The concept of nano near open sets was originally proposed by Thivagar and Richard (Int. J. Math. Stat. Inven 1:31-37). The main aspect of this paper is to introduce a new sort of nano near open sets namely, nano ∧β-sets. Fundamental properties of these sets are studied and compared to the previous one. It turns out that every nano β-open set is a nano ∧β-set. So, nano ∧β-sets are an extension of the previous nano near open sets, such as nano regular open, nano α-open, nano semi-open, nano pre-open, nano γ-open, and nano β-open sets. Meanwhile, it is shown that the concepts of nano ∧β-sets and nano δβ-open sets are different and independent. Based on these new sets, nano ∧β-continuous functions are defined and some results involving their characterizations are derived. In addition, the concepts of nano ∨β-closure and nano ∧β-interior are presented. Their properties are used to introduce and study the nano ∧β-continuous functions.

中文翻译：

---

纳米∧β-集合和纳米∧β-连续性

纳米近开集的概念最初是由 Thivagar 和 Richard (Int. J. Math. Stat. Inven 1:31-37) 提出的。本文的主要方面是介绍一种新的纳米近开集，即纳米∧β-集。研究了这些集合的基本特性，并与前一个特性进行了比较。事实证明，每个nano β-open 集都是nano ∧β-set。因此，nano ∧β-集是之前nano 近开集的扩展，例如nano 规则开集、nano α-开集、nano 半开集、nano 预开集、nano γ-开集和nano β-开集. 同时，证明了nano ∧β-sets 和nano δβ-open set 的概念是不同的和独立的。基于这些新集合，定义了nano ∧β-连续函数，并推导出了一些涉及其表征的结果。此外，还介绍了nano ∨β-closure 和nano ∧β-interior 的概念。