Semi-separation axioms of the infinite Khalimsky topological sphere

Han, Sang-Eon

doi:10.1016/j.topol.2019.107006

Cited by 6 publications

(3 citation statements)

References 18 publications

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“…The exploration of topological concepts is further enriched by Han's (2020) study on the semi-separation axioms of the infinite Khalimsky topological sphere. This work provides an excellent example of how theoretical topology intersects with geometric structures, offering a nuanced view of separation properties in complex topological spaces.…”

Section: Introductionmentioning

confidence: 99%

Infinite Patterns: A Journey into the Heart of Topology

Pasupuleti

2024

Beyond Dimensions: A Journey Through Topological Mathematics

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"Infinite Patterns: A Journey into the Heart of Topology" is an insightful exploration into the intricate world of topology, focusing on its application in understanding infinite-dimensional spaces and the complex patterns that emerge within them. This chapter delves into the fundamental concepts of topology, including key principles like separation axioms and the topology of infinite-dimensional manifolds. It also discusses the practical implications of these concepts through the lens of equivariant topology, generative design in engineering, and their interplay with algebraic structures. By integrating theoretical research with practical applications, the chapter showcases the profound impact of topological insights on advancing mathematical sciences and their significant crossover into real-world problem-solving. Through detailed analysis of cutting-edge research and foundational theories, this chapter provides a comprehensive overview of the current state and future potential of topological studies, highlighting how topology continues to be a pivotal element in the broader narrative of scientific advancement. Keywords: Topology Infinite-Dimensional Spaces,Equivariant Topology,Generative Design,Separation Axioms,Topological Manifolds,Mathematical Patterns,Algebraic Topology,Geometric Topology and Applied Topology.

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Section: Introductionmentioning

confidence: 99%

Infinite Patterns: A Journey into the Heart of Topology

Pasupuleti

2024

Beyond Dimensions: A Journey Through Topological Mathematics

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“…Defining a new type of generalized open sets and utilizing it to define new topological concepts is now a very hot research topic [1][2][3][4][5][6][7][8][9][10][11]. As a new type of generalized open sets, Al-Ghour and Samarah in [12] defined coc-open sets as follows: A subset A of a topological space (X, τ) is called coc-open set if A is a union of sets of the form V − C, where V ∈ τ and C is a compact subset of X.…”

Section: Introductionmentioning

confidence: 99%

Co-Compact Separation Axioms and Slight Co-Continuity

Ghour

Moghrabi

2020

Symmetry

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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.

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“…Besides, | • | means the cardinality of the given set. To study the FPP problems of the infinite K-sphere and the infinite K-circle [7], we need to define the following sets. Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

The fixed point property of the infinite K-sphere in the set Con((Z2))

Hana

2020

Filomat

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In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

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Semi-separation axioms of the infinite Khalimsky topological sphere

Cited by 6 publications

References 18 publications

Infinite Patterns: A Journey into the Heart of Topology

Infinite Patterns: A Journey into the Heart of Topology

Co-Compact Separation Axioms and Slight Co-Continuity

The fixed point property of the infinite K-sphere in the set Con((Z2))

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Semi-separation axioms of the infinite Khalimsky topological sphere

Cited by 6 publications

References 18 publications

Infinite Patterns: A Journey into the Heart of Topology

Infinite Patterns: A Journey into the Heart of Topology

Co-Compact Separation Axioms and Slight Co-Continuity

The fixed point property of the infinite K-sphere in the set Con*((Z2)*)

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The fixed point property of the infinite K-sphere in the set Con((Z2))