Cantor Cube Research Articles

Extending homeomorphisms on Cantor cubes

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum Dτ. For every set P⊂Dτ let LP be the set of cardinality λ such that the λ-interior of P is not empty. It is established that any homeomorphism f between two proper closed subsets P and K of Dτ can be extended to an autohomeomorphism of Dτ provided the sets LP and LK do not have so many points of discontinuity and f preserves the λ-interiors of P and K.

On Hausdorff operators in ZF$\mathsf {ZF}$

AbstractA Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

On homogeneity of Cantor cubes

AbstractWe discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$ . It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$ .

Error recognition in the Cantor cube

Abstract. Based on the notion of thin sets introduced recently by T. Banakh, Sz. Głąb, E. Jabłońska and J. Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube ℤ 2 ω ). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D. Niwiński and E. Kopczyński in 2014.

NULL SETS AND COMBINATORIAL COVERING PROPERTIES

AbstractA subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $ , a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

On metrizability and compactness of certain products without the Axiom of Choice

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15].Furthermore, we strengthen a result in the above paper, namely “Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice Principle”, by first observing that the above topological statement implies the weak choice principle MC(ℵ0,ℵ0), i.e. “Every countably infinite set of countably infinite sets has a multiple choice function”, and then by establishing that van Douwen's choice principle is strictly weaker than MC(ℵ0,ℵ0) in ZF. The above independence result also resolves an open problem in P. Howard and J.E. Rubin (1998) [7], and strengthens a result by P. Howard, K. Keremedis, H. Rubin, and J.E. Rubin (1998) [6].

Copies of $c_0(\tau )$ spaces in projective tensor products

This paper is dedicated to the memory of our friend Eve Oja. Abstract. Let X and Y be Banach spaces and consider the projective tensor product X ⊗ π Y. Suppose that τ is an infinite cardinal, X has the bounded approximation property and the density character of X is strictly smaller than the cofinality of τ. We prove the following c 0 (τ) generalizations of classical c 0 results due to Oja (1991) and Kwapien (1974) respectively: (i) If c 0 (τ) is isomorphic to a complemented subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a complemented subspace of Y. (ii) If c 0 (τ) is isomorphic to a subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a subspace of Y. We also show that the result (i) is optimal for regular cardinals τ and Banach spaces X without copies of c 0 (τ). In order to do so, we provide a c 0 (τ) extension of a classical c 0 result due to Emmanuele (1988) concerning the c 0 (τ) complemented subspaces of Lp(D τ , Y) spaces, 1 ≤ p ≤ ∞, where D τ is the Cantor cube. Finally, as a consequence of (i) we conclude that under the continuum hypothesis, the space c 0 (ℵα), with α > 1, is not isomorphic to a complemented subspace of l∞ ⊗ π l∞(ℵα).

An example of a capacity for which all positive Borel sets are thick

On the Cantor cube {0,1}N with the standard product topology we construct a finite Choquet capacity with respect to the family of all compact sets such that every compact set of positive capacity contains continuum many pairwise disjoint compact subsets of positive capacity.

Quasi-metrizability of products in ZF and equivalences of CUT(fin)

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product XJ is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT(fin) are deduced. It is shown that, in every model of ZF+¬CUT(fin), for an uncountable set J, the space NJ can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

Extension operators and twisted sums of c0 and C(K) spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c0 and C(K), i.e., does there exist a Banach space X containing a non-complemented copy Y of c0 such that the quotient space X/Y is isomorphic to C(K)?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2ω1 or K is a separable scattered compact space of height 3 and weight ω1, then every twisted sum of c0 and C(K) is trivial.We also construct nontrivial twisted sums of c0 and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K⊆L which do not admit an extension operator C(K)→C(L).

On variants of the principle of consistent choices, the minimal cover property and the 2-compactness of generalized Cantor cubes

In set theory without the Axiom of Choice (AC), we study the deductive strength of variants of the Principle of Consistent Choices (PCC) and their relationship with the minimal cover property, the 2-compactness of generalized Cantor cubes, and with certain weak choice principles. (Complete definitions are given in Section “Notation and terminology”.) Among other results, we establish the following:1.“For every infinite set X, the generalized Cantor cube 2X has the minimal cover property” (MCP) implies “PCC restricted to families of 2-element sets” (F2), which in turn implies “for every infinite set X, 2X is 2-compact” (Q(2)). Moreover, ‘MCP implies F2’ is not reversible in ZFA (i.e., ZF, the Zermelo–Fraenkel set theory minus AC, with the Axiom of Extensionality weakened in order to permit the existence of atoms).The above results strengthen related results in Howard–Tachtsis “On the minimal cover property in ZF” and “On the set-theoretic strength of the n-compactness of generalized Cantor cubes”.2.“Every Dedekind-finite set is finite” (DF=F) implies the Principle of Partial Consistent Choices (PPCC) – the latter principle being introduced here – which in turn implies ACfinℵ0 (i.e., the axiom of choice for denumerable families of non-empty finite sets). None of the previous implications is reversible in ZF.3.The Principle of Countable Consistent Choices (PCCℵ0), which is introduced here, is equivalent to ACfinℵ0.4.Rado's Lemma (RL) + “every infinite set has an infinite linearly orderable subset” implies PPCC. In addition, RL does not imply PPCC in ZFA, PPCC does not imply RL in ZF, and PPCC does not imply “every infinite set has an infinite linearly orderable subset” in ZFA.5.PPCC does not imply “there are no amorphous sets” in ZFA.6.F2 implies “there are no amorphous sets” and the implication is not reversible in ZF. This clarifies the relationship between the latter two statements, whose status is mentioned as unknown in Howard–Rubin “Consequences of the Axiom of Choice”.7.“Every infinite partially ordered set has either an infinite chain or an infinite antichain” (CAC) does not imply X, where X∈{PPCC,F2}, in ZF. In addition, F2 does not imply CAC in ZFA.

Classifying homogeneous cellular ordinal balleans up to coarse equivalence

For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $\gamma$ we shall define a ballean $2^{<\gamma}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(\gamma)$ plays a role analogous to the role of the Cantor cube $2^{\kappa}$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<\gamma}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^\omega$.

On the set-theoretic strength of the $n$-compactness of generalized Cantor cubes

On the set-theoretic strength of the $n$-compactness of generalized Cantor cubes

Ai-maximal independent families and irresolvable Baire spaces

A topological space is almost irresolvable if it cannot be written as a countable union of subsets with empty interior. Given a cardinal κ, denote by (⋆κ) the statement ‘‘the Cantor cube 22κ has a dense subspace of size κ which is almost irresolvable and whose dispersion character is equal to κ.’’ In this paper we prove:(1)(⋆κ) is equivalent to the existence of a dense subspace of 22κ which is Baire submaximal and whose cardinality and dispersion character are both equal to κ. In particular, (⋆κ) implies that κ is measurable in an inner model of ZFC.(2)If the Continuum Hypothesis holds, (⋆κ) fails for all κ.(3)(⋆κ) is equivalent to the existence of an ω1-complete ideal I on κ containing all sets of cardinality <κ and such that the quotient Boolean algebra P(κ)/I is isomorphic to the complete Boolean algebra that adjoins 2κ Cohen reals.

Noetherian type in topological products

The cardinal invariant Noetherian type Nt(X) of a topological space X was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces.We prove in Section 2: (1) There are spaces X and Y such that Nt(X×Y)< min{Nt(X), Nt(Y)}. (2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight \({\aleph _\omega }\) with the countable box topology, \({({2^{{\aleph _\omega }}})_\delta }\), is shown in Section 3 to be closely related to the combinatorics of covering collections of countable subsets of \({\aleph _\omega }\). We discuss the influence of principles like \({\square _{{\aleph _\omega }}}\) and Chang’s conjecture for \({\aleph _\omega }\) on this number and prove that it is not decidable in ZFC (relative to the consistency of ZFC with large cardinal axioms).Within PCF theory we establish the existence of an (ℵ4, ℵ1)-sparse covering family of countable subsets of \({\aleph _\omega }\) (Theorem 3.20). From this follows an absolute upper bound of ℵ4 on the Noetherian type of \({({2^{{\aleph _\omega }}})_\delta }\). The proof uses a method that was introduced by Shelah in 1993 [33].

A nodec regular analytic topology

A topological space X is said to be maximal if its topology is maximal among all T1 topologies over X without isolated points. It is known that a space is maximal if, and only if, it is extremely disconnected, nodec and every open set is irresolvable. We present some results about the complexity of those properties on countable spaces. A countable topological space X is analytic if its topology is an analytic subset of P(X) identified with the Cantor cube {0,1}X. No extremely disconnected space can be analytic and every analytic space is hereditarily resolvable. However, we construct an example of a nodec regular analytic space.

Note on K-analyticity and normality in function spaces

We prove that whenever X is zero-dimensional metrizable with σ-compact set of accumulation points and K is compact metrizable, the function space KX endowed with the compact-open topology is a compact-covering image of the product of the irrationals and the Cantor cube. In particular, for any metrizable E, the iterated function space E(KX) is perfectly normal and paracompact. However, there is a closed subgroup G of {0,1}X with X as above whose space of characters G∧ is not normal.

Every scattered space is subcompact

We prove that every scattered space is hereditarily subcompact and any finite union of subcompact spaces is subcompact. It is a long-standing open problem whether every Čech-complete space is subcompact. Moreover, it is not even known whether the complement of every countable subset of a compact space is subcompact. We prove that this is the case for linearly ordered compact spaces as well as for ω-monolithic compact spaces. We also establish a general result for Tychonoff products of discrete spaces which implies that dense Gδ-subsets of Cantor cubes are subcompact.

On a topological choice principle by Murray Bell

In ZF set theory, we investigate the deductive strength of Murray Bellʼs principle (C):For every set{Ai:i∈I}of non-empty sets, there exists a set{Ti:i∈I}such that for everyi∈I,Tiis a compactT2topology onAi, with regard to various choice forms.Among other results, we prove the following:(1)The Axiom of Multiple Choice (MC) does not imply statement (C) in ZFA set theory.(2)If κ is an infinite well-ordered cardinal number, then (C) + “Every filter base on κ can be extended to an ultrafilter” implies “For every family A={Ai:i∈κ} such that for all i∈κ, |Ai|⩾2, there is a function (called a Kinna–Wagner function) f with domain A such that for all A∈A, ∅≠f(A)⊊A” and “For every natural number n⩾2, every family A={Ai:i∈κ} of non-empty sets each of which has at most n elements has a choice function”.(3)If κ is an infinite well-ordered cardinal number, then (C) + “There exists a free ultrafilter on κ” implies “For every family A={Ai:i∈κ} such that for all i∈κ, |Ai|⩾2, there is an infinite subset B⊆A with a Kinna–Wagner function” and “For every natural number n⩾2, every family A={Ai:i∈κ} of non-empty sets each of which has at most n elements has an infinite subfamily with a choice function”.(4)(C) + “Every compact T2 space is effectively normal” implies MC restricted to families of non-empty sets each expressible as a countable union of finite sets, and “For every family A={Ai:i∈ω} such that for all i∈ω, 2⩽|Ai|<ℵ0, there is an infinite subset B⊆A with a Kinna–Wagner function”.(5)(C) + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” implies ACℵ0, i.e., the axiom of choice for countable families of non-empty sets.(6)(C) restricted to countable families of non-empty sets + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” is equivalent to ACℵ0 + “There exists a free ultrafilter on ω”.(7)(C) restricted to countable families of non-empty sets + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” implies the statements: “The Tychonoff product of a countable family of compact spaces is compact” and “For every infinite set X, the (generalized) Cantor cube 2X is countably compact”.(8)(C) restricted to countable families of non-empty sets does not imply “There exists a free ultrafilter on ω” in ZF.(9)(C) + “The axiom of choice for countable families of non-empty sets of reals” implies “There exists a non-Lebesgue-measurable set of reals”.(10)The conjunction of the Countable Union Theorem (the union of a countable family of countable sets is countable) and “Every infinite set is Dedekind-infinite” does not imply (C) restricted to countable families of non-empty sets, in ZFA set theory.

Measurability in $$C({2^\kappa })$$ } and Kunen cardinals

A cardinal κ is called a Kunen cardinal if the σ-algebra on κ × κ generated by all products A×B, where A,B ⊂ κ, coincides with the power set of κ×κ. For any cardinal κ, let \(C({2^\kappa })\) be the Banach space of all continuous real-valued functions on the Cantor cube \(C({2^\kappa })\) . We prove that κ is a Kunen cardinal if and only if the Baire σ-algebra on \(C({2^\kappa })\) for the pointwise convergence topology coincides with the Borel σ-algebra on \(C({2^\kappa })\) for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.