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Journal articles on the topic 'Nowhere dense sets'

Author: Grafiati

Published: 5 June 2025

Last updated: 1 August 2025

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1

Yupapin, Preecha, Vadakasi Subramanian, and Yasser Farhat. "On Nowhere Dense Sets." European Journal of Pure and Applied Mathematics 15, no. 2 (2022): 403–14. http://dx.doi.org/10.29020/nybg.ejpam.v15i2.4283.

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We introduce two types of strongly nowhere dense sets, namely (s, v)-strongly nowhere dense set, (s, v)*-strongly nowhere dense set and analyze their characteristics in a bigeneralized topological space (BGTS). Further, it is also given some relations between these two types of strongly nowhere dense sets along with its various properties for (s, v)*-strongly nowhere dense set. Finally, the necessary and sufficient condition is found between \mu-strongly nowhere dense set and (s, v)*-strongly nowhere dense set in a BGTS.

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2

N.Raksha, Ben, and Hari siva annam G. "An Overview on \mu_N Strongly Nowhere Dense Sets." Asia Mathematika 6, no. 2 (2022): 48——55. https://doi.org/10.5281/zenodo.7120723.

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In this article we have introduced some new types of sets such as \(\mu_N\)  strongly dense, \(\mu_N\)  strongly nowhere dense, \(\mu_N\) strongly first category sets, \(\mu_N\)  strongly nowhere residual sets and their attributes are explained briefly. Also by making use of these we have retrieved \(\mu_N\)  strongly Baire space and its properties are to be described.

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3

Fejzić. "F"UTILITY ON NOWHERE DENSE SETS." Real Analysis Exchange 23, no. 1 (1997): 87. http://dx.doi.org/10.2307/44152827.

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4

N. Raksha Ben, G. Hari Siva Annam та G.Helen Rajapushpam. "A Perspective Note on μ_N σ Baire’s Space". Neutrosophic Systems with Applications 7 (22 липня 2023): 54–60. http://dx.doi.org/10.61356/j.nswa.2023.29.

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In this article, we have introduced some new types of sets such as strongly dense, strongly nowhere dense, strongly first category sets , strongly nowhere residual sets and their attributes are explained briefly. Also by making use of these we have retrieved strongly Baire space and its properties are to be described.

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5

Yasser, Farhat, and Vadakasi Subramanian. "Generalized Dense set in Bigeneralized Topological Spaces." European Journal of Pure and Applied Mathematics 16, no. 4 (2023): 2049–65. http://dx.doi.org/10.29020/nybg.ejpam.v16i4.4911.

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In this article, in a bigeneralized topological space, we introduce an interesting tool namely, $(s, v)$-dense set, and examine the significance of this set. Also, we give the relations between nowhere-dense sets defined in generalized and bigeneralized topological space and give some of their properties by using functions. Finally, we give some applications for $(s, v)$-dense and $(s, v)$-nowhere dense sets in a soft set theory.

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6

Poongothai, E., and S. Divyapriya. "On Fuzzy soft Regularly nowhere dense sets." Journal of Physics: Conference Series 1850, no. 1 (2021): 012083. http://dx.doi.org/10.1088/1742-6596/1850/1/012083.

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7

Aponte, Elvis, Vadakasi Subramanian, Jhixon Macías, and Muthumari Krishnan. "On Semi-Continuous and Clisquish Functions in Generalized Topological Spaces." Axioms 12, no. 2 (2023): 130. http://dx.doi.org/10.3390/axioms12020130.

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In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature of cliquish functions in generalized metric spaces and provide the characterization theorem for cliquish functions in terms of nowhere dense sets.

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8

LINDNER, SEBASTIAN. "RESOLVABILITY PROPERTIES OF SIMILAR TOPOLOGIES." Bulletin of the Australian Mathematical Society 92, no. 3 (2015): 470–77. http://dx.doi.org/10.1017/s0004972715001021.

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We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.

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9

Drakakis, Konstantinos. "On Costas Sets and Costas Clouds." Abstract and Applied Analysis 2009 (2009): 1–23. http://dx.doi.org/10.1155/2009/467342.

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We abstract the definition of the Costas property in the context of a group and study specifically dense Costas sets (named Costas clouds) in groups with the topological property that they are dense in themselves: as a result, we prove the existence of nowhere continuous dense bijections that satisfy the Costas property on , , and , the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on , , and , which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb construction methods for Costas arrays to apply on and , and we prove that

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10

Cao, Jiling, та Sina Greenwood. "The ideal generated by σ-nowhere dense sets". Applied General Topology 7, № 2 (2006): 253. http://dx.doi.org/10.4995/agt.2006.1928.

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11

Baggs, Ivan. "Nowhere dense sets and real-valued functions with closed graphs." International Journal of Mathematics and Mathematical Sciences 12, no. 1 (1989): 1–8. http://dx.doi.org/10.1155/s0161171289000013.

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Closed and nowhere dense subsets which coincide with the points of discontinuity of real-valued functions with a closed graph on spaces which are not necessarily perfectly normal are investigated. CertainGδsubsets of completely regular and normal spaces are characterized. It is also shown that there exists a countable connected Urysohn space X with the property that no closed and nowhere dense subset of X coincides with the points of discontinuity of a real-valued function on X with a closed graph.

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12

Ameen, Zanyar A., and Mesfer H. Alqahtani. "Congruence Representations via Soft Ideals in Soft Topological Spaces." Axioms 12, no. 11 (2023): 1015. http://dx.doi.org/10.3390/axioms12111015.

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This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft

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13

Scheepers, Marion. "Meager nowhere-dense games (IV): n-tactics (continued)." Journal of Symbolic Logic 59, no. 2 (1994): 603–5. http://dx.doi.org/10.2307/2275411.

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AbstractWe consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [

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14

Renfro. "POROSITY, NOWHERE DENSE SETS AND A THEOREM OF DENJOY." Real Analysis Exchange 21, no. 2 (1995): 572. http://dx.doi.org/10.2307/44152667.

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15

Banakh, Taras, and Dušan Repovš. "Universal nowhere dense and meager sets in Menger manifolds." Topology and its Applications 161 (January 2014): 127–40. http://dx.doi.org/10.1016/j.topol.2013.09.012.

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16

Balka, Richárd, Márton Elekes, Viktor Kiss, Donát Nagy, and Márk Poór. "Compact sets with large projections and nowhere dense sumset." Nonlinearity 36, no. 10 (2023): 5190–215. http://dx.doi.org/10.1088/1361-6544/acebae.

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Abstract We answer a question of Banakh, Jabłońska and Jabłoński by showing that for d ⩾ 2 there exists a compact set K ⊆ R d 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 ℓ -fold sumsets. We obtain numerous

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17

Brendle, Jörg, and Diana Carolina Montoya. "A base-matrix lemma for sets of rationals modulo nowhere dense sets." Archive for Mathematical Logic 51, no. 3-4 (2012): 305–17. http://dx.doi.org/10.1007/s00153-012-0274-y.

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18

Al Ghour, Samer. "Boolean Algebra of Soft Q-Sets in Soft Topological Spaces." Applied Computational Intelligence and Soft Computing 2022 (August 28, 2022): 1–9. http://dx.doi.org/10.1155/2022/5200590.

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We define soft Q-sets as soft sets whose soft closure and soft interior are commutative. We show that the soft complement, soft closure, and soft interior of a soft Q-set are all soft Q-sets. We show that a soft subset K of a given soft topological space is a soft Q-set if and only if K is a soft symmetric difference between a soft clopen set and a soft nowhere dense set. And as a corollary, the class of soft Q-sets contains simultaneously the classes of soft clopen sets and soft nowhere dense sets. Also, we prove that the class of soft Q-sets is closed under finite soft intersections and fini

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19

Khalil, Shuker Mahmood, Mayadah Ulrazaq, Samaher Abdul-Ghani та Abu Firas Al-Musawi. "σ-Algebra and σ-Baire in Fuzzy Soft Setting". Advances in Fuzzy Systems 2018 (2 липня 2018): 1–10. http://dx.doi.org/10.1155/2018/5731682.

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We first introduce some new notions of Baireness in fuzzy soft topological space (FSTS). Next, their characterizations and basic properties are investigated in this work. The notions of fuzzy soft dense, fuzzy soft nowhere dense, fuzzy soft meager, fuzzy soft second category, fuzzy soft residual, fuzzy soft Baire, fuzzy soft δ-sets, fuzzy soft λσ-sets, fuzzy soft σ-nowhere dense, fuzzy soft σ-meager, fuzzy soft σ-residual, fuzzy soft σ-Baire, fuzzy soft σ-second category, fuzzy soft σ-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy so

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20

Porter, Jack R., and R. Grant Woods. "Accumulation Points of Nowhere Dense Sets in H-Closed Spaces." Proceedings of the American Mathematical Society 93, no. 3 (1985): 539. http://dx.doi.org/10.2307/2045630.

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21

Porter, Jack R., and R. Grant Woods. "Accumulation points of nowhere dense sets in $H$-closed spaces." Proceedings of the American Mathematical Society 93, no. 3 (1985): 539. http://dx.doi.org/10.1090/s0002-9939-1985-0774019-7.

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22

Mizera, Ivan. "Continuous chaotic functions of an interval have generically small scrambled sets." Bulletin of the Australian Mathematical Society 37, no. 1 (1988): 89–92. http://dx.doi.org/10.1017/s0004972700004172.

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It is shown that continuous self-mappings of a compact interval, chaotic in the sense of Li and Yorke, have generically, in the uniform topology, only scrambled sets which are nowhere dense and of zero Lebesgue measure.

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23

Thangaraj, Ganesan, and Natarajan Raji. "On Fuzzy Baire-Separated Spaces and Related Concepts." Pure and Applied Mathematics Journal 13, no. 1 (2024): 1–8. http://dx.doi.org/10.11648/j.pamj.20241301.11.

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In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy F&lt;sub&gt;σ&lt;/sub&gt;- sets in a fuzzy Baire-separated space. Conditions under which fuzzy topological spaces become fuzzy Baire-separated spaces are established. Fuzzy nowhere dense sets are fuzzy closed sets in fuzzy nodec spaces and subsequently a question will arise. Which fuzzy topolog

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24

KOLYADA, SERGII˘, L’UBOMÍR SNOHA, and SERGEI˘ TROFIMCHUK. "Proper minimal sets on compact connected 2-manifolds are nowhere dense." Ergodic Theory and Dynamical Systems 28, no. 3 (2008): 863–76. http://dx.doi.org/10.1017/s0143385707000740.

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AbstractLet $\mathcal {M}^2$ be a compact connected two-dimensional manifold, with or without boundary, and let $f:{\mathcal {M}}^2\to \mathcal {M}^2$ be a continuous map. We prove that if $M \subseteq \mathcal {M}^2$ is a minimal set of the dynamical system $(\mathcal {M}^2,f)$ then either $M = \mathcal {M}^2$ or M is a nowhere dense subset of $\mathcal {M}^2$. Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case $\mathcal {M}^2$ is a torus or a Klein bottle.

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25

Kwela, Marta, and Andrzej Nowik. "Ideals of nowhere dense sets in some topologies on positive integers." Topology and its Applications 248 (October 2018): 149–63. http://dx.doi.org/10.1016/j.topol.2018.08.015.

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26

Hejduk, Jacek, and Anna Loranty. "On Lower and Semi-Lower Density Operators." gmj 14, no. 4 (2007): 661–71. http://dx.doi.org/10.1515/gmj.2007.661.

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Abstract This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

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27

Mıah, Chhapikul, Monoj Kumar Das, and Shyamapada Modak. "Variants of sets and functions with primals." Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 74, no. 2 (2025): 180–90. https://doi.org/10.31801/cfsuasmas.1476888.

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Through this paper, we will discuss the limit points of a set and its complement set. To do this the authors will consider Acherjee et al.’s mathematical structure primal. These limit points via primal will further express the representation of nowhere dense sets in the literature. Expression of ⋄-local function and its associated set-valued set function will also be discussed here. Levine’s semi-open sets will also be further represented by these limit points and will be decomposed.

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28

Yaremenko, Mykola. "The irregular Cantor sets Ce ([0, 1]) and Cπ ([0, 1]), and the Cantor- Lebesgue irregular functions Ge and Gπ". PROOF 3 (15 вересня 2023): 29–31. http://dx.doi.org/10.37394/232020.2023.3.5.

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In this article, we introduce and study a new class of perfect nowhere-dense sets, which are not selfsimilar in any subset, also, we constructed the correspondent singular functions. We construct a twodimensional irregular Cantor set Ce,π ([0, 1]) on the real plane.

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29

Nasefa, Arafa A., and R. Mareay. "Ideal and some applications of simply open sets." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 3 (2017): 7264–71. http://dx.doi.org/10.24297/jam.v13i3.6204.

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Recently there has been some interest in the notion of a locally closed subset of a topo- logical space. In this paper, we introduce a useful characterizations of simply open sets in terms of the ideal of nowhere dense set. Also, we study a new notion of functions in topo- logical spaces known as dual simply-continuous functions and some of their fundamental properties are investigated. Finally, a new type of simply open sets is introduced.

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30

Alikhani-Koopaei, Aliasghar. "On common fixed points, periodic points, and recurrent points of continuous functions." International Journal of Mathematics and Mathematical Sciences 2003, no. 39 (2003): 2465–73. http://dx.doi.org/10.1155/s0161171203205366.

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It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. we had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove thatSis a nowhere dense subset of[0,1]if and only if{f∈C([0,1]):Fm(f)∩S¯≠∅}is a nowhere dense subset ofC([0,1]). We also give some results about the common fixed, periodic, and recurrent points of functions. We consider the class of functionsfwith continuousωfst

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31

Pilipczuk, Michał, and Sebastian Siebertz. "Kernelization and approximation of distance-r independent sets on nowhere dense graphs." European Journal of Combinatorics 94 (May 2021): 103309. http://dx.doi.org/10.1016/j.ejc.2021.103309.

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32

Miller, Harry I., and Leila Miller-Van Wieren. "Translates of Sequences for Some Small Sets." Sarajevo Journal of Mathematics 7, no. 2 (2024): 201–5. http://dx.doi.org/10.5644/sjm.07.2.06.

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D. Borwein and S. Z. Ditor have found a measurable subset $A$ of the real line having positive Lebesgue measure and a decreasing sequence $(d_n)$ of reals converging to $0$ such that, for each $x$, $x+d_n \notin A$ for infinitely many $n$. The set they constructed is nowhere dense. This result prompted us to further explore the question of subsets of $R$ and $R^{2}$ that are of "small size" and the existence of null sequences with the described property and hence attain some related results. 2000 Mathematics Subject Classification. 40D25, 40G99, 28A12

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33

Wachowicz, Artur. "Preimages of Residual Sets of Continuous Functions under Operation of Superposition." gmj 12, no. 4 (2005): 763–68. http://dx.doi.org/10.1515/gmj.2005.763.

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Abstract Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

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34

Agadzhanov, А. N. "Peano-type curves, Liouville numbers, and microscopic sets." Доклады Академии наук 485, no. 1 (2019): 7–10. http://dx.doi.org/10.31857/s0869-565248417-10.

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Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is give

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35

Alikhani-Koopaei, Aliasghar. "On the sets of fixed points of bounded Baire one functions." Asian-European Journal of Mathematics 12, no. 03 (2019): 1950040. http://dx.doi.org/10.1142/s1793557119500402.

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In this paper, we present some results on typical properties of the sets of fixed points of bounded Baire one functions. In particular, we show that typical elements of a uniformly closed subclass [Formula: see text] of such class of functions have nowhere dense set of fixed points. We also show that typical elements of the class of bounded Baire one functions have [Formula: see text], where [Formula: see text] is an arbitrary continuous Borel measure on the unit interval.

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36

Atamuratov, A. A., and K. K. Rasulov. "On Shimoda's Theorem." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 33, no. 1 (2023): 17–31. http://dx.doi.org/10.35634/vm230102.

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The present work is devoted to Shimoda's Theorem on the holomorphicity of a function $f(z,w)$ which is holomorphic by $w\in V$ for each fixed $z\in U$ and is holomorphic by $z\in U$ for each fixed $w\in E$, where $E\subset V$ is a countable set with at least one limit point in $V$. Shimoda proves that in this case $f(z,w)$ is holomorphic in $U\times V$ except for a nowhere dense closed subset of $U\times V$. We prove the converse of this result, that is for an arbitrary given nowhere dense closed subset of $U$, $S\subset U$, there exists a holomorphic function, satisfying Shimoda's Theorem on

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37

Kurilić, Miloš S. "Cohen-stable families of subsets of integers." Journal of Symbolic Logic 66, no. 1 (2001): 257–70. http://dx.doi.org/10.2307/2694920.

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AbstractA maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A ∈ are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stab

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38

Tamilselvan, K., and V. Visalakshi. "Identifying the Severity of Criminal Activity in Society Using Picture Fuzzy Baire Space." International Journal of Analysis and Applications 22 (April 8, 2024): 67. http://dx.doi.org/10.28924/2291-8639-22-2024-67.

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In this paper, the idea of picture fuzzy Baire space is explored and its properties are examined. The features of picture fuzzy semi-closed and semi-open sets, picture fuzzy nowhere dense sets, picture fuzzy first and second category sets, picture fuzzy residual sets, picture fuzzy submaximal spaces, picture fuzzy strongly irresolvable spaces, picture fuzzy Gδ set, picture fuzzy Fσ set, and picture fuzzy regular closed sets are analyzed. To understand the concepts, some examples are provided. An algorithm using picture fuzzy Baire space is developed to address real-world scenarios. This method

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39

Ðinh, Sĩ Tiệp, and Zbigniew Jelonek. "Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values." Discrete & Computational Geometry 65, no. 1 (2019): 279–304. http://dx.doi.org/10.1007/s00454-019-00087-w.

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AbstractLet $$X\subset {\mathbb {C}}^n$$ X ⊂ C n be an affine variety and $$f:X\rightarrow {\mathbb {C}}^m$$ f : X → C m be the restriction to X of a polynomial map $${\mathbb {C}}^n\rightarrow {\mathbb {C}}^m$$ C n → C m . We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of $${\mathbb {C}}^m$$ C m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

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40

Moore, Justin Tatch, та Sławomir Solecki. "A Gδ ideal of compact sets strictly above the nowhere dense ideal in the Tukey order". Annals of Pure and Applied Logic 156, № 2-3 (2008): 270–73. http://dx.doi.org/10.1016/j.apal.2008.07.003.

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41

DIJKSTRA, JAN J., and JAN VAN MILL. "ON SETS THAT MEET EVERY HYPERPLANE IN n-SPACE IN AT MOST n POINTS." Bulletin of the London Mathematical Society 34, no. 3 (2002): 361–68. http://dx.doi.org/10.1112/s0024609301008979.

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A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the pla

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42

Staiger, Ludwig. "How Large is the Set of Disjunctive Sequences?" JUCS - Journal of Universal Computer Science 8, no. (2) (2002): 348–62. https://doi.org/10.3217/jucs-008-02-0348.

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We consider disjunctive sequences, that is, infinite sequences -words) having all finite words as infixes. It is shown that the set of all disjunctive sequences can be described in an easy way using recursive languages and, besides being a set of measure one, is a residual set in Cantor space. Moreover, we consider the subword complexity of sequences: here disjunctive sequences are shown to be sequences of maximal complexity. Along with disjunctive sequences we consider the set of real numbers having disjunctive expansions with respect to some bases and to all bases. The latter are called abso

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43

Banakh, Taras, Iryna Banakh, and Eliza Jabłońska. "Products of K-Analytic Sets in Locally Compact Groups and Kuczma–Ger Classes." Axioms 11, no. 2 (2022): 65. http://dx.doi.org/10.3390/axioms11020065.

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We prove that for any K-analytic subsets A,B of a locally compact group X if the product AB has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set AAAA has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was prov

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Ravsky, A. V., and T. O. Banakh. "On pseudobounded and premeage paratopological groups." Matematychni Studii 56, no. 1 (2021): 20–27. http://dx.doi.org/10.30970/ms.56.1.20-27.

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Let $G$ be a paratopological group.Following F.~Lin and S.~Lin, we say that the group $G$ is pseudobounded,if for any neighborhood $U$ of the identity of $G$,there exists a natural number $n$ such that $U^n=G$.The group $G$ is $\omega$-pseudobounded,if for any neighborhood $U$ of the identity of $G$, the group $G$ is aunion of sets $U^n$, where $n$ is a natural number.The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of$G$ and any positive integer $n$.In this paper we investigate relations between the above classes of groups andanswer some questions posed by F. Lin, S.

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Fremlin, D. H., and P. J. Nyikos. "Saturating ultrafilters on N." Journal of Symbolic Logic 54, no. 3 (1989): 708–18. http://dx.doi.org/10.2307/2274735.

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AbstractWe discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that

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Geraldo, Jean, Luis Bladismir Ruiz Leal, and Sergio Muñoz. "Homeomorphisms of the real line with singularities." Proyecciones (Antofagasta) 43, no. 6 (2024): 1229–52. http://dx.doi.org/10.22199/issn.0717-6279-6381.

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Given a real number a ≠ 0, we consider the set of homeomorphisms f: R\{0}→ R \{a} where{(x, y):x=0}is a vertical asymtote, {(x, y):y=a} is a horizontal asymtote and f is strictly increasing in each connected component (−∞,0) and (0,+∞). In this context, similar to circle homeomorphisms, all possible dynamics are shown. It is established the relationship between existence of periodic orbits and the limit sets. Also, whenever f−n(0) ≠a for all n ∈ N, then the non-existence of periodic orbits leads to a non-trivial limit set, which is either the whole line R or perfect and nowhere dense. It is sh

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Veksler, A. I. "Maximal Nowhere Dense Sets and their Applications to Problems of Existence of Remote Points and of Weak P-Points." Mathematische Nachrichten 150, no. 1 (1991): 263–75. http://dx.doi.org/10.1002/mana.19911500119.

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Živaljević, Boško. "U-meager sets when the cofinality and the coinitiality of U are uncountable." Journal of Symbolic Logic 56, no. 3 (1991): 906–14. http://dx.doi.org/10.2307/2275060.

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AbtractWe prove that every countably determined set C is U-meager if and only if every internal subset A of C is U-meager, provided that the cofinality and coinitiality of the cut U are both uncountable. As a consequence we prove that for such cuts a countably determined set C which intersects every U-monad in at most countably many points is U-meager. That complements a similar result in [KL]. We also give some partial solutions to some open problems from [KL]. We prove that the set , where H is an infinite integer, cannot be expressed as a countable union of countably determined sets each of

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Newelski, Ludomir. "On partitions of the real line into compact sets." Journal of Symbolic Logic 52, no. 2 (1987): 353–59. http://dx.doi.org/10.2307/2274384.

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The problem mentioned in the title has already been investigated by J. Baumgartner, J. Stern, A. Miller and many others (see [2] and [5]). We prove here some generalizations of theorems of Miller and Stern from [2] and [5]. We use standard set-theoretical notation. LetOne can check that in the above definition we can replace “compact subset of ωω” by “closed nowhere dense subset of ω2” or “Fσ and meager subset of ω2” (as any Fσ subset of ω2 can be presented as a disjoint countable union of compact sets).For functions f, g ϵ ωω we define f ≼ g if for all but finitely many n ϵ ω we have f(n) ≤ g

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DE LA HARPE, PIERRE, and JEAN-PHILIPPE PRÉAUX. "C*-SIMPLE GROUPS: AMALGAMATED FREE PRODUCTS, HNN EXTENSIONS, AND FUNDAMENTAL GROUPS OF 3-MANIFOLDS." Journal of Topology and Analysis 03, no. 04 (2011): 451–89. http://dx.doi.org/10.1142/s1793525311000659.

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We establish sufficient conditions for the C*-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their nontrivial subnormal subgroups; for example normal subgroups of Baumslag–Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser–Milnor and JSJ decompositions. Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better calle

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