Amply soft set and its topologies: AS and PAS topologies

Orhan Göçür; Orhan Göçür

doi:10.3934/math.2021189

AIMS Mathematics

2021, Volume 6, Issue 4: 3121-3141. doi: 10.3934/math.2021189

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Research article

Amply soft set and its topologies: AS and PAS topologies

Orhan Göçür ^,

Department of Statistics and Computer Sciences, Faculty of Science and Literature, Bilecik Seyh Edebali University, 11000 Bilecik-Turkey

Received: 14 November 2020 Accepted: 28 December 2020 Published: 13 January 2021
MSC : 54A10, 54A05, 03E72, 03E05, 03G25, 54A40, 54D10, 54D15

Do the topologies of each dimension have to be same of any space? I show that this is not necessary with amply soft topology produced by classical topologies. For example, an amply soft topology produced by classical topologies may have got any indiscrete topologies, discrete topologies or any topological spaces in each different dimension. The amply soft topology allows to write elements of different classical topologies in its each parameter sets. The classical topologies may be finite, infinite, countable or uncountable. This situation removes the boundary in soft topology and cause it to spread over larger areas. Amply soft topology produced by classical topologies is a special case of an amply soft topology. For this, I define a new soft topology it is called as an amply soft topology. I introduce amply soft open sets, amply soft closed sets, interior and closure of an amply soft set and subspace of any amply soft topological space. I gave parametric separation axioms which are different from T_i separation axioms. T_i questions the relationship between the elements of space itself while P_i questions the strength of the connection between their parameters. An amply soft topology is built on amply soft sets. Amply soft sets use any kind of universal parameter set or initial universe (such as finite or infinite, countable or uncountable). Also, subset, superset, equality, empty set, whole set on amply soft sets are defined. And operations such as union, intersection, difference of two amply soft sets and complement of an amply soft set are given. Then three different amply soft point such as amply soft whole point, amply soft point and monad point are defined. And also I give examples related taking a universal set as uncountable.
- soft set theory,
- amply soft set,
- monad point,
- AS open set,
- AS closed set,
- AS topology,
- subspace,
- interior,
- closure,
- PAS topology,
- halime space,
- orhan space,
- P_i, i = 0, 1, 2, 3, 4
Citation: Orhan Göçür. Amply soft set and its topologies: AS and PAS topologies[J]. AIMS Mathematics, 2021, 6(4): 3121-3141. doi: 10.3934/math.2021189

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Abstract

Do the topologies of each dimension have to be same of any space? I show that this is not necessary with amply soft topology produced by classical topologies. For example, an amply soft topology produced by classical topologies may have got any indiscrete topologies, discrete topologies or any topological spaces in each different dimension. The amply soft topology allows to write elements of different classical topologies in its each parameter sets. The classical topologies may be finite, infinite, countable or uncountable. This situation removes the boundary in soft topology and cause it to spread over larger areas. Amply soft topology produced by classical topologies is a special case of an amply soft topology. For this, I define a new soft topology it is called as an amply soft topology. I introduce amply soft open sets, amply soft closed sets, interior and closure of an amply soft set and subspace of any amply soft topological space. I gave parametric separation axioms which are different from T_i separation axioms. T_i questions the relationship between the elements of space itself while P_i questions the strength of the connection between their parameters. An amply soft topology is built on amply soft sets. Amply soft sets use any kind of universal parameter set or initial universe (such as finite or infinite, countable or uncountable). Also, subset, superset, equality, empty set, whole set on amply soft sets are defined. And operations such as union, intersection, difference of two amply soft sets and complement of an amply soft set are given. Then three different amply soft point such as amply soft whole point, amply soft point and monad point are defined. And also I give examples related taking a universal set as uncountable.

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