On dual Soft compact space

Introduction

Compactness is one of the central and most influential concepts in general topology, playing a fundamental role in the structural analysis of topological spaces and their associated properties. Owing to its theoretical significance, numerous generalizations and variations of compactness both weaker and stronger have been developed to broaden the scope of topological investigations and to deepen the understanding of relationships among different classes of spaces [1,5] . Among these generalizations is H-closedness, a notion originating from the seminal work of Alexandrov and Urysohn in 1929 [4] , which laid the groundwork for the study of spaces exhibiting closure properties beyond classical compactness. In the continued search for weaker forms of compactness. Nearly Compact spaces were presented in 1969 by M. K. Sinegal and Ashea Mathur [5] , providing an alternative framework for analyzing spaces that do not satisfy compactness in its traditional sense. Subsequently, in 1976, Travis Thompson [6] proposed the concept of S-compact spaces, which has since attracted considerable attention in the topological literature. Over time, various other forms of compactness have been explored, each contributing to a richer hierarchical structure within topology. Motivated by this ongoing development, the present work introduces and investigates a new class of topological spaces that represents a weaker form of compactness.

Objective of the Study

This study aims to investigate the fundamental properties of this new notion, establish its relationships with existing compact-type spaces, and illustrate the theory through relevant examples and supporting results. This contribution aims to extend the current understanding of compactness-related concepts and to provide a foundation for further research in this direction.

Preliminaries

Definition 2.1 [3]: Let V 1 and V 2 are initial universes sets and O be the set of all potential boundaries under consideration as for V 1 andV 2 . Whereas parameters are descriptions, features or properties of members of the initials universes sets. The triple (A D , W, G D ) is dual soft set over V 1 and V 2 where,W, G D are functions from O ̃ to power of V 1 and V 2 respectively, so

The set consisting of all dual soft sets constitutes a dual soft space, written as.

Definition 2.3: [1] Let (X Ds , StT E ) be a Dual soft topological space over X Ds . A Dual soft set (A D , W, G) over X Ds is said to be a Dual Soft Close set in X Ds , if its relative complement (A D , W, G) c belong to StT E . Definition 2.4: [1]

There for f(FG K ) dual Soft compact .

Results and Discussion

Results

This Analysis of a dual soft compact space (X Ds , StT E ) confirms the structure of dual soft topology to classical topological properties as established in the triple (AD, W, GD) defined by the authors. The results shown in Section 3 are consistent with the foundational structure of compactness being preserved under dual soft union and intersection therefore establishing the axiomatic consistency of dual soft topology. Additionally, Theorem 3.4 states that all dual soft compact spaces are also dual soft locally compact by using properties of closure on a finite subcover generated from the original compactness property. Thus since the closure of the dual soft open neighborhood is also compact as shown in Section 4 meets the criteria of Remark 3.5 and the classical method that a global property of compactness will provide a pointwise property of local compactness which provides continuity for future uses of the theory for use with convergent and filter bases in X Ds

Discussion

The dual soft compact property is demonstrated by Theorem 3.8, which shows that dual soft closed subsets inherit their parent spaces' compactness. The method used to prove this includes taking the relative complement of the closed subset in an open cover (similar to how proof of the standard compactness works) using dual soft closed sets defined as the complements of all elements in the topology StTEStTE [6] is essential in the demonstration of this fact. Therefore, this result asserts that a dual soft compact set's intersection with a dual soft closed set will also be a dual soft compact space (Theorem 3.8, Part ii). The consequence of this is that these intersections are vital when attempting to establish separation axioms for dual soft spaces; e.g., if the dual soft space satisfies a higher degree of separation (dual soft Hausdorff), then all of its dual soft compact subsets will also be dual soft closed. The structure of the mappings was used to study the behaviour of dual soft compactness under morphisms by considering compositions of the mappings. Theorem 3.14 demonstrated that the composition of two dual soft compact functions is therefore still dual soft compact, which gives a categorical consistency in the mapping structures between the spaces (XDs, StTE) and (YDs, StTE′) [7] . Theorem 3.16 showed that dual soft compactness is also respected under dual soft homeomorphisms as defined in [8] . The proof procedures involved using properties of the continuity of bijective mappings to return to the original open covers. This shows that the image of a dual soft compact subset under a dual soft homeomorphism will also be dual soft compact. However, further review of the presented theorems revealed limitations in that the entire exposition in the current format does not explicitly support the more rigorous and general theorem that the continuous images of a dual soft compact set must continue to have the property of dual soft compactness. This would be a fundamental result according to the general and/or classical theory of compactness and suggests a premise for further theoretical expansion. Future research should look at developing and refining Theorem 3.13 to demonstrate the general behaviour of continuous surjections as opposed to the case of homeomorphisms, thereby completing the characterization of dual soft compactness as a proper topological invariant under the mapping structures defined by [9] .

Conclusion and future study

The purpose of this document is to present a new class of Topological Spaces referred to as dual soft compact spaces. The study of dual soft compact spaces has been thoroughly researched in regards to dual soft topology. We utilize the fundamental notions of dual soft sets and dual soft neighborhoods in order to describe a "weakened" version of compactness that generalizes the traditional compactness definition into a wider/directly related dual soft topology equivalence. This work has produced many new and interesting results. For example, it can be shown that all dual soft compact spaces are due soft locally compact spaces, thus these spaces can be combined with previously existing classes of compact spaces. Furthermore, all closed dual soft subsets of dual soft compact spaces are also dual soft compact, while finite sets are always counted as dual soft compact. In addition, significant work has been completed exploring how dual soft compactness operates with respect to mappings, thus, it was demonstrated that the dual soft compactness of a set is preserved in the image of a dually soft homeomorphism. Furthermore, it was shown that the composition of two or more dual soft compact functions results in a dual soft compact function. Lastly, it was established how the dual soft compactness of an open subspace influences the dual soft compactness of the surrounding space. This research adds to our understanding of behaviours based on compactness within dual soft topological spaces and provides a solid basis for future theory development. The research and results presented have relied upon much of the work of other mathematicians, whilst offering new insights about structural features in soft set theory and its applications [10,12] . The introduction of dual soft compactness opens up a variety of new possibilities in research. Subsequent studies may compare the differences in weak vs. strong forms of compactness in the context of dual soft topology (e.g., dual soft Lindelöf spaces) such as dual soft countably compact spaces, dual soft sequentially compact spaces and how those spaces relate to one another. One additional area for future research concerns the study of dual soft compactness, in conjunction with the different types of dual soft separation axioms (e.g., dual soft Hausdorff, dual soft regular, or dual soft normal). There is a need to determine whether dual soft compactness will lead to the establishment of dual soft closure in spaces that are dual soft Hausdorff. Another broad region for future research relates to discovering an appropriate definition of product topology on dual soft spaces, and examining Tychonoff-type theorems associated with dual soft compactness. Finally, we can use dual soft compactification in combination with the one-point compactification and the Stone-Čech compactification, as well as investigate both uniqueness and universality characteristics. It appears further studies should be done on potential uses that are beyond theoretical progress made for problems with decision-making or analyses of data/uncertainty modelling as a consequence of the outcome produced from dual soft compact type spaces of two interrelated world types. Future work may also look at ways to sustain dual soft compactness through different function types (example: perfect mappings, irreducible mappings, quotient mappings) using dual soft sets [18] . Lastly, the concept of dual soft compact could be expanded by including dual fuzzy soft topological spaces; dual intuitionistic soft topological spaces; and dual neutrosophic soft topological spaces. In other words, this would include both the concept of soft topology and more applications for soft topology.

Graphical abstract

This project presents and develops a new concept of dual soft topology, which extends the classical theory of soft sets (created by Molodtsov in 1999) to dual topological contexts. The research is divided into four major areas: (1) study of dual soft sets; (2) description of dual soft neighborhoods; ِ (3) their role in defining compact spaces in dual soft topology, and (4) an examination of the main properties of the new structure, as well as its connections to more traditional topological and soft set ideas. A major result of the research will be the establishment of a framework for studying compact spaces within the dual soft topological framework. The work was done in cooperation with P. Knight, as shown in the summary.

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