IP-Open Sets, IP-Cont., IP-Con. and IP-Sep. Axioms in Topological Spaces

In this work we introduce and study a new class of open sets by means of preopen that we call Ip-open set. By the above mentioned set, several new concepts such as Ip-continuous functions, almost and weakly Ip-continuous functions, Ip-open and Ip-closed functions, Ip-connected and Ip-separation axioms are defined and studied. In the light of this work, some of our main results can be listed as follows: If a space (X, τ) is hyperconnected, then IpO(X) pO(X), and The following statements are equivalents for the function f: (X, τ) ® (Y, σ): f is Ip-continuous, the inverse image of every open set in Y is Ip-open set in X, the inverse image of every closed set in Y is Ip-closed set in X,for each AÌX, f (Ipcl(A))Ì clf (A), for each AÌX, intf (A) Ì f (Ipint(A)), for each BÌY, Ipcl(f ˉ¹ (B))Ì f ˉ¹ (clB), for each BÌY and f ˉ¹ (intB)Ì Ipint(f ˉ¹ (B)). Moreover let f: (X, τ) --(Y, σ) be a function and let {A : } be pre-open cover of X. If the restriction f|A :A Y is Ip-continuous function for each , then f is Ip-continuous function. and also a function f: (X, τ)--(Y, σ) is an Ip -open function if and only if for every BÌY, f ˉ¹ (Ip Cl (B)) Ì Clf ˉ¹ (B).