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Journal articles on the topic 'Infinite intersection of open sets'

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Published: 5 June 2025
Last updated: 2 August 2025

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1

DURAND, ARNAUD. "Sets with large intersection and ubiquity." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 1 (2008): 119–44. http://dx.doi.org/10.1017/s0305004107000746.

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AbstractA central problem motivated by Diophantine approximation is to determine the size properties of subsets of$\R^d$ ($d\in\N$)of the formwhere ‖⋅‖ denotes an arbitrary norm,Ia denumerable set, (xi,ri)i∈ Ia family of elements of$\R^d\$× (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that ifFId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subsetVof$\R^d\$, the setFϕbelongs to a class Gh(V) of sets with large intersection inVwith respect to a given gauge functionh. We establish that this class is closed under counta
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2

Weihrauch, Klaus, and Tanja Grubba. "Elementary Computable Topology." JUCS - Journal of Universal Computer Science 15, no. (6) (2009): 1381–422. https://doi.org/10.3217/jucs-015-06-1381.

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We revise and extend the foundation of computable topology in the framework of Type-2 theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. We start from a computable topological space, which is a T0-space with a notation of a base such that intersection is computable, and define a number of multi-representations of the points and of the open, the closed and the compact sets and study their properties and relations. We study computability of b
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3

Howard, Paul E. "Definitions of compact." Journal of Symbolic Logic 55, no. 2 (1990): 645–55. http://dx.doi.org/10.2307/2274654.

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Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following:Definition. A topological space X is1. Compact(1) if every open cover of X has a finite subcover.2. Compact(2) if every infinite subset E of X has a complete accumulation point (i.e., a point x0 ∈ X such that for every neighborhood U of x0, |E ∩ U| = |E|).3. Compact(3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover.4. Compact(4) if each nest of closed, nonempty sets has a nonempty intersection.5. Compac
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4

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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5

VELDMAN, WIM. "THE FINE STRUCTURE OF THE INTUITIONISTIC BOREL HIERARCHY." Review of Symbolic Logic 2, no. 1 (2009): 30–101. http://dx.doi.org/10.1017/s1755020309090121.

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In intuitionistic analysis, a subset of a Polish space like ℝ or ${\cal N}$ is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assumi
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6

Hajnal, A., and I. Juhász. "Intersection properties of open sets." Topology and its Applications 19, no. 3 (1985): 201–9. http://dx.doi.org/10.1016/0166-8641(85)90001-x.

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7

Williams, N. H. "An order property for families of sets." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 3 (1988): 294–310. http://dx.doi.org/10.1017/s1446788700032110.

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AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrict
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8

JUHÁSZ, I., ZS NAGY, L. SOUKUP, and Z. SZENTMIKLÓSSY. "Intersection Properties of Open Sets. II." Annals of the New York Academy of Sciences 788, no. 1 General Topol (1996): 147–59. http://dx.doi.org/10.1111/j.1749-6632.1996.tb36806.x.

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9

González, Osvaldo Guzmán. "P-points, MAD families and Cardinal Invariants." Bulletin of Symbolic Logic 28, no. 2 (2022): 258–60. http://dx.doi.org/10.1017/bsl.2021.24.

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AbstractThe main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\math
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10

Balanda, Kevin P. "Families of partial representing sets." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 38, no. 2 (1985): 198–206. http://dx.doi.org/10.1017/s1446788700023053.

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AbstractAssume GCH. Let κ, μ, Σ be cardinals, with κ infinite. Let be a family consisting of λ pairwise almost disjoint subsets of Σ each of size κ, whose union is Σ. In this note it is shown that for each μ with 1 ≤ μ ≤min(λ, Σ), there is a “large” almost disjoint family of μ-sized subsets of Σ, each member of having non-empty intersection with at least μ members of the family .
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11

Goberna, Miguel A., Mercedes Larriqueta, and Virginia N. Vera de Serio. "Stability of the intersection of solution sets of semi-infinite systems." Journal of Computational and Applied Mathematics 217, no. 2 (2008): 420–31. http://dx.doi.org/10.1016/j.cam.2007.02.009.

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12

Mehmood, Arif, Farkhanda Afzal, Saleem Abdullah, Muhammad Imran Khan, and Saeed Gul. "An Absolutely New Attempt to Vague Soft Bitopological Spaces Basis at Newly Defined Operations regarding Vague Soft Points." Mathematical Problems in Engineering 2021 (December 14, 2021): 1–11. http://dx.doi.org/10.1155/2021/4408754.

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In this study, new operations of union, intersection, and complement are defined with the help of vague soft sets in a new way that is in both true and false statements, union is defined with maximum, and intersection is defined with minimum. On the basis of these operations, vague soft topology is defined. Pairwise vague soft open sets and pairwise vague soft closed sets are defined in vague soft bitopological structures (VSBTS). Moreover, generalized vague soft open sets are introduced in VSBTS concerning soft points of the space. On the basis of generalized vague soft open sets, separation
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Adams, Ashleigh, Carole Hall та Eric Stucky. "Classifications of ℓ-Zero-Sumfree Sets". PUMP Journal of Undergraduate Research 2 (17 вересня 2019): 179–98. http://dx.doi.org/10.46787/pump.v2i0.1805.

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 The set of all ℓ-zero-sumfree subsets of ℤ/nℤ is a simplicial complex denoted by Δn, ℓ. We create an algorithm via defining a set of integer partitions we call (n,ℓ)-congruent partitions in order to compute this complex for moderately-sized parameters n and ℓ. We also theoretically determine Δn, ℓ for several infinite families of parameters, and compute the intersection posets and the characteristic polynomials of the corresponding coordinate subspace arrangements.
 
 
 
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14

Das, Birojit, Jayasree Chakraborty та Baby Bhattacharya. "On fuzzy γµ-open sets in generalized fuzzy topological spaces". Proyecciones (Antofagasta) 41, № 3 (2022): 733–49. http://dx.doi.org/10.22199/issn.0717-6279-4784.

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In this paper, we explore the existence of operation approach on open sets in a generalized fuzzy topological space. We introduce the concept of fuzzy γµ-open set and study some basic properties of it. We obtain an interesting result that the intersection of two fuzzy γµ-open sets may not be a fuzzy γµ-open set, but if the operation is regular then the intersection becomes a fuzzy γµ-open set. We also initiate the notions of fuzzy minimal γµ-open set and fuzzy γµ-locally finite space and establish various results related to these.
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15

Hussain, Sabir, and Mohammad Ahmad Alghamdi. "Some properties of weak form of $\gamma$-semi-open sets." Tamkang Journal of Mathematics 43, no. 3 (2012): 329–38. http://dx.doi.org/10.5556/j.tkjm.43.2012.771.

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In this paper, we introduce and explore fundamental properties of weak form of $\gamma$-semi-open sets namely maximal $\gamma$-semi-open sets in topological spaces such as decomposition theorem for maximal $\gamma$-semi-open set. Basic properties of intersection of maximal $\gamma$-semi-open sets are established, such as the $\gamma$-semi-closure law of $\gamma$-semi-radical.
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16

Künzi, Hans-Peter A., and Stephen Watson. "A nontrivial T1-space admitting a unique quasi-proximity." Glasgow Mathematical Journal 38, no. 2 (1996): 207–13. http://dx.doi.org/10.1017/s0017089500031451.

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AbstractWe construct a T1-space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities.
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17

Kincaid, Rex, Allison Oldham, and Gexin Yu. "Optimal open-locating-dominating sets in infinite triangular grids." Discrete Applied Mathematics 193 (October 2015): 139–44. http://dx.doi.org/10.1016/j.dam.2015.04.024.

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18

DONG, PANDENG, SEBASTIÁN DONOSO, ALEJANDRO MAASS, SONG SHAO, and XIANGDONG YE. "Infinite-step nilsystems, independence and complexity." Ergodic Theory and Dynamical Systems 33, no. 1 (2011): 118–43. http://dx.doi.org/10.1017/s0143385711000861.

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AbstractAn∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some∞-step nilsystem. The question if such a system is uniquely ergod
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19

Towle, Eli, and James Luedtke. "Intersection Disjunctions for Reverse Convex Sets." Mathematics of Operations Research 47, no. 1 (2022): 297–319. http://dx.doi.org/10.1287/moor.2021.1132.

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We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that
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20

Al Ghour, Samer. "Soft Rω-Open Sets and the Soft Topology of Soft δω-Open Sets". Axioms 11, № 4 (2022): 177. http://dx.doi.org/10.3390/axioms11040177.

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The author devotes this paper to defining a new class of soft open sets, namely soft Rω-open sets, and investigating their main features. With the help of examples, we show that the class of soft Rω-open sets lies strictly between the classes of soft regular open sets and soft open sets. We show that soft Rω-open subsets of a soft locally countable soft topological space coincide with the soft open sets. Moreover, we show that soft Rω-open subsets of a soft anti-locally countable coincide with the soft regular open sets. Moreover, we show that the class of soft Rω-open sets is closed under fin
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21

ECHI, OTHMAN, and MOHAMED OUELD ABDALLAHI. "ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE." Journal of Algebra and Its Applications 10, no. 04 (2011): 687–99. http://dx.doi.org/10.1142/s0219498811004847.

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An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives
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22

Shi, Yanfeng, and Shuo Qiu. "Delegated Key-Policy Attribute-Based Set Intersection over Outsourced Encrypted Data Sets for CloudIoT." Security and Communication Networks 2021 (March 18, 2021): 1–11. http://dx.doi.org/10.1155/2021/5595243.

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Private set intersection (PSI) is a fundamental cryptographic primitive, allowing two parties to calculate the intersection of their data sets without exposing additional private information. In cloud-based IoT system, IoT-enabled devices would like to outsource their data sets in their encrypted form to the cloud. In this scenario, how to delegate the set intersection computation over outsourced encrypted data sets to the cloud and how to achieve the fine-grained access control for PSI without divulging any additional information to the cloud are still open problems. With that in mind, in thi
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23

LE ROUX, FRÉDÉRIC. "Bounded recurrent sets for planar homeomorphisms." Ergodic Theory and Dynamical Systems 19, no. 4 (1999): 1085–91. http://dx.doi.org/10.1017/s0143385799141646.

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We answer a question proposed by Barge and Franks in 1993 by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.
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24

Jean, Devin C., and Suk J. Seo. "Error-correcting open-locating-dominating sets." Congressus Numerantium 235 (January 11, 2025): 23–40. https://doi.org/10.61091/cn235-03.

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An open-locating-dominating set of a graph models a detection system for a facility with a possible “intruder” or a multiprocessor network with a possible malfunctioning processor. A “sensor” or “detector” is assumed to be installed at a subset of vertices where each can detect an intruder or a malfunctioning processor in its neighborhood, but not at its own location. We consider a fault-tolerant variant of an open-locating-dominating set called an error-correcting open-locating-dominating set, which can correct a false-positive or a false-negative signal from a detector. In particular, we pro
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25

Reich, Simeon, and Alexander J. Zaslavski. "Porosity and the bounded linear regularity property." Journal of Applied Analysis 20, no. 1 (2014): 1–6. http://dx.doi.org/10.1515/jaa-2014-0001.

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Abstract.H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of
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26

Arwini, Khadiga, and Huda Almqtouf Mira. "Further Remarks On Somewhere Dense Sets." Journal of Pure & Applied Sciences 21, no. 1 (2022): 46–48. http://dx.doi.org/10.51984/jopas.v21i1.1630.

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In this article, we prove that a topological space X is strongly hyperconnected iff any somewhere dense set in X is open, in addition we investigate some conditions that make sets somewhere dense in subspaces, finally, we show that any topological space defined on infinite set X has SD-cover with no proper subcover.
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27

Xiang, Li. "Everywhere Nonrecursive r.e. Sets in Recursively Presented Topological Spaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 1 (1988): 105–28. http://dx.doi.org/10.1017/s1446788700031402.

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AbstractRecursively presented topological spaces are topological spaces with a recursive system of basic neighbourhoods. A recursively enumerable (r.e.) open set is a r.e. union of basic neighbourhoods. A set is everywhere r.e. open if its intersection with each basic neighbourhood is r.e. Similarly we define everywhere creative, everywhere simple, everywhere r.e. non-recursive sets and show that there exist sets both with and without these everywhere properties.
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28

KELETI, T., and D. PREISS. "The balls do not generate all Borel sets using complements and countable disjoint unions." Mathematical Proceedings of the Cambridge Philosophical Society 128, no. 3 (2000): 539–47. http://dx.doi.org/10.1017/s0305004199004090.

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We prove that in a separable infinite dimensional Hilbert space the Dynkin system generated by the family of all open balls (that is, the smallest collection containing the open balls and closed under complements and countable disjoint unions) does not contain all Borel sets.
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29

TACHTSIS, ELEFTHERIOS. "ON RAMSEY’S THEOREM AND THE EXISTENCE OF INFINITE CHAINS OR INFINITE ANTI-CHAINS IN INFINITE POSETS." Journal of Symbolic Logic 81, no. 1 (2016): 384–94. http://dx.doi.org/10.1017/jsl.2015.47.

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AbstractRamsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle.In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementione
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30

Guo, Hongwen, and Dihe Hu. "The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping." International Journal of Mathematics and Mathematical Sciences 31, no. 1 (2002): 11–21. http://dx.doi.org/10.1155/s0161171202110337.

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We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.
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31

BESSA, MÁRIO, and MARIA CARVALHO. "NON-UNIFORM HYPERBOLICITY FOR INFINITE DIMENSIONAL COCYCLES." Stochastics and Dynamics 13, no. 03 (2013): 1250026. http://dx.doi.org/10.1142/s0219493712500268.

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Let [Formula: see text] be an infinite dimensional Hilbert space, X a compact Hausdorff space and f : X → X a homeomorphism which preserves a Borel ergodic probability measure which is positive on non-empty open sets. We prove that non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic cocycles with non-trivial unstable bundles.
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32

Bal, P. "A Countable Intersection Like Characterization of Star-Lindelöf Spaces." Researches in Mathematics 31, no. 2 (2023): 3. http://dx.doi.org/10.15421/242308.

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There have been various studies on star-Lindelöfness but they always explain it in terms of open coverings. So, we have demonstrated in this study a connection between star-Lindelöfness and the family of closed sets that resembles countable intersection property of Lindelöf space. We show that a topological space $X$ is star-Lindelöf if and only if every closed subset's family of $X$ not having the modified non-countable intersection property have non-empty intersection.
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33

SHINMOTO, JUNKO, and FUKIKO TAKEO. "THE HAUSDORFF DIMENSION OF SUB-SELF-SIMILAR SETS." Fractals 11, no. 01 (2003): 9–18. http://dx.doi.org/10.1142/s0218348x03001549.

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This paper gives a method to obtain Hausdorff dimensions of sub-self-similar sets, where the ratios of similitudes are different. By using Gibbs measure and pressure in the sense of thermodynamic formalism, we obtain the Hausdorff dimensions of sub-self-similar sets in the space of infinite sequences. We then consider sub-self-similar sets in Rd and show how to calculate their Hausdorff dimensions under an open set condition.
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34

Taddele, Guash Haile, and Songpon Sriwongsa. "An Approximation Technique for General Split Feasibility Problems Based on Projection onto the Intersection of Half-spaces." Nonlinear Convex Analysis and Optimization: An International Journal on Numerical, Computation and Applications 2, no. 1 (2023): 1–29. http://dx.doi.org/10.58715/ncao.2023.2.1.

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This paper presents a novel relaxed CQ algorithm for solving the multiple-sets split feasibility problem with multiple output sets (MSSFPMOS) in infinite-dimensional real Hilbert spaces. The proposed method replaces the projection to half-space with the projection to the intersection of two half-spaces, resulting in accelerated convergence by utilizing previous half-spaces. The present study introduces a novel algorithm that dynamically determines the stepsize, without any a priori knowledge of the operator norm required. Furthermore, the algorithm is proven to exhibit strong convergence to th
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35

Al-Badry, Ayed E. Hashoosh. "PRE-ALEXANDROFF SPACE." University of Thi-Qar Journal of Science 3, no. 4 (2013): 150–55. http://dx.doi.org/10.32792/utq/utjsci/v3i4.532.

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In this paper we introduce a new definition of the topological space weaker of Alexandroff space, namely pre-Alexandroff space. These spaces are which arbitrary intersection of an open set is a pre-open set. In addition to give a new definition of minimal pre-open sets and investigate about some of its properties, also we get some theorems and result related to this space.
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36

Duffus, D., V. Rodl, N. Sauer, and R. Woodrow. "Coloring Ordered Sets to Avoid Monochromatic Maximal Chains." Canadian Journal of Mathematics 44, no. 1 (1991): 91–103. http://dx.doi.org/10.4153/cjm-1992-005-1.

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AbstractThis paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains of X meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality N3. Variants of the problem are also considered and open problems specified.
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37

Dawod, Aveen B., та Sabih W. Askandar. "Soft Generalized αii-Closed Sets in Soft Topological Spaces". BIO Web of Conferences 97 (2024): 00163. http://dx.doi.org/10.1051/bioconf/20249700163.

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This work presented a novel kind of soft generalized closed sets and soft generalized open sets, which we will call soft generalized αii-closed sets and soft generalized αii-open sets in right now paper. The relationships between these two families and several other forms of soft sets, like soft generalized closed, soft generalized ii-closed, soft generalized α-closed, and soft semi-generalized closed sets, are studied and clarified through proofs and evidences such as such as, Theorem8 (If (T,Z) ⊆ (W,Z) ⊆ (X,η,Z) is such that (T,Z) is a sGαICs in (X,η,Z), then (T,Z) is sGαIICs in relation to
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38

Zaslavski, Alexander J. "Generic Existence of Solutions of Symmetric Optimization Problems." Symmetry 12, no. 12 (2020): 2004. http://dx.doi.org/10.3390/sym12122004.

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In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution.
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39

Hamant, Kumar Hamant. "ON C* GENERALIZED eta-CLOSED SETS IN TOPOLOGICAL SPACES." EPRA International Journal of Multidisciplinary Research (IJMR) 8, no. 4 (2022): 255–60. https://doi.org/10.5281/zenodo.14880903.

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The aim of this paper is to introduce the notion of c* generalized eta-closed sets in topological spaces and study their basic properties. It is the weaker form of closed and generalized c*-closed sets. Further, we shall see that the collection of c* generalized eta-closed sets is not closed under finite intersection but it is closed under arbitrary union. Also, we establish the relationship between this new class of closed sets with other existing classes of generalized closed sets in general topology. 
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40

Akhlil, khalid. "Infinite dimensional reflecting ornstein-uhlenbeck stochastic process on non convex open sets." Afrika Statistika 12, no. 1 (2017): 1117–46. http://dx.doi.org/10.16929/as/2017.1117.95.

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41

Miller, Joseph S., and André Nies. "Randomness and Computability: Open Questions." Bulletin of Symbolic Logic 12, no. 3 (2006): 390–410. http://dx.doi.org/10.2178/bsl/1154698740.

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It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a
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42

Altoumi, Nadiy A., and Fatma A. Toumi. "On P^* g- Closed Set in topological Spaces." International Science and Technology Journal 36, no. 1 (2025): 1–10. https://doi.org/10.62341/nafa3003.

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Closed sets play a fundamental role in topological spaces. Notably, a topology on a set can even be characterized by specifying the properties of its closed sets. In 1970, N. Levine introduced the concept of generalized closed sets, defined: A subset S of a topological space X is considered generalized closed if the closure of A is contained in U, cl(A)⊆U whenever A⊆U and U is open set. In this study, we define and explore novel classes of sets termed pre star generalized closed sets (P^* g-closed), pre star generalized open sets (P^* g-open) within the context of topological spaces. The relat
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43

Dammak, Jamel, and Rahma Salem. "Graphic topology on tournaments." Advances in Pure and Applied Mathematics 9, no. 4 (2018): 279–85. http://dx.doi.org/10.1515/apam-2018-0024.

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Abstract Alexandroff spaces are the topological spaces in which the intersection of arbitrary many open sets is open. Let T be an indecomposable tournament. In this paper, first, we associate a trivial topology to T. Then we define another topology on T, called the graphic topology of T, and we show that it is an Alexandroff topology. Our motivation is to investigate some properties of this topology.
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44

Mamman, Ali Bulama, Abdul Iguda, Enoch Suleiman, Haruna Usman Idriss, and Kaze Atsi. "On Semi Generalization of Compatible Ideals Regarding Semi Generalised Open Sets." International Journal of Computational and Applied Mathematics & Computer Science 4 (June 18, 2024): 27–32. http://dx.doi.org/10.37394/232028.2024.4.3.

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In this paper, we use the exiting semi-generalized open set to define semi generalised local function as intersection between any subset of a topological space X and semi generalized-open neighboorhood of any point of X that is not belong to ideal I and investigate its properties in ideal topological space. It is a generalization of the existing generalized local function. We also use the exiting semi-generalized open set to define semi generalised compatible ideals as for every A ⊆ X such that for every x ∈ A, there exist semi generalized-open set U containing x such that U ∪ A ∈ I, then A ∈
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45

Ganster, M., and I. L. Reilly. "Locally closed sets andLC-continuous functions." International Journal of Mathematics and Mathematical Sciences 12, no. 3 (1989): 417–24. http://dx.doi.org/10.1155/s0161171289000505.

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In this paper we introduce and study three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-LC-continuity. All three notions are defined by using the concept of a locally closed set. A subset S of a topological space X is locally closed if it is the intersection of an open and a closed set. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sub-LC-continuous and nearly continuous in the sense of Ptak. Several examples are provided to illustrate the behavior of these new
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46

Dougherty, Randall. "Monotone but not positive subsets of the Cantor space." Journal of Symbolic Logic 52, no. 3 (1987): 817–18. http://dx.doi.org/10.1017/s0022481200029790.

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A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone
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47

Stephan, Frank. "On the structures inside truth-table degrees." Journal of Symbolic Logic 66, no. 2 (2001): 731–70. http://dx.doi.org/10.2307/2695042.

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AbstractThe following theorems on the structure inside nonrecursive truth-table degrees are established: Dëgtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-
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48

Kaviyarasu, M., M. Rajeshwari, and Mohammed Alqahtani. "Neutrosophic Hypersoft Topological Framework for Agricultural Decision-Making." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5905. https://doi.org/10.29020/nybg.ejpam.v18i2.5905.

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This paper presents a novel neutrosophic Agricultural Topology based MCDM approach for solving decision-making problems in agriculture, with uncertainties and infinite variables. Neutrosophic sets are introduced to formalize uncertainties and the neutrosophic sub-base can produce a topology for a dual study of the issue via open sets. The work analyses further properties of the neutrosophic hypersoft(NH) topology; basis, subspace, interior, and closure properties. Two algorithms are proposed: one employing NH sets and the other employing NH topology. Numerical examples derived from real-life a
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Al-shami, Tareq M., Abdelwaheb Mhemdi, Alaa M. Abd El-latif, and Fuad A. Abu Shaheen. "Finite soft-open sets: characterizations, operators and continuity." AIMS Mathematics 9, no. 4 (2024): 10363–85. http://dx.doi.org/10.3934/math.2024507.

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<abstract><p>In this paper, we present a novel family of soft sets named "finite soft-open sets". The purpose of investigating this kind of soft sets is to offer a new tool to structure topological concepts that are stronger than their existing counterparts produced by soft-open sets and their well-known extensions, as well as to provide an environment that preserves some topological characteristics that have been lost in the structures generated by celebrated extensions of soft-open sets, such as the distributive property of a soft union and intersection for soft closure and inter
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50

SMANIA, DANIEL. "Shy shadows of infinite-dimensional partially hyperbolic invariant sets." Ergodic Theory and Dynamical Systems 39, no. 5 (2017): 1361–400. http://dx.doi.org/10.1017/etds.2017.65.

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Let ${\mathcal{R}}$ be a strongly compact $C^{2}$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_{F}{\mathcal{R}}$ is dense for every $F$. Let $\unicode[STIX]{x1D6FA}$ be a compact, forward invariant and partially hyperbolic set of ${\mathcal{R}}$ such that${\mathcal{R}}:\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$ is onto. The $\unicode[STIX]{x1D6FF}$-shadow $W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$ of $\unicode[STIX]{x1D6FA}$ is the union of the sets $$\begin{eqnarray}W_{\unicode[STIX]{x1D6FF}}^{s}(G)
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