scholarly journals Topological structures in computer science

1987 ◽  
Vol 1 (1) ◽  
pp. 25-40 ◽  
Author(s):  
Efim Khalimsky
Keyword(s):  
Computer Science ◽  
Computer Tomography ◽  
Discrete Mathematics ◽  
Topological Spaces ◽  
Picture Processing ◽  
Topological Methods ◽  
Intermediate Value Theorem ◽  
Robotic Vision ◽  
Finite Spaces ◽  
Intermediate Value

Topologies of finite spaces and spaces with countably many points are investigated. It is proven, using the theory of ordered topological spaces, that any topology in connected ordered spaces, with finitely many points or in spaces similar to the set of all integers, is an interval-alternating topology. Integer and digital lines, arcs, and curves are considered. Topology of N-dimensional digital spaces is described. A digital analog of the intermediate value theorem is proven. The equivalence of connectedness and pathconnectedness in digital and integer spaces is also proven. It is shown here how methods of continuous mathematics, for example, topological methods, can be applied to objects, that used to be investigated only by methods of discrete mathematics. The significance of methods and ideas in digital image and picture processing, robotic vision, computer tomography and system's sciences presented here is well known.

scholarly journals Pβ-connectedness in topological spaces

2017 ◽  
Vol 50 (1) ◽  
pp. 299-308 ◽  
Author(s):  
Brij K. Tyagi ◽  
Sumit Singh ◽  
Manoj Bhardwaj
Keyword(s):  
Topological Spaces ◽  
Connected Space ◽  
Intermediate Value Theorem ◽  
Intermediate Value

Abstract A new property called Pβ-connectedness is introduced which is stronger than connectedness and equivalent to pre-connectedness. The properties of this notion are explored and its relationship with other forms of connectedness, for example hyperconnectedness etc. are discussed. Locally pre-indiscrete spaces are defined as the spaces in which pre-open sets are closed. In such spaces connectedness becomes equivalent to pre-connectedness and hence to Pβ-connectedness, and semi-connectedness becomes equivalent to Pβ- connectedness. The notion of locally Pβ-connected space is introduced. The behavior of Pβ-connectedness under several types of mappings is investigated. An intermediate value theorem is obtained.

scholarly journals The Reverse of the Intermediate Value Theorem in Some Topological Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Oussama Kabbouch ◽  
Mustapha Najmeddine
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Topological Spaces ◽  
Closed Graph ◽  
Intermediate Value Theorem ◽  
Hausdorff Topological Space ◽  
Intermediate Value

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

scholarly journals On some properties of the conformable fractional derivative

2020 ◽  
Vol 6 (2) ◽  
pp. 210-217
Author(s):  
Radouane Azennar ◽  
Driss Mentagui
Keyword(s):  
Fractional Derivative ◽  
Conformable Fractional Derivative ◽  
Intermediate Value Theorem ◽  
Definition Of ◽  
Intermediate Value

AbstractIn this paper, we prove that the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].

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