{"product_id":"ip-open-sets-ip-cont-ip-con-and-ip-sep-axioms-in-topological-spaces-von-diyar-muhsen-und-layla-saadulla","title":"Ip-Open Sets,Ip-Cont.,Ip-Con. and Ip-Sep. axioms in topological spaces","description":"\u003cp\u003eIn 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).\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783838390024\"\u003e\u003ch3\u003eTopological Spaces\u003c\/h3\u003e\u003c\/div\u003e","brand":"Autorenwelt Shop","offers":[{"title":"Softcover - 9783838390024","offer_id":39486570627165,"sku":"9783838390024","price":59.0,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/7b9cc5bb-67a5-404a-95f2-66bd80db454a.jpg?v=1773209612","url":"https:\/\/shop.autorenwelt.de\/products\/ip-open-sets-ip-cont-ip-con-and-ip-sep-axioms-in-topological-spaces-von-diyar-muhsen-und-layla-saadulla","provider":"Autorenwelt Shop","version":"1.0","type":"link"}