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 semiopen, 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.