International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/3553 2020 2020 Department of Mathematics, Faculty of Natural Sciences and Mathematics, Escuela Superior Polit´ecnica del Litoral, ESPOL, Campus Gustavo Galindo, km. 30.5 v´ıa Perimetral, Guayaquil, Ecuador Jos Jos´e Department of Mathematics, Faculty of Natural Sciences and Mathematics, Escuela Superior Polit´ecnica del Litoral, ESPOL, Campus Gustavo Galindo, km. 30.5 v´ıa Perimetral, Guayaquil, Ecuador Jorge Vielma Department of Mathematics, Faculty of Education and Sciences, Universidad de Sucre, Sincelejo, Colombia Jos´e Sanabria Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla, Colombia Ennis Rosas In this paper, we employ the classical topological preorder to introduce the concept of topologically bounded sets, in order to relate it to the Collatz conjecture problem. In addition, this preorder allows us to derive some results about topologically convex sets, showing that these form a convex structure. Finally, using this topological preorder, we define the neutrosophic continuous sets and establish the necessary conditions to identify the points that are connected to these sets, which form a topological convex set. 2025 2025 425 432 10.54216/IJNS.250436 https://www.americaspg.com/articleinfo/21/show/3553