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By using -operator we define a new type of dense set, codense ideal, many type of maps and studied their properties and relationships, and carries them from a space to other space by using -map and their generalizations . We will define and study many definitions of resolvable spaces, I-open, pre-open and pre-I-open sets by using -operator and their properties and relationships between them. A new definition of compatible and many theorems have been introduced and by using -operator a new definition of perfect set. Also new type of -operator denoted by -operator. The behavior of this operator has been studied through section three, and the relationships with our new definition in section one have been introduced. And new definition of hereditarily irresolvable and many generalizations by using -operator . finally we use -operator to define Baire set and we show how to link these concepts to ideal topological groups.