sets

Remark 4 . 8 .
Assuming cov(M ) = c, the union of any family K of compact zero-dimensional sets in I N , with |K | < c, is zero-dimensional.Proof.Let K be a family of compact zero-dimensional sets in I N , with |K | < c.For any K ∈ K , let us consider(1)

Claim 1 .
There exists a punctiform G δ -set S in K and distinct points a, b ∈ S such that each relatively closed set in S separating S between a and b has dimension at least 2. Indeed, since dim K ≥ 4, K contains a punctiform G δ -set W with dim W ≥ 3.
and H = M -the closure of M in I N × I, one obtains sets M ⊆ H satisfying the assertion of Lemma 4.7.
[12,ied to a pair of disjoint compact sets A and B in K such that for any set N in K with dim N ≤ 2 there is a continuum in K joining A and B and missing N (cf.[12,Theorem4.2]), and to a continuous map π :K → [−1, 1] sending A to −1 and B to 1.ß Now, since dim W ≥ 3, there is a ∈ W such that all sufficiently small neighbourhoods of a in W have boundaries of dimension ≥ 2.An argument in [8, proof of Theorem 8, p. 172] gives a point b ∈ K \ {a} such that for S = W ∪ {b}, every relatively closed set in S separating S between a and b has dimension ≥ 2. Since S is a punctiform G δ -set in K, it satisfies Claim 1.Claim 2. There are embeddings h n