Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

A pseudocompact Tychonoff space all countable subsets of which are closed and C1-embedded

Let X be a Tychonoff space and let all subsets of cardinality ⩽τ of X be closed (and C¹-embedded). Then X can be embedded as a closed subspace in a pseudocompact Tychonoff space Y such that all subsets of cardinality ⩽τ of Y are closed (and C¹-embedded). As an application of this result we construct a pseudocompact connected left-separated Tychonoff space having all its subsets of cardinality ⩽τ closed and C¹-embedded.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Productivity numbers in topological spaces

The κ-productivity of classes C of topological spaces closed under quotients and disjoint sums is characterized by means of Cantor spaces. The smallest infinite cardinals κ such that such classes are not κ-productive are submeasurable cardinals. It follows that if a class of topological spaces is closed under quotients, disjoint sums and countable products, it is closed under products of non-sequentially many spaces (thus under all products, if sequential cardinals do not exist).     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Which compacta are noncommutative ARs?

We give a short answer to the question in the title: dendrits. Precisely we show that the C^*-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C¹ of all (not necessarily commutative) unital C^*-algebras if and only if X is an absolute retract of dimension dimX?1 or, equivalently, that X is a dendrit.     

Monoidal intervals of clones on infinite sets

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2^κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2^κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ?2^κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2^κ, as well as monoidal intervals of cardinality 2^λ, for all λ?2^κ.     

Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X. We show that a κ-uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

Moscow spaces and selection theory

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We present here the selection version of this result and also (by means of two different approaches) we use selection theory to characterize the open bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.     

On the calculation of generalized binomial coefficients

The generalized binomial coefficients (_κ^λ) are defined by

$\text{Cλ(I}{}_{\mathrm{m}}\text{+ R)}\text{Cλ(I}{}_{\mathrm{m}}\text{)}\text{=}\text{Σ}\text{k=0}\text{l}\text{Σ}\text{κ}\text{(}\text{λ}\text{κ}\text{)}\text{C}{}_{\mathrm{\kappa }}\text{(R)}\text{C}{}_{\mathrm{\kappa }}\text{(I}{}_{\mathrm{m}}\text{)}$

, where the C_k(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function f∈C(Y,Z) with Y⊆Y^′ and Z⊆Z^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=κ(XI)=_∏i∈IXi in the κ-box topology (that is, with basic open sets of the form _∏i∈IUi with Ui open in Xi and with Ui≠Xi for <κ-many i∈I). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this.

Theorem. Letω?κ?α (that means: κ<α, and[β<αandλ<κ]⇒βλ<α) with α regular,be a set of non-empty spaces with eachd(Xi)<α,π[Y]=XJfor each non-emptyJ⊆Isuch that|J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets ofZ×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for everyf∈C(Y,Z)there isJ∈[I]^<αsuch thatf(x)=f(y)wheneverxJ=yJ, and (c) every such f extends to.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product red_? on Mred from a strongly invariant star product ? on M. The new questions which are addressed in this paper concern the existence of natural ^∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ^∗-algebra.We assume that ? is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ^∗-involution for red_? on the reduced space. Looking into the question whether the corresponding ^∗-involution is the complex conjugation (which is a ^∗-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C^∞(M)?λ?,?)-submodule and a right (C^∞(Mred)?λ?,red_?)-submodule of C^∞(C)?λ?; we define on it a C^∞(Mred)?λ?-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^∞(Mred)?λ? and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ^∗-representations of (C^∞(Mred)?λ?,red_?) on pre-Hilbert right D-modules to those of (C^∞(M)?λ?,?), for any auxiliary coefficient ^∗-algebra D over C?λ?.     

On sieve-complete and compact-like spaces

It is shown that a completely regular space X is sieve-complete (or, equivalenty, X is the open image of a paracompact ?ech-complete space) iff βX?X is compact-like, i.e., Player I has a winning strategy in the topological game G(C, βX?X) of [13].     

Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.     

Topological stability for conservative systems

We prove that the C¹ interior of the set of all topologically stable C¹ incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

Compact embeddings and proper mappings

Sufficient conditions are given to assert that a C¹-proper mapping defined on a compact embedding between Banach spaces over K has a fixed point. The proof of the result is based upon continuation methods.