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In this paper, we introduce the concept of quasi-coincidences in MBJ-neutrosophic sets and obtain some of its properties. Also by defining the pre-image and the image of an MBJ-neutrosophic set under a mapping, we confirm that their properties are naturally extensions of the classical case. Next, we define two types of MBJ-neutrosophic neighborhood system and discuss their various properties and introduce the notion of ◦-[resp. ∗-]MBJ-neutrosophic bases and deal with some of their properties. Finally, we define ◦-CI , ∗-CI , ◦-CII and ∗-CII corresponding to the first countability and the second countability in classical topological spaces, and we obtain the relationships between them.