International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/3003 2020 2020 Department of Mathematics & Statistics, Mutah University, Al-Karak, Jordan admin admin The objective of this paper is to introduce for the first time the concept of two-fold neutrosophic and fuzzy topological space defined over real numbers, where we combine the two-fold neutrosophic sets with real numbers to get a novel topological space based on them. Also, we present many of its elementary properties and special subsets such as two-fold neutrosophic open sets, two-fold neutrosophic closed sets, and two-fold neutrosophic closure. Many examples and theorems will be provided to clarify the validity of our approach. 2025 2025 418 427 10.54216/IJNS.250137 https://www.americaspg.com/articleinfo/21/show/3003