New results for certain types of contra pre-continuous functions in topological space

Abstract

The aim of this work, introduce some properties of another type of contra pre-continuous functions in topological space up on some generalization of closed sets, namely pre-closed and preopen sets as well as showing some theorems related about to composition and restriction for these types of functions, also submitted another types ofcontra-continuous which are bi-contra-continuous functions with several theorems about this concept have been given.