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25 pages, 367 KiB

Open AccessFeature PaperArticle

New Interval-Valued Soft Separation Axioms

by Jong Il Baek, Tareq M. Al-shami, Saeid Jafari, Minseok Cheong and Kul Hur

Axioms 2024, 13(7), 493; https://doi.org/10.3390/axioms13070493 - 22 Jul 2024

Cited by 1 | Viewed by 1008

Our research’s main aim is to study two viewpoints: First, we define partial interval-valued soft T

_{i}

(j)-spaces (i = 0, 1, 2, 3, 4; j = i, ii), study some of their properties and some of relationships among them, and give [...] Read more.

Our research’s main aim is to study two viewpoints: First, we define partial interval-valued soft T

_{i}

(j)-spaces (i = 0, 1, 2, 3, 4; j = i, ii), study some of their properties and some of relationships among them, and give some examples. Second, we introduce the notions of partial total interval-valued soft T

_{j}

(i)-spaces (i = 0, 1, 2, 3, 4; j = i, ii) and discuss some of their properties. We present some relationships among them and give some examples. Full article

(This article belongs to the Special Issue Advances in Octahedron Sets and Its Applications)

15 pages, 326 KiB

Open AccessArticle

Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology

by Salem Saleh, Tareq M. Al-shami, A. A. Azzam and M. Hosny

Mathematics 2023, 11(23), 4801; https://doi.org/10.3390/math11234801 - 28 Nov 2023

Cited by 4 | Viewed by 1201

Abstract

It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like [...] Read more.

It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-open sets, can contribute to the growth of fuzzy topology. In the present work, we create some classifications of fuzzy topologies which enable us to obtain several desirable features and relationships. At first, we introduce and analyze stronger forms of fuzzy pre-separation and regularity properties in fuzzy topology called fuzzy pre-

T_{i}, i = 0, \frac{1}{2}, 1, 2, 3, 4,

fuzzy pre-symmetric, and fuzzy pre-

R_{i}, i = 0, 1, 2, 3

by utilizing the concepts of fuzzy pre-open sets and quasi-coincident relation. We investigate more novel properties of these classes and uncover their unique characteristics. By presenting a wide array of related theorems and interconnections, we structure a comprehensive framework for understanding these classes and interrelationships with other separation axioms in this setting. Moreover, the relations between these classes and those in some induced topological structures are examined. Additionally, we explore the hereditary and harmonic properties of these classes. Full article

(This article belongs to the Special Issue Recent Advances on Fuzzy Topology)

8 pages, 255 KiB

Open AccessArticle

Quasihomeomorphisms and Some Topological Properties

by Khedidja Dourari, Alaa M. Abd El-latif, Sami Lazaar, Abdelwaheb Mhemdi and Tareq M. Al-shami

Mathematics 2023, 11(23), 4748; https://doi.org/10.3390/math11234748 - 24 Nov 2023

Viewed by 1204

Abstract

In this paper, we study the properties of topological spaces preserved by quasihomeomorphisms. Particularly, we show that quasihomeomorphisms preserve Whyburn, weakly Whyburn, submaximal and door properties. Moreover, we offer necessary conditions on continuous map

q : X \to Y

where Y is Whyburn [...] Read more.

In this paper, we study the properties of topological spaces preserved by quasihomeomorphisms. Particularly, we show that quasihomeomorphisms preserve Whyburn, weakly Whyburn, submaximal and door properties. Moreover, we offer necessary conditions on continuous map

q : X \to Y

where Y is Whyburn (resp., weakly Whyburn ) in order to render X Whyburn (resp., weakly Whyburn). Also, we prove that if

q : X \to Y

is a one-to-one continuous map and Y is submaximal (resp., door), then X is submaximal (resp., door). Finally, we close this paper by studying the relation between quasihomeomorphisms and k-primal spaces. Full article

(This article belongs to the Special Issue Topological Space and Its Applications)

11 pages, 271 KiB

Open AccessArticle

A New Approach to Soft Continuity

by Sandeep Kaur, Tareq M. Al-shami, Alkan Özkan and M. Hosny

Mathematics 2023, 11(14), 3164; https://doi.org/10.3390/math11143164 - 19 Jul 2023

Cited by 9 | Viewed by 1314

Abstract

The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach [...] Read more.

The concept of continuity in topological spaces has a very important place. For this reason, a great deal of work has been done on continuity, and many generalizations of continuity have been obtained. In this work, we seek to find a new approach to the study of soft continuity in soft topological spaces in connection with an induced mapping based on soft sets. By defining the *-image of a soft set, we define an induced soft mapping and present its related properties. To elaborate on the obtained results and relationships, we furnish a number of illustrative examples. Full article

(This article belongs to the Special Issue Recent Advances on Fuzzy Topology)

15 pages, 336 KiB

Open AccessArticle

On Primal Soft Topology

by Tareq M. Al-shami, Zanyar A. Ameen, Radwan Abu-Gdairi and Abdelwaheb Mhemdi

Mathematics 2023, 11(10), 2329; https://doi.org/10.3390/math11102329 - 16 May 2023

Cited by 22 | Viewed by 2397

Abstract

In a soft environment, we investigated several (classical) structures such as ideals, filters, grills, etc. It is well known that these structures are applied to expand abstract concepts; in addition, some of them offer a vital tool to address some practical issues, especially [...] Read more.

In a soft environment, we investigated several (classical) structures such as ideals, filters, grills, etc. It is well known that these structures are applied to expand abstract concepts; in addition, some of them offer a vital tool to address some practical issues, especially those related to improving rough approximation operators and accuracy measures. Herein, we contribute to this line of research by presenting a novel type of soft structure, namely “soft primal”. We investigate its basic properties and describe its behaviors under soft mappings with the aid of some counterexamples. Then, we introduce three soft operators

{(\cdot)}^{⋄}

,

C l^{⋄}

and

{(\cdot)}^{□}

inspired by soft primals and explore their main characterizations. We show that

C l^{⋄}

satisfies the soft Kuratowski closure operator, which means that

C l^{⋄}

generates a unique soft topology we call a primal soft topology. Among other obtained results, we elaborate that the set of primal topologies forms a natural class in the lattice of topologies over a universal set and set forth some descriptions for primal soft topology under specific types of soft primals. Full article

(This article belongs to the Special Issue Recent Advances in Fuzzy Deep Learning for Uncertain Medicine Data)

12 pages, 309 KiB

Open AccessArticle

The Relationship between Ordinary and Soft Algebras with an Application

by Zanyar A. Ameen, Tareq M. Al-shami, Radwan Abu-Gdairi and Abdelwaheb Mhemdi

Mathematics 2023, 11(9), 2035; https://doi.org/10.3390/math11092035 - 25 Apr 2023

Cited by 18 | Viewed by 1687

Abstract

This work makes a contribution to the theory of soft sets. It studies the concepts of soft semi-algebras and soft algebras, along with some operations. Then, it examines the relations of soft algebras set to their ordinary (crisp) counterparts. Among other things, we [...] Read more.

This work makes a contribution to the theory of soft sets. It studies the concepts of soft semi-algebras and soft algebras, along with some operations. Then, it examines the relations of soft algebras set to their ordinary (crisp) counterparts. Among other things, we show that every algebra of soft sets induces a collection of ordinary algebras of sets. By using the formulas (in Theorem 7 and Corollary 1), we present a novel construction, allowing us to construct a soft algebra from a system of ordinary algebras of sets. Two examples are presented to show how these formulas can be used in practice. This approach is general enough to be applied to many other (soft) algebraic properties and shows that ordinary algebras contain instruments enabling us to construct soft algebras and to study their properties. As an application, we demonstrate how elements of the generated soft algebra can be used to describe the weather conditions of a region. Full article

(This article belongs to the Special Issue Data Driven Decision-Making Under Uncertainty (D3U), 2nd Edition)

17 pages, 356 KiB

Open AccessArticle

Five Generalized Rough Approximation Spaces Produced by Maximal Rough Neighborhoods

by A. A. Azzam and Tareq M. Al-shami

Symmetry 2023, 15(3), 751; https://doi.org/10.3390/sym15030751 - 18 Mar 2023

Cited by 8 | Viewed by 1692

Abstract

In rough set theory, the multiplicity of methods of calculating neighborhood systems is very useful to calculate the measures of accuracy and roughness. In line with this research direction, in this article we present novel kinds of rough neighborhood systems inspired by the [...] Read more.

In rough set theory, the multiplicity of methods of calculating neighborhood systems is very useful to calculate the measures of accuracy and roughness. In line with this research direction, in this article we present novel kinds of rough neighborhood systems inspired by the system of maximal neighborhood systems. We benefit from the symmetry between rough approximations (lower and upper) and topological operators (interior and closure) to structure the current generalized rough approximation spaces. First, we display two novel types of rough set models produced by maximal neighborhoods, namely, type 2

m_{ξ}

-neighborhood and type 3

m_{ξ}

-neighborhood rough models. We investigate their master properties and show the relationships between them as well as their relationship with some foregoing ones. Then, we apply the idea of adhesion neighborhoods to introduce three additional rough set models, namely, type 4

m_{ξ}

-adhesion, type 5

m_{ξ}

-adhesion and type 6

m_{ξ}

-adhesion neighborhood rough models. We establish the fundamental characteristics of approximation operators inspired by these models and discuss how the properties of various relationships relate to one another. We prove that adhesion neighborhood rough models increase the value of the accuracy measure of subsets, which can improve decision making. Finally, we provide a comparison between Yao’s technique and current types of adhesion neighborhood rough models. Full article

16 pages, 351 KiB

Open AccessArticle

A Novel Framework for Generalizations of Soft Open Sets and Its Applications via Soft Topologies

by Tareq M. Al-shami, Abdelwaheb Mhemdi and Radwan Abu-Gdairi

Mathematics 2023, 11(4), 840; https://doi.org/10.3390/math11040840 - 7 Feb 2023

Cited by 39 | Viewed by 2012

Abstract

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach [...] Read more.

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions. Full article

(This article belongs to the Special Issue Recent Advances on Fuzzy Topology)

21 pages, 461 KiB

Open AccessEditor’s ChoiceArticle

Generalized Frame for Orthopair Fuzzy Sets: (m,n)-Fuzzy Sets and Their Applications to Multi-Criteria Decision-Making Methods

by Tareq M. Al-shami and Abdelwaheb Mhemdi

Information 2023, 14(1), 56; https://doi.org/10.3390/info14010056 - 16 Jan 2023

Cited by 86 | Viewed by 4278

Abstract

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of [...] Read more.

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “

(m, n)

-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank

(m, n)

-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with

(m, n)

-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of

(m, n)

-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example. Full article

(This article belongs to the Special Issue New Trend on Fuzzy Systems and Intelligent Decision Making Theory: A Themed Issue Dedicated to Dr. Ronald R. Yager)

► Show Figures

Figure 1

18 pages, 358 KiB

Open AccessArticle

Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms

by Tareq M. Al-shami, José Carlos R. Alcantud and A. A. Azzam

Mathematics 2022, 10(23), 4488; https://doi.org/10.3390/math10234488 - 28 Nov 2022

Cited by 17 | Viewed by 1771

Abstract

This paper contributes to the field of supra-soft topology. We introduce and investigate supra

p p

-soft

T_{j}

and supra

p t

-soft

T_{j}

-spaces

(j = 0, 1, 2, 3, 4)

. These are [...] Read more.

This paper contributes to the field of supra-soft topology. We introduce and investigate supra

p p

-soft

T_{j}

and supra

p t

-soft

T_{j}

-spaces

(j = 0, 1, 2, 3, 4)

. These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra

t p

-soft

T_{j}

and supra

t t

-soft

T_{j}

-spaces

(j = 0, 1, 2, 3, 4)

. This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for

i = 0, 1

, possessing the

T_{i}

property by a parametric supra-topological space implies possessing the

p p

-soft

T_{i}

property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of

p p

-soft

T_{i}

-spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft

S^{Full article(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)13 pages,                777 KiB Open Access Article Generating Soft Topologies via Soft Set Operators by A. A. Azzam, Zanyar A. Ameen, Tareq M. Al-shami and Mohammed E. El-Shafei Symmetry 2022, 14 (5), 914; https://doi.org/10.3390/sym14050914 - 29 Apr 2022 Cited by 43 | Viewed by 2591 Abstract As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating [...] Read more. As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology. Full article (This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning)24 pages,                645 KiB Open Access Article Some Topological Approaches for Generalized Rough Sets and Their Decision-Making Applications by Radwan Abu-Gdairi, Mostafa A. El-Gayar, Tareq M. Al-shami, Ashraf S. Nawar and Mostafa K. El-Bably Symmetry 2022, 14 (1), 95; https://doi.org/10.3390/sym14010095 - 7 Jan 2022 Cited by 34 | Viewed by 3150 Abstract The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we [...] Read more. The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we present new topological approaches as a generalization of Pawlak’s theory by using j -adhesion neighborhoods and elucidate the relationship between them and some other types of approximations with the aid of examples. Topologically, we give another generalized rough approximation using near open sets. Also, we generate generalized approximations created from the topological models of j -adhesion approximations. Eventually, we compare the approaches given herein with previous ones to obtain a more affirmative solution for decision-making problems. Full article (This article belongs to the Topic Topology vs. Geometry in Data Analysis/Machine Learning) ► ▼ Show Figures Figure 115 pages,                1003 KiB Open Access Article Caliber and Chain Conditions in Soft Topologies by José Carlos R. Alcantud, Tareq M. Al-shami and A. A. Azzam Mathematics 2021, 9 (19), 2349; https://doi.org/10.3390/math9192349 - 22 Sep 2021 Cited by 21 | Viewed by 2531 Abstract In this paper, we contribute to the growing literature on soft topology. Its theoretical underpinning merges point-set or classical topology with the characteristics of soft sets (a model for the representation of uncertain knowledge initiated in 1999). We introduce two types of axioms [...] Read more. In this paper, we contribute to the growing literature on soft topology. Its theoretical underpinning merges point-set or classical topology with the characteristics of soft sets (a model for the representation of uncertain knowledge initiated in 1999). We introduce two types of axioms that generalize suitable concepts of soft separability. They are respectively concerned with calibers and chain conditions. We investigate explicit procedures for the construction of non-trivial soft topological spaces that satisfy these new axioms. Then we explore the role of cardinality in their study, and the relationships among these and other properties. Our results bring to light a fruitful field for future research in soft topology. Full article (This article belongs to the Special Issue Fuzzy Sets and Artificial Intelligence) ► ▼ Show Figures Figure 118 pages,                2151 KiB Open Access Article Classification of Reservoir Recovery Factor for Oil and Gas Reservoirs: A Multi-Objective Feature Selection Approach by Qasem Al-Tashi, Emelia Akashah Patah Akhir, Said Jadid Abdulkadir, Seyedali Mirjalili, Tareq M. Shami, Hitham Alhusssian, Alawi Alqushaibi, Ayed Alwadain, Abdullateef O. Balogun and Nasser Al-Zidi J. Mar. Sci. Eng. 2021, 9 (8), 888; https://doi.org/10.3390/jmse9080888 - 18 Aug 2021 Cited by 10 | Viewed by 4159 Abstract The accurate classification of reservoir recovery factor is dampened by irregularities such as noisy and high-dimensional features associated with the reservoir measurements or characterization. These irregularities, especially a larger number of features, make it difficult to perform accurate classification of reservoir recovery factor, [...] Read more. The accurate classification of reservoir recovery factor is dampened by irregularities such as noisy and high-dimensional features associated with the reservoir measurements or characterization. These irregularities, especially a larger number of features, make it difficult to perform accurate classification of reservoir recovery factor, as the generated reservoir features are usually heterogeneous. Consequently, it is imperative to select relevant reservoir features while preserving or amplifying reservoir recovery accuracy. This phenomenon can be treated as a multi-objective optimization problem, since there are two conflicting objectives: minimizing the number of measurements and preserving high recovery classification accuracy. In this study, wrapper-based multi-objective feature selection approaches are proposed to estimate the set of Pareto optimal solutions that represents the optimum trade-off between these two objectives. Specifically, three multi-objective optimization algorithms—Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Grey Wolf Optimizer (MOGWO) and Multi-Objective Particle Swarm Optimization (MOPSO)—are investigated in selecting relevant features from the reservoir dataset. To the best of our knowledge, this is the first time multi-objective optimization has been used for reservoir recovery factor classification. The Artificial Neural Network (ANN) classification algorithm is used to evaluate the selected reservoir features. Findings from the experimental results show that the proposed MOGWO-ANN outperforms the other two approaches (MOPSO and NSGA-II) in terms of producing non-dominated solutions with a small subset of features and reduced classification error rate. Full article (This article belongs to the Special Issue Applications of AI Tools in Petroleum Industry from Geosciences to Engineering) ► ▼ Show Figures Figure 1 <div class='openpopupgallery' data-imgindex='1' data-target='article-616600-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/jmse/jmse-09-00888/article_deploy/html/images/jmse-09-00888-g002-550.jpg?1629451024'><p>Figure 2</p></div>
                            ---                                                                                <div class='openpopupgallery' data-imgindex='2' data-target='article-616600-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/jmse/jmse-09-00888/article_deploy/html/images/jmse-09-00888-g003-550.jpg?1629451024'><p>Figure 3</p></div>
                            ---                                                                                <div class='openpopupgallery' data-imgindex='3' data-target='article-616600-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/jmse/jmse-09-00888/article_deploy/html/images/jmse-09-00888-g004-550.jpg?1629451024'><p>Figure 4</p></div>
                            ---                                                                                <div class='openpopupgallery' data-imgindex='4' data-target='article-616600-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/jmse/jmse-09-00888/article_deploy/html/images/jmse-09-00888-g005-550.jpg?1629451024'><p>Figure 5</p></div>
                            ---                                                                                <div class='openpopupgallery' data-imgindex='5' data-target='article-616600-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/jmse/jmse-09-00888/article_deploy/html/images/jmse-09-00888-g006-550.jpg?1629451024'><p>Figure 6</p></div>13 pages,                803 KiB Open Access Article Connectedness and Local Connectedness on Infra Soft Topological Spaces by Tareq M. Al-shami and El-Sayed A. Abo-Tabl Mathematics 2021, 9 (15), 1759; https://doi.org/10.3390/math9151759 - 26 Jul 2021 Cited by 23 | Viewed by 2697 Abstract This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We [...] Read more. This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components. 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                        var checkboxId    = $(this).attr('id');
                        var link          = $('.remove-filter-link[data-filterid="'+checkboxId+'"');
                        var linkContainer = link.closest('.remove-filter-container');

                        $(this).prop('checked', false);
                        linkContainer.addClass('remove-filter-container--hidden');
                    });

                    filterContainer.find('.filter-actions-container--empty').removeClass('filter-actions-container--hidden');
                    filterContainer.find('.filter-actions-container--filled').addClass('filter-actions-container--hidden');
                });

                processResetAllVisibility();
                $('.filter-count').hide();
                $('.refineSearch').removeClass('button--grey').addClass('button--default');
            });

            $('.js-filter').on('keyup', function(e) {
                var modal  = $(this).closest(".reveal-modal");
                var search = $(this).val().toLowerCase();

                modal.find(".js-data-filter").each(function() {
                    var filterContainer = $(this);
                    if ("" == search || filterContainer.find(".refine_checkbox:first").prop('checked') || filterContainer.data('filter').includes(search)) {
                        filterContainer.show();
                    }
                    else {
                        filterContainer.hide();
                    }
                });
            });

            setTimeout(function(){
                $(document).foundation('equalizer', 'reflow');
            }, 35)
        });}

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