Amply soft set and its topologies: AS and PAS topologies

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 <i>T_i</i> separation axioms. <i>T_i</i> questions the relationship between the elements of space itself while <i>P_i</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.