In 1982 the ω-closed set was first introduced by H. Z. Hdeib and he defined it as Definition 1. A sub set of any topological space is ω-closed if it contains all its condensation points and the ω-open set is the complement of the ω-closed set.
Levine introduced the definition of the generalized closed set. And he defined it as follows Definition 2. A sub set A of any topological space is called generalized closed set if cl(A)⊆U whenever A⊆U and U is open, where cl(A) is the closure of the set A.
Modefining Definition2 by using Definition1 we get a new type of generalized closed set call it the generalized ω-closed set (g- ω-closed). Define it as follows Definition 3. A sub set A of any topological space is called generalized ω-closed set (g- ω-closed) if 〖cl〗_ω (A)⊆U whenever A⊆U and U is ω-open set, where 〖cl〗_ω (A) is the ω-closure of the set A.
By making use of the generalized ω-closed set above we can define new types of weak continuities such as approximately ω-continuity, ultra ω-continuity, as follows Definition 4. A map f:X⟶Y is said to be approximately ω-continuous if 〖cl〗_ω (F)⊆f^(-1) (U), whenever U is ω-open subset of Y and F is g- ω-closed subset of X such that F⊆f^(-1) (U).
Definition 5. A map f:X⟶Y is said to be ultraω-continuous if f^(-1) (V) is ω-closed set in X for each ω-open set V in Y.
Also we define new types of weak openness and closeness such as approximately ω-closed mapping, approximately ω-open mapping , ultra ω-closed mapping, and ultra ω-open mapping as follows Definition 6. A map f:X⟶Y is said to be approximately ω-closed if 〖f(U)⊆int〗_ω (V), whenever V is a generalized ω-open subset of Y, F is ω-closed subset of X and f(F)⊆V. We mean by〖int〗_ω (V) is the union of all ω-open subset of V.
Definition 7. A map f:X⟶Y is said to be approximately ω-open if 〖cl〗_ω (F)⊆f(U), , whenever U is an ω-open subset of X, F is a generalized ω-closed subset of Y and F⊆f(U).
Definition 8. A map f:X⟶Y is said to be ultra ω-closed if for every ω-closed subset A of X, f(A) is ω-open set in Y.
Definition 9. A map f:X⟶Y is said to be ultra ω-open if for every ω-open subset A of X, f(A) is ω-closed set in Y.
Using Definition 6, Definition 7, Definition 8, and Definition 9 above we can find and prove relationships between ultra ω-continuity and the approximately ω-continuity, ultra ω-closed mappings and the approximately ω-closed mappings, and the ultra ω-open mappings and the approximately ω-open mappings.By the Definitions of α-ω-open set, pre –ω-open set, b-ω-open set, and β-ω-open set defined by T. Noiri, A. Al-Omari, M. S. M. Noorani and Definition2, we can define generalized α-ω-closed set, generalized pre –ω-closed set, generalized b-ω-closed set, and generalized β-ω-closed set. And then use them to define other types of ultra, approximately weak types of continuity, openness and closeness. Then make use of them to find the relationships mentioned above.
Aims and Scope:
- ω-closed
- generalized ω-closed set
- approximation space
- ultraω-continuous
- approximately ω-open
- approximately ω-continuous
- approximately ω-closed
- ultra ω-open