Ip-Open Sets,Ip-Cont.,Ip-Con. and Ip-Sep. axioms in topological spaces
Topological Spaces
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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, ) is hyperconnected, then IpO(X) pO(X), and The following statements are equivalents for the function f: (X, ) ® (Y, ): f is Ip-continuous, the inverse image of every open set in...
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, ) is hyperconnected, then IpO(X) pO(X), and The following statements are equivalents for the function f: (X, ) ® (Y, ): 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 ¹ (B))Ì f ¹ (clB), for each BÌY and f ¹ (intB)Ì Ipint(f ¹ (B)). Moreover let f: (X, ) --(Y, ) 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, )--(Y, ) is an Ip -open function if and only if for every BÌY, f ¹ (Ip Cl (B)) Ì Clf ¹ (B).
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