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Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra,…
We study several types of multivalued functions in digital topology.
Digital topology has its own working conditions and sometimes differs from the normal topology. In the area of topological robotics, we have important counterexamples in this study to emphasize this red line between a digital image and a…
We study topological properties of the graph topology.
In this study, we improve the topological complexity computations on digital images with introducing the digital topological complexity computations of a surjective and digitally continuous map between digital images. We also reveal…
In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies.…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces and homotopy invariant fixed point theory. We show that in many cases,…
Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have…
We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…
Some boundedness properties of function spaces (considered as topological groups) are studied.
We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of…
For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain…
The soft topological spaces and some their related concepts have stud- ied in [7]. In this paper, we introduce and study the notions of soft connected topological spaces after a review of preliminary definitions.
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…