Generalized topology and the family of monotonic maps $$\Gamma (X)$$

Abstract

In this paper, interesting properties of the generalized topological spaces, generated by the monotonic maps $$\sigma = (cl_{\delta }\circ int_{\delta }),$$ σ = ( c l δ ∘ i n t δ ) , $$\alpha = (int_{\delta }\circ cl_{\delta }\circ int_{\delta }),$$ α = ( i n t δ ∘ c l δ ∘ i n t δ ) , $$\pi = (int_{\delta }\circ cl_{\delta })$$ π = ( i n t δ ∘ c l δ ) and $$\beta = (cl_{\delta }\circ int_{\delta }\circ cl_{\delta }),$$ β = ( c l δ ∘ i n t δ ∘ c l δ ) , for any generalized topological space $$(X,g_{\delta })$$ ( X , g δ ) are deduced and analyzed. Special subfamilies of the family of monotonic maps $$\Gamma (X)$$ Γ ( X ) are studied and interesting results regarding generalized topologies are obtained.

On pre­ Open Regular Spaces

“…Throughout this paper Z means a topological space (Z,𝜏) without separation axioms, unless it is explicitly referenced. The nature of this work is to explore and extract a specific new genre, the generalized separation axioms have been constructed by several authors [1][2][3][4] . A pre-open was studied in 1982 by Mashhour, Abd El -Monsef and El-Deeb 5 , a subset 𝑀 is a pre-open set if 𝑀 ⊆ int(Cl(𝑀)), where the Cl(𝑀) and the int(𝑀) are the closure and the interior operators of a set 𝑀 respectively 6 .…”