Certain Concepts by Using ωpre-Open Sets

Abstract

The aim of this paper is to add new types of continuous functions and results in the specialization field, where we merge two important terms of open sets that are pre-open and ω-open to get a new set named ωp-open set, where we introduce new functions by using this set, such as Mωpc, ωpMc, ωpMωpc, ωp-continuous, ωp*-continuous, and ωp-irresolute function. We clarify the relationship between these types and illustrate their relationship with some other types of continuous functions. Also we define some new types of spaces such as, ωp-compact, ωp-Lindelöf, and Cωp-Lindelöf. In addition, we submit important results like, the ωp*-continuous image of compact space being ωp-compact, and if X is a ωp-regular space, hence it is Cωp-Lindelöf, as well as we provide the composition between most of the new functions that we defined. Many other results have been found in this our work and they are supported by many examples.

Contra Wpre−Continuous Functions

“…For a subset 𝐴 its interior and closure are denoted by 𝑖𝑛𝑡(𝐴) and 𝑐𝑙(𝐴) , respectively. Also, 𝐴 is said b-open if 𝐴 ⊆ 𝑖𝑛𝑡(𝑐𝑙(𝐴)) ∪ 𝑐𝑙(𝑖𝑛𝑡(𝐴)) and A is ω-open if for every point in it, there is an open set 𝑈 containing 𝑥 with 𝑈 − 𝐴 is countable [6], while A is 𝜔 𝑝𝑟𝑒open (shortly 𝜔 𝑝 -open) whenever replacing the open set to be pre-open [7] and every pre-open, 𝜔-open set is 𝜔 𝑝 -open. The 𝜔 𝑝 -closed and 𝜔 𝑝 -interior defined as in 𝑐𝑙(𝐴) and 𝑖𝑛𝑡(𝐴) , respectively.…”