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The goal of this a book is to recall some types of weak open sets, prove some of its properties and use them to define new kinds of separation axioms. Let us state below some of our important main theorems Let (X,s) and (Y, ) be two topological spaces satisfy the -condition then the map f:(X,s) (Y, ) is continuous if and only if it is -continuous. ( This result is not true without -condition ), also Let (X,s) and (Y, ) be two topological spaces satisfy the -B_ -condition then the map f:(X,s) (Y, ) is continuous if and only if it is - -continuous. And Let (X,s) and (Y, ) be two topological spaces satisfy the -B-condition then the map f:(X,s) (Y, ) is continuous if and only if it is pre- -continuous. Addition Let (X,s) and (Y, ) be two door topological spaces and f:(X,s) (Y, ) be a map, then f is pre- -continuous if and only if it is -continuous. And f is beta- -continuous if and only if it is b- -continuous.