Baire Category Research Articles

Complexity profiles and generic Muchnik reducibility

We study the relative computational complexity of expansions of Cantor and Baire space in terms of generic Muchnik reducibility. We show that no expansion of Cantor space by countably many unary or closed relations can give a generic Muchnik degree strictly between the degree of Cantor space and the degree of Baire space. Similarly, assuming Δ21-Wadge determinacy we show that no expansion of Baire space by countably many unary or closed relations can give a generic Muchnik degree strictly between the degree of Baire space and the Borel complete degree. On the other hand, we provide a construction of a degree strictly between the degree of Cantor space and the degree of Baire space and also between the degree of Baire space and the Borel complete degree.

Turing independence and Baire category

We show that it is relatively consistent with ZFC that there is a non-meager set of reals [Formula: see text] such that for every non-meager [Formula: see text], there exist distinct [Formula: see text] such that [Formula: see text] is computable from the Turing join of [Formula: see text] and [Formula: see text].

Some results on Baireness in generalized topological spaces

"The aim of this paper is to extend some results on the Baire category in generalized topological spaces. We will apply the Banach-Mazur game to characterize Baireness in generalized topological spaces. Moreover, we will introduce a new separation axiom for generalized topological spaces which provides opportunity to generalize the Banach category theorem for locally compact generalized topological spaces."

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold M⊂Rd which is a Lipschitz neighborhood retract of dimension m+n, of class Ch,β and an oriented, closed submanifold Γ⊂M of dimension m−1, which is a boundary in integral homology, we construct a complete metric space B of Ch,α-perturbations of Γ inside M, with α<β, enjoying the following property. For the typical element b∈B, in the sense of Baire categories, there exists a unique m-dimensional integral current in M which solves the corresponding Plateau problem and it has multiplicity one.

A Universal Space for the Bourbaki-Complete Spaces and Further Examples

In this paper it is provided an explicit embedding for complete metric spaces having a star-finite base of its uniformity into the universal space kappa ^{omega _0} times mathbb {R}^{omega _0}, where kappa ge omega _0 is the cellularity of the space and kappa ^{omega _0} is the Baire space of weight kappa . From this result, similar embeddings for general Bourbaki-complete uniform spaces as well as for metric spaces are derived. In particular, these embeddings partially preserve the uniform structure of the original space. Several examples of Bourbaki-complete metric spaces having no star-finite base for its metric uniformity are constructed.

On Topological and Metric Properties of ⊕-sb-Metric Spaces

In this paper, we study ⊕-sb-metric spaces, which were introduced to generalize the concept of strong b-metric spaces. In particular, we study the properties of the topology induced via an ⊕-sb metric (separation properties, countability axioms, etc.), prove the continuity of the ⊕-sb-metric, establish the metrizability of the ⊕-sb-metric spaces of countable weight, discuss the convergence structure of an ⊕-sb-metric space and prove the Baire category type theorem for such spaces. Most of the results obtained here are new already for strong b-metric spaces, i.e., in the case where an arithmetic sum “+” is taken in the role of ⊕.

Baire category and the relative growth rate for partial quotients in continued fractions

Baire category and the relative growth rate for partial quotients in continued fractions

Generic uniqueness of optimal transportation networks

We prove that for the generic boundary, in the sense of Baire categories, there exists a unique minimizer of the associated optimal branched transportation problem.

Baire Category Soft Sets and Their Symmetric Local Properties

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft σ-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points.

Compact sets with large projections and nowhere dense sumset

We answer a question of Banakh, Jabłońska and Jabłoński by showing that for there exists a compact set such that the projection of K onto each hyperplane is of non-empty interior, but K + K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for -fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

Polish topologies on endomorphism monoids of relational structures

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid End(A) of a countable relational structure A. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space NN. We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.

Generalized Polish spaces at regular uncountable cardinals

AbstractIn the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish‐like spaces and standard ‐Borel spaces for an uncountable (regular) cardinal satisfying . As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from Coskey and Schlicht (Fund. Math. 232 (2016) 227–248), Galeotti (Ph.D. thesis, Universität Hamburg, 2019), and Lücke and Schlicht (Israel J. Math. 209 (2015) 421–461), and answer some questions contained therein.

A Perspective Note on μ_N σ Baire’s Space

This paper presents an introduction to many novel types of sets, including μN strongly dense sets, μN strongly nowhere dense sets, μN strongly first category sets, and μN strongly nowhere residual sets. The features of these sets are briefly elucidated. In addition, by the use of these techniques, we have successfully obtained the highly Baire space μN, and it is imperative to elucidate its inherent features.

A note on the interior and dimensions of typical sumsets

Let K \mathcal {K} be the collection of all non-empty compact subsets of the unit cube [ 0 , 1 ] d [0,1]^d . Fixing any F σ F_\sigma set F ⊂ [ 0 , 1 ] d F\subset [0,1]^d of Lebesgue measure zero and 0 &gt; c &gt; 1 0&gt;c&gt;1 , we show that for typical K ∈ K K\in \mathcal {K} with L d ( K ) ≥ c \mathcal {L}^d(K)\geq c (in the sense of Baire category), the sumset K + F K+F has empty interior. We also find the Hausdorff, lower and upper box and packing dimensions of K + F K+F for typical K ∈ K K\in \mathcal {K} .

On Successive Approximations for Compact-Valued Nonexpansive Mappings

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory.

Strong independence and its spectrum

For μ,κ infinite, say A⊆[κ]κ is a (μ,κ)-maximal independent family if whenever A0 and A1 are pairwise disjoint non-empty in [A]<μ then ⋂A0﹨⋃A1≠∅, A is maximal under inclusion among families with this property, and moreover all such Boolean combinations have size κ. We denote by spi(μ,κ) the set of all cardinalities of such families, and if non-empty, we let iμ(κ) be its minimal element. Thus, iμ(κ) (if defined) is a natural higher analogue of the independence number on ω for the higher Baire spaces. In this paper, we study spi(μ,κ) for μ,κ uncountable. Among others, we show that:(1)The property spi(μ,κ)≠∅ cannot be decided on the basis of ZFC plus large cardinals.(2)Relative to a measurable, it is consistent that:(a)(∃κ>ω)iκ(κ)<2κ.(b)(∃κ>ω)κ+<iω1(κ)<2κ. To the best knowledge of the authors, (2b) is the first example of a (μ,κ)-maximal independent family of size strictly between κ+ and 2κ, for uncountable κ.(3)spi(μ,κ) cannot be quite arbitrary.

Further properties of soft somewhere dense continuous functions and soft Baire spaces

Further properties of soft somewhere dense continuous functions and soft Baire spaces

Generic algebraic properties in spaces of enumerated groups

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin’s question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.

Continuous functions with impermeable graphs

AbstractWe construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.