scholarly journals More on the cardinality of a topological space

2018 ◽  
Vol 19 (2) ◽  
pp. 269 ◽  
Author(s):  
M. Bonanzinga ◽  
N. Carlson ◽  
M. V. Cuzzupè ◽  
D. Stavrova
Keyword(s):  
Hausdorff Space ◽  
Negative Answer ◽  
Topological Spaces ◽  
Compact Sets ◽  
Hausdorff Spaces ◽  
Separation Axioms ◽  
Covering Properties ◽  
Previous Inequality ◽  
The Impact ◽  
The Continuum

<p>In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”.</p><p>Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995.  The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)<sup>aL</sup><sup>c</sup><sup>(X)ψ(X)</sup>, for Hausdorff spaces and (2) |X| ≤ Hpsw(X)<sup>ω</sup><sup>L</sup><sup>c</sup><sup>(X)ψ(X)</sup>, where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2<sup>c(X)χ(X)</sup> for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2<sup>Uc(X)πχ(X)</sup> if X is a power homogeneous Urysohn space.</p>

scholarly journals Intuitionistic Fuzzy Topological Spaces Model

2020 ◽  
Vol 55 (2) ◽  
Author(s):  
Hind Fadhil Abbas
Keyword(s):  
Hausdorff Space ◽  
Basic Structure ◽  
Normal Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Ideal ◽  
Scientific Phenomenon ◽  
Fuzzy Topological Spaces

The fusion of technology and science is a very complex and scientific phenomenon that still carries mysteries that need to be understood. To unravel these phenomena, mathematical models are beneficial to treat different systems with unpredictable system elements. Here, the generalized intuitionistic fuzzy ideal is studied with topological space. These concepts are useful to analyze new generalized intuitionistic models. The basic structure is studied here with various relations between the generalized intuitionistic fuzzy ideals and the generalized intuitionistic fuzzy topologies. This study includes intuitionistic fuzzy topological spaces (IFS); the fundamental definitions of intuitionistic fuzzy Hausdorff space; intuitionistic fuzzy regular space; intuitionistic fuzzy normal space; intuitionistic fuzzy continuity; operations on IFS, the compactness and separation axioms.

scholarly journals Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft ∗ b Open Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Arif Mehmood ◽  
Saleem Abdullah ◽  
Mohammed M. Al-Shomrani ◽  
Muhammad Imran Khan ◽  
Orawit Thinnukool
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Soft Topology ◽  
Open Set ◽  
Topological Features ◽  
Separation Axioms ◽  
Countable Space ◽  
Countably Compact

In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.

scholarly journals Co-Compact Separation Axioms and Slight Co-Continuity

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1614
Author(s):  
Samer Al Ghour ◽  
Enas Moghrabi
Keyword(s):  
Topological Space ◽  
Topological Property ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Separation Axioms ◽  
Preservation Theorems

Via co-compact open sets we introduce co-T2 as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co-T2 which forms a symmetry. We show that co-T2 propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co-T2 topological space is co-T2. We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions.

Borsuk–Ulam theorem for filtered spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos Biasi ◽  
Alice Kimie Miwa Libardi ◽  
Denise de Mattos ◽  
Sergio Tsuyoshi Ura
Keyword(s):  
Wide Class ◽  
Hausdorff Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Equivariant Map ◽  
Ulam Theorem ◽  
Free Involution

Abstract Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 )   for  ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

scholarly journals Martin's axiom and a regular topological space with uncountable net weight whose countable product is hereditarily separable and hereditarily Lindelöf

1987 ◽  
Vol 52 (2) ◽  
pp. 396-399
Author(s):  
Krzysztof Ciesielski
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Negative Answer ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countable Product ◽  
Generic Filter ◽  
Cohen Forcing ◽  
The Continuum

In [1, p. 51] A. V. Arhangel'skiĭ, in connection with the problems of L-spaces and S-spaces, examined further the notions of hereditary separability and hereditary Lindelöfness. In particular he considered the following property P: “Every regular topological space has a countable net weight provided its countable product is hereditarily Lindelöf and hereditarily separable.” He noticed that the continuum hypothesis implies negation of the property P and posed a question: “Do Martin's Axiom and the negation of the continuum hypothesis imply P?” The purpose of this paper is to give a negative answer to this question.The set-theoretical and topological notation that we use is standard and can be found in [6] and [5] respectively.Throughout the paper we will use the notation H(X, Y) to denote the set of all finite functions from a set X to Y.Theorem. Con(ZFC) → Con(ZFC + MA + ¬CH + there exists a 0-dimensional Hausdorff space X such that nw(X) = с and nw(Y) = ω for any Y ϵ [X]<с).Proof. Let M be a model of ZFC satisfying CH and let F be an M-generic filter over the Cohen forcing {H(ω2 × ω2, 2), ⊃). Then f = ⋃F is a function and f: ω2 × ω2 → 2.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

scholarly journals £-Single Valued Extremally Disconnected Ideal Neutrosophic Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 53
Author(s):  
Fahad Alsharari
Keyword(s):  
Topological Spaces ◽  
Neutrosophic Sets ◽  
Extremally Disconnected ◽  
Fundamental Properties ◽  
Separation Axioms ◽  
New Concepts ◽  
Definition Of

This paper aims to mark out new concepts of r-single valued neutrosophic sets, called r-single valued neutrosophic £-closed and £-open sets. The definition of £-single valued neutrosophic irresolute mapping is provided and its characteristic properties are discussed. Moreover, the concepts of £-single valued neutrosophic extremally disconnected and £-single valued neutrosophic normal spaces are established. As a result, a useful implication diagram between the r-single valued neutrosophic ideal open sets is obtained. Finally, some kinds of separation axioms, namely r-single valued neutrosophic ideal-Ri (r-SVNIRi, for short), where i={0,1,2,3}, and r-single valued neutrosophic ideal-Tj (r-SVNITj, for short), where j={1,2,212,3,4}, are introduced. Some of their characterizations, fundamental properties, and the relations between these notions have been studied.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

Where did Italian peasants live?

1979 ◽  
Vol 25 ◽  
pp. 1-25 ◽  
Author(s):  
P.D.A. Garnsey
Keyword(s):  
Characteristic Feature ◽  
Rural Economy ◽  
Negative Answer ◽  
Second Century ◽  
Small Farmers ◽  
Scholarly Literature ◽  
Wage Labour ◽  
Economic Forces ◽  
Tenant Farmers ◽  
The Impact

Until recently this was a question that was not asked. It was not asked because there was a prior question that was asked, and that received a negative answer: Did peasant proprietors survive in significant numbers in the late Republic or early Empire?The consensus of opinion has been that they were always to be found, but that they were relatively few. As the traditional rural economy of which they had been the characteristic feature gave way under the impact of new economic forces, they became a residual phenomenon. Moreover, this development had already occurred by the late second century B.C.It is to be noted that peasant proprietors, small farmers working the land they owned, rather than free cultivators as a whole, have usually been the object of inquiry. The roles of tenancy in the late Republic and of wage labour in all periods have rarely been positively evaluated. Again, the idea that small ownercultivators, tenant-farmers and day-labourers were overlapping categories in ancient Italy has been little developed in the scholarly literature.

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