m-Convex Topological Stabilization and mcₙ-Convex Functions in Preconvex Spaces

Introduction

This research lies at the intersection of several key mathematical disciplines, including Functional Analysis, General Topology, and Abstract Convexity Theory. Specifically, it contributes to the study of generalized convex structures and stabilized closure operations in preconvexity spaces. According to the Mathematics Subject Classification (MSC2020), the research is primarily categorized under: 46A19 (Convex sets and functions in topological vector spaces), 54A05 (Topological structures), 52A41 (Convex functions and convex analysis), and 46N10 (Applications of functional analysis in optimization and convex analysis). The study of generalized convexity structures has gained increasing attention in recent decades, particularly in the context of topological and functional spaces that lack classical linearity or continuity assumptions. Among these, m-convexity originally introduced by Min (2011) [1] offers a framework for describing convex-like behavior without relying on traditional convexity axioms. However, classical treatments of m-convexity often omit the effects of iterative closure and the structural consequences of stabilization within such spaces. This paper aims to bridge that gap by introducing a new topological structure, denoted as τₘ, generated by the family of stable m-convex sets, which remain invariant under iterative application of the m-closure operator. By exploring the stabilization behavior of m-convex closures, we develop foundational theorems that extend and refine the properties of preconvexity spaces. Moreover, we generalize the notion of convex functions to define mcₙconvex functions, which preserve m-convexity structure across different closure levels. Through rigorous analysis, theoretical development, and illustrative examples, this research contributes a coherent framework that unifies m-convexity, closure operations, and topological continuity. The proposed topology τₘ not only enhances our understanding of abstract convexity but also opens pathways to new applications in optimization theory, fuzzy systems, and infinite-dimensional functional analysis.

Related Work

Several studies have explored generalized convexity and closure operators in non-classical topological spaces.

In particular, Min (2011) [1] introduced the notion of mconvex sets and their corresponding closure operator, providing foundational results for the development of mconvex analysis. Further contributions include work on weak convexity, generalized convex hulls, and fixed-point applications within preconvexity frameworks. However, these approaches often treat the closure operator in isolation, without addressing stabilization behavior or the possibility of generating a topology from fixed points. This paper advances the theory by systematically incorporating iterative closure stabilization as a structural foundation for new topological constructs. Modular function spaces and fixed point results therein have played a significant role in the analysis of generalized convex structures. The work by Khamsi et al. [4] provides foundational results that inspire further applications of mcₙconvex functions in abstract spaces. Recent contributions in the study of generalized convex structures include works by Zelinskii and Shpakivskyi, particularly in the context of weak m-convex sets and their applications to topological and geometric shadow problems. These works emphasize the interplay between convex hull constructions and continuity-like properties in abstract convex systems. Additionally, studies on generalized closure operators in Orlicz spaces and topologies generated by fixed point sets provide further foundational motivation for our approach.

Motivation

The motivation behind this work stems from the need to better understand the behavior of closure operations under iteration and their impact on generalized topologies. In many applications particularly in abstract optimization, nonclassical geometry, and fuzzy systems it is desirable to construct topologies where closure-stable sets play a central role. By formalizing a topology induced by stable m-convex sets, we provide a novel framework that unifies convexity, closure, and topological continuity. This leads to new insights into the behavior of generalized convex functions and sets the stage for further extensions in layered and hierarchical convexity. Such stabilized convex structures may provide a suitable framework for optimization problems defined on non-linear or discrete spaces. This research also aims to bridge the gap between fixed-point topological behavior and functional convexity, particularly in applications involving infinite-dimensional analysis, such as functional spaces, where iterative closure behavior is crucial. Furthermore, connections with earlier works on semicompactness and weak closure systems justify the importance of developing a stabilized m-convex topology that supports generalized functional continuity.

Basic Definitions and Generalizations

The m-closure of A is defined by

Proof

We prove by mathematical induction that for all k ∈ ℕ:

Base Case (k = 1)

By definition, mh(A) is the intersection of all m-convex supersets of A. Since each such set contains A, we have A ⊆ mh(A).

Inductive Step

Assume mhᵏ(A) ⊆ mhᵏ⁺¹(A) for some k ≥ 1. Then, by the monotonicity of mh, we get: mhᵏ⁺¹(A) = mh (mhᵏ(A)) ⊆ mh (mhᵏ⁺¹(A)) = mhᵏ⁺²(A) Thus, the sequence is non-decreasing.

Theorem 5.2 (Stabilization Criterion)

If mhⁿ(A) = mhⁿ⁺¹(A), then mhᵏ(A) = mhⁿ(A) for all k ≥ n.

Proof

By induction, since mhⁿ(A) = mhⁿ⁺¹(A) implies fixed point.

Proof

We use induction on k ≥ n.

Base Case (k = n)

This is given: mhⁿ(A) = mhⁿ⁺¹(A).

Inductive Step

Assume mhᵏ(A) = mhⁿ(A) for some k ≥ n.

Then mhᵏ⁺¹(A) = mh(mhᵏ(A)) = mh(mhⁿ(A))

But since mhⁿ(A) = mhⁿ⁺¹(A), we know mh(mhⁿ(A)) = mhⁿ(A), i.e., it is a fixed point.

Hence, mhᵏ⁺¹(A) = mhⁿ(A).

By induction, mhᵏ(A) = mhⁿ(A) for all k ≥ n.

Theorem 5.3 (Relation with Convex Hull):

For all k ∈ ℕ: G(A) ⊆ mhᵏ(A)

Step 1: Determine m-convexity of f⁻¹(U) Assume under the preconvexity σ that {a, b} is not m-convex.

Step 2: Apply the m-closure operator mh¹(f⁻¹(U)) = mh({a, b}) = {a, b}, mh²(f⁻¹(U)) = mh(mh({a, b})) = {a, b}, So the closure stabilizes at level 1.

Step 3: Compare with image closure mh¹(f⁻¹(U)) = {a, b}, f⁻¹(mh¹(U)) = f⁻¹({b}) = {a, b}, So mh¹(f⁻¹(U)) = f⁻¹(mh¹(U)).

Although the set {a, b} is not m-convex initially, its closure stabilizes, and the function meets the mc₂-convex condition.

Conclusion

F is not mc₁-convex, but is exactly mc₂-convex. This study introduced a topological framework based on the stabilization of m-convex closures in preconvexity spaces. By defining the stabilized topology τₘ and characterizing its structure via closure-fixed m-convex sets, we established a foundation for analyzing generalized continuity and convexity. The introduction of mcₙ-convex functions adds an additional layer of abstraction, enabling a nuanced understanding of convexity-preserving mappings across iterative structures. Future directions may include integration with fuzzy convexity, duality theory, and topological fixedpoint theorems.

References

1. Min H. M-Convexity and generalized closure operators in preconvex spaces. J Nonlinear Anal Topol. 2011;5(3):215-228.
2. Zelinskii YB, Shpakivskyi VS. On weakly m-convex sets and their applications to geometric covering problems. In: Proceedings of the Conference on Convex Geometry and Analysis, Lviv, Ukraine; 2015.
3. Hudzik H, Szymaszkiewicz L. Basic topological and geometric properties of Orlicz spaces over an arbitrary set of atoms. J Anal Appl. 2008;27:425-449.
4. Khamsi MA, Kozlowski WM, Reich S. Fixed point theory in modular function spaces. Nonlinear Anal Theory Methods Appl. 1990;14(11):935-953. https://doi.org/10.1016/0362-546X(90)90111-S
5. Rubinov AM. Abstract convexity and global optimization. Dordrecht: Kluwer Academic Publishers; 2000.