Special Type of Subset Topological Spaces

In this book we construct special subset topological spaces using subsets from semigroups or groups or rings or semirings. Such study is carried out for the first time and it is both interesting and innovative. Suppose P is a semigroup and S is the collection of all subsets of P together with the empty set, then S can be given three types of topologies and all the three related topological spaces are distinct and results in more types of topological spaces. When the semigroup is finite, S gives more types of finite topological spaces. The same is true in case of groups also. Several interesting properties enjoyed by them are also discussed in this book. In case of subset semigroup using semigroup P we can have subset set ideal topological spaces built using subsemigroups. The advantage of this notion is we can have as many subset set ideal topological spaces as the number of semigroups in P. In case of subset semigroups using groups we can use the subset subsemigroups to build subset set ideal topological spaces over these subset semigroups. This is true in case of subset semigroups which are built using semigroups also. Finally these special subset topological spaces can also be non commutative depending on the semigroup or the group