Pretopological space

A pretopological space is a mathematical structure consisting of an ordered pair (X, 𝜏), where X is a set and 𝜏 is a pseudo-closure operator. The pseudo-closure operator satisfies two axioms, which ensure that it behaves in a way that is similar to a closure operator. However, unlike a closure operator, a pseudo-closure operator need not be idempotent or satisfy the exchange law. The purpose of a pretopological space is to provide a framework for studying topological spaces, which are a more restrictive type of mathematical structure.From: Stochastic pretopology as a tool for complex networks analysis [2019]

Fuzzy-Genetic Approach to Epidemiology

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Published in Jyoti Mishra, Ritu Agarwal, Abdon Atangana, Mathematical Modeling and Soft Computing in Epidemiology, 2020

Minakshi Biswas Hathiwala, Jignesh Pravin Chauhan, Gautam Suresh Hathiwala

In mathematics, general topology, viz., point set topology or ordinary topology, deals with the characteristics of a space that are preserved under continuous distortions, such as stretching and bending, but not tearing or gluing. General topology, which is based on the crisp set, establishes the foundational aspects of other branches of topology. Pretopological spaces are the generalization of topological spaces. Two elements may be close to a third element via some relation; however, there is not sufficient structure to say which one of them is closer or nearer. Such a space is called a topological space. It has a suitable structure to embrace the concept of boundary. If we remove the underlying behavior of boundary from this structure, the weakest notion of nearness is uncovered, and thus, we obtain a pretopological space [7]. It is customary to define topology on a set through a class of open sets or a class of closed sets [8]. However, the concept of topology on a set can also be presented through operators such as closures, neighborhoods, and interiors instead of the conventional class of open or closed sets [9].

Stochastic pretopology as a tool for complex networks analysis

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Published in Journal of Information and Telecommunication, 2019

Quang Vu Bui, Soufian Ben Amor, Marc Bui

A pretopological space is an ordered pair , where X is a set and is a pseudo-closure operator, satisfying the two following axioms: (P1): (Preservation of Nullary Union)(P2): (Extensivity)

Pretopological space

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Fuzzy-Genetic Approach to Epidemiology

Stochastic pretopology as a tool for complex networks analysis