This book introduces computational proximity (CP) as an algorithmic approachto finding nonempty sets of points thatare either close to each other or far apart Typically in computational proximity, the bookstarts with some form of proximity space (topological space equipped with aproximity relation) that has an inherent geometry In CP,two types of near setsare considered, namely, spatially near sets and descriptivelynear sets It is shown that connectedness, boundedness, mesh nerves, convexity, shapes and shape theory are principal topics in the study of nearness and separation of physical aswell as abstract sets CP has a hefty visual content Applications of CP in computer vision, multimedia, brain activity, biology, social networks, and cosmology are included The book has been derived from the lectures of the author in a graduate course on thetopology of digital images taught over the past several years Many of thestudents have provided important insights and valuable suggestions The topics in this monograph introduce many forms of proximities with acomputational flavour (especially, what has become known as the strong contactrelation), many nuances of topological spaces, and point-free geometry

Computational Proximity: Excursions in the Topology of Digital Images

In this paper, we generalize three types of rough set models based on j-neighborhood space (i.e, type 1 j-neighborhood rough set, type 2 j-neighborhood rough set, and type 3 j-neighborhood rough set), and investigate some of their basic properties. Also, we present another three types of rough set models based on j-adhesion neighborhood space (i.e, type 4 j-adhesion neighborhood rough set, type 5 j-adhesion neighborhood rough set, and type 6 j-adhesion neighborhood rough set). The fundamental properties of approximation operators based on j-adhesion neighborhood space are established. The relationship between the properties of these types is explained. Finally, according to j-adhesion neighborhood space, we give a comparison between the Yao’s approach and our approach.

Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space

Determination of entropy measures for the ordinal scale-based linguistic models

Recently, Sahin studied the warped product semi-slant submanifolds of locally product Riemannian manifolds. In this paper, we obtain some geometric properties of such submanifolds with an example. Also, we establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant angle. The equality case is also considered.

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An inequality for warped product submanifolds of a locally product Riemannian manifold

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

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Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

On I−Proximity Spaces

Fuzzy soft connected sets in fuzzy soft topological spaces II

In this paper, we introduce the notion of -separated soft sets based on the soft space ) , , ( 12 E X  which generate by soft bitopological space ) , , , ( 2 1 E X   and study some of its properties. Based on this notion we introduce the notions of -soft connected(disconnected) spaces and study some of their characterizations and properties. Also, we study the connected of -soft sets by using the soft space ) , , ( 12 E X  . Some examples have given to support these concepts.

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O. A. El-Tantawy

Papers

On I−Proximity Spaces

Fuzzy soft connected sets in fuzzy soft topological spaces II

Pairwise Soft Connected in Soft Bitopological Spaces

Generalized Rough Sets

γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces