Set Iff Research Articles

Strongly ultrametric preserving functions

An ultrametric preserving function f is said to be strongly ultrametric preserving if ultrametrics d and f∘d define the same topology on X for each ultrametric space (X,d). The set of all strongly ultrametric preserving functions is characterized by several distinctive features. In particular, it is shown that an ultrametric preserving f belongs to this set iff f preserves the property to be compact.

Nice Embedding in Classical Logic

It is shown that a set of semi-recursive logics, including many fragments of CL (Classical Logic), can be embedded within CL in an interesting way. A logic belongs to the set iff it has a certain type of semantics, called nice semantics. The set includes many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for CL that are goal directed with respect to CL are turned into partial decision methods that are goal directed with respect to the embedded logics.

Separable connected metric spaces need not have continuum size in ZF

We show: (i) It is relatively consistent with ZF that there exists a connected, separable subspace C of R2 with |C|<|R|.(ii) It is relatively consistent with ZF that there exists a separable, connected, compact, pseudometric (X,d) with |X|<|R|.(iii) It is relatively consistent with ZF that there exists a separable, compact, connected, pseudometric space (X,d) whose size is strictly less than the power of its metric reflection (X⁎,d⁎).(iv) It is relatively consistent with ZF that there exists a connected, locally connected, non-pathwise connected, compact, non-separable pseudometric space (A,d) such that A is Dedekind-finite and its metric reflection is the interval [0,1].(v) It is relatively consistent with ZF0 (=ZF minus the axiom of regularity) that there exists a non-separable, compact, connected, locally connected metric space.(vi) Every subspace X of Hilbertʼs cube [0,1]N such that X¯\\X is meager in X¯ is separable. In particular, every connected subspace X of [0,1]N such that X¯\\X is meager in X¯ is separable.(vii) Every connected subspace X of [0,1]N such that X¯\\X is meager in X¯ has continuum size.(viii) The countable axiom of choice restricted to subsets of the real line R, CAC(R) is equivalent to the proposition: “Every connected subspace X of R2 is separable”.(ix) Every family A=(Ai)i∈R of non-empty sets has a choice set iff every connected, locally connected, compact pseudometric space is pathwise connected.

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

AbstractGiven a set X, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} denotes the statement: “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$[X]^{&lt;\omega }\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set” and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$\end{document} denotes the family of all closed subsets of the topological space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {2}^{X}$\end{document} whose definition depends on a finite subset of X. We study the interrelations between the statements \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{&lt;\omega })},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} and “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document}has a choice set”. We show: \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{&lt;\omega } )}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$\end{document}. \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} (\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}$\end{document} restricted to families of finite sets) iff for every set X, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set. \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} does not imply “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set(\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {X})$\end{document} is the family of all closed subsets of the space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}$\end{document}) \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$\end{document} implies \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} but \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} does not imply \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document}. We also show that “For every setX, “\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document}has a choice set” iff “for every setX, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document}has a choice set” iff “for every product\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}$\end{document}of finite discrete spaces,\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set”.

Post's Programme for the Ershov Hierarchy

This article extends Post's; programme to finite levels of the Ershov hierarchy of Δ2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284–316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n ≥ 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets. Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.

Free generating sets of lattice-ordered abelian groups

We prove that the generators g1,…,gn of a lattice-ordered abelian group G form a free generating set iff each ℓ-ideal generated by any n−1 linear combinations of the gi is strictly contained in some maximal ℓ-ideal of G.

Degrees of unsolvability of continuous functions

Abstract.We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a &lt;Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

Geometric structure in stochastic approximation

LetJ be the zero set of the gradientfx of a functionf∶Rn→R. Under fairly general conditions the stochastic approximation algorithm ensuresd(f(xk), f(J))→0, ask→∞. First of all, the paper considers this problem: Under what conditions the convergence\(d(f(x_k ),f(J))\mathop \to \limits_{k \to \infty } 0\) implies\(d(x_k ,J)\mathop \to \limits_{k \to \infty } 0\). It is shown that such implication takes place iffx is continuous andf(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: Iff is aC1 function and either i)J is a compact set or ii) for any bounded setB, f−1(B) is bounded, thef(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove thatf(J) is a finite set iff is a polynomial. Hencef(J) is nowhere dense in the polynomial case.

Strongly meager sets and their uniformly continuous images

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly set Y, the image f[X Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by + we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a measure zero (or negligible) set always denotes a Lebesgue set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, zero) set Y, XnY is at most countable. We say that a set X is of strong (respectively, strongly meager) iff for each meager (respectively, zero) set Y, X Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong set Y, X Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real-valued function f defined on a convex set K is an approximately convex function iff it satisfiesfx+y2⩽f(x)+f(y)2+1. A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function E vanishing on the vertices of a simplex. A set A in a normed space is an approximately convex set iff for all a, b∈A the distance of the midpoint (a+b)/2 to A is ⩽1. The bounds on approximately convex functions are used to show that in Rn with the Euclidean norm, for any approximately convex set A, any point z of the convex hull of A is at a distance of at most [log2(n−1)]+1+(n−1)/2[log2(n−1)] from A. Examples are given to show this is the sharp bound. Bounds for general norms on Rn are also given.

On the covering and the additivity number of the real line

We show that the real line R cannot be covered by k many nowhere dense sets iff whenever D = { D i : i ∈ k } D = \{ {D_i}:i \in k\} is a family of dense open sets of R there exists a countable dense set G of R such that | G ∖ D i | &gt; ω |G\backslash {D_i}| &gt; \omega for all i ∈ k i \in k . We also show that the union of k meagre sets of the real line is a meagre set iff for every family D = { D i : i ∈ k } D = \{ {D_i}:i \in k\} of dense open sets of R and for every countable dense set G of R there exists a dense set Q ⊆ G Q \subseteq G such that | Q ∖ D i | &gt; ω |Q\backslash {D_{i}}| &gt; \omega for all i ∈ k i \in k .

Rings of Stable Rank 2 are Barbilian Rings

Let MR a free unitary modul. A non-empty subset B of MR is by definition a Barbilian set iff B* ≔ F ⊂ B ¦ F is a free generating system of MRand #F > 1 satisfies: (B1) Each u1 ϵ B may be completed to 1, u2, … ϵ B* (B1) 1, u2, … ϵ B* implies 1 + u2α, u2, … ϵ B* for all α ϵ R. For each Barbilian set B ⊂ MR holds: B ⊂ Bmax= ϵ MR ¦ u may be completed to a free generating system F of MRand #F > 1 and Bmax is a Barbilian set iff Bmax ≠ .

An equivalent of the axiom of choice in finite models of the powerset axiom.

It is shown that in a finite model for the set-theoretical Powerset axiom every set s has a Choice set iff every set s has a Meet set ∩s. Moreover, the Choice set of s is unique and is equal to ∩s, where ∩s is a singleton and ∩s ∈ s

Local maximum property and q-plurisubharmonic functions in uniform algebras

There is proven a formula expressing higher order Shilov boundaries of the tensor products of uniform algebras in terms of boundaries of the factor algebras. Main step: Cartesian product of k- and l-maximum sets is a ( k + l + 1)-maximum set. (Definition: locally closed X⊂ C n is a k-maximum set if polynomials have local maximum property on intersections of X with (n-k)-dimensional complex planes.) Other properties and characterizations of k-maximum sets are given, e.g., X⊂ C n is a k-maximum set iff each k-plurisubharmonic function has local maximum property on X.

Continuous functions on polar sets

Let Ω \Omega be a second countable Brelot harmonic space with a positive potential. If K K is a compact subset of Ω \Omega with more than one point, then K K is a polar set iff every positive continuous function on K K can be extended to a continuous potential on Ω = R n ( n ⩾ 3 ) \Omega = {{\mathbf {R}}^n}(n \geqslant 3) . This is a generalization of the result proved by H. Wallin for the special case Ω = R n ( n ⩾ 3 ) \Omega = {{\mathbf {R}}^n}(n \geqslant 3) with Laplace harmonic space.

When the continuum has cofinalityω1

In this paper we consider models of set theory in which the continuum has cofinality ωv We show that it is consistent with -,CH that any complete boolean algebra B of cardinality less than or equal to c (continuum) there exists an ωλ-generated ideal / in P(ω) (power set of ω) such that B is isomorphic to P(ω)mod/. We also show that the existence of generalized Luzin sets every ωλ -saturated ideal in the Borel sets does not imply Martin's axiom. Introduction. In §1 we prove our main result that it is consistent with -ηCH that every complete boolean algebra of cardinality < c is isomorphic to P(ω)mod / some / ω^generated. We think of this as generalizing Kunen's theorem that it is consistent with -πCH that there is an ω1 generated nonprincipal ultrafilter on ω. For / an ideal in the Borel subsets of the reals we say that a set of reals X is a κ-/-Luzin set iff X has cardinality K and every A in /, A n X has cardinality less than /c. If c is regular, then it follows easily from Martin-Solovay (9) that MA is equivalent to the statement for every ωΓsaturated σ-ideal / in the Borels there is a c-/-Luzin In §2 we show that the regularity of c is necessary. This answers a question of Fremlin (5). We also show that it is consistent with -,CH that every such / there exists an ω Γ/-Luzin set. These results can be thought of as a weak form of the following conjecture.

Metric Characterizations of Dimension for Separable Metric Spaces

A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with $B = \{ z:d(x,z) = d(y,z)\}$. It is shown that if X is a separable metrizable space, then $\dim (X) \leqslant n$ iff X has an admissible metric d for which $\dim (B) \leqslant n - 1$ whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.

Metric characterizations of dimension for separable metric spaces

A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with B = { z : d ( x , z ) = d ( y , z ) } B = \{ z:d(x,z) = d(y,z)\} . It is shown that if X is a separable metrizable space, then dim ⁡ ( X ) ⩽ n \dim (X) \leqslant n iff X has an admissible metric d for which dim ⁡ ( B ) ⩽ n − 1 \dim (B) \leqslant n - 1 whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.