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13 pages, 334 KiB

Open AccessArticle

Homological Properties and Digital Relative Homology Groups of MA-Spaces

by Emel Ünver Demir

Symmetry 2025, 17(6), 862; https://doi.org/10.3390/sym17060862 - 1 Jun 2025

Viewed by 358

This study investigates certain homological properties of Marcus–Wyse-based digital spaces, particularly the construction of digital relative homology groups and the excision property within MA-spaces. The Marcus–Wyse topology (for brevity M-) facilitates the definition of continuity, connectedness, and neighborhood relations in pixel-based spaces, thereby enabling the adaptation of algebraic topology methods to digital image data. Within this framework, the functorial structure of digital singular homology on MA-spaces is established, showing that homology groups can be defined and computed in a categorical setting. These results not only strengthen the theoretical foundation of digital topology but also contribute to practical applications such as topological feature extraction, image segmentation, and shape analysis in discrete environments. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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11 pages, 331 KiB

Open AccessArticle

The Fixed Point Property of the Infinite M-Sphere

by Sang-Eon Han and Selma Özçağ

Mathematics 2020, 8(4), 599; https://doi.org/10.3390/math8040599 - 15 Apr 2020

Cited by 3 | Viewed by 2180

Abstract

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space

(Z^{2}, γ)

. This compactification is called the infinite M-topological sphere and denoted by [...] Read more.

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space

(Z^{2}, γ)

. This compactification is called the infinite M-topological sphere and denoted by

({(Z^{2})}^{*}, γ^{*})

, where

{(Z^{2})}^{*} : = Z^{2} \cup {*}, * \notin Z^{2}

and

γ^{*}

is the topology for

{(Z^{2})}^{*}

induced by the topology

γ

Z^{2}

. With the topological space

({(Z^{2})}^{*}, γ^{*})

, since any open set containing the point

“ * ”

has the cardinality

ℵ_{0}

, we call

({(Z^{2})}^{*}, γ^{*})

the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does

({(Z^{2})}^{*}, γ^{*})

have the fixed point property (FPP, for short)? The present paper proves that

({(Z^{2})}^{*}, γ^{*})

has the FPP in the category

M o p (γ^{*})

whose object is the only

({(Z^{2})}^{*}, γ^{*})

and morphisms are all continuous self-maps g of

({(Z^{2})}^{*}, γ^{*})

such that

| g ({(Z^{2})}^{*}) | = ℵ_{0}

with

* \in g ({(Z^{2})}^{*})

g ({(Z^{2})}^{*})

is a singleton. Since

({(Z^{2})}^{*}, γ^{*})

can be a model for a digital sphere derived from the M-topological space

(Z^{2}, γ)

, it can play a crucial role in topology, digital geometry and applied sciences. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

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