aspg-21-3553-1791468663 20261008141103 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 25 4 Elvis Aponte 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 Natural Sciences and Mathematics, Escuela Superior Polit´ecnica del Litoral, ESPOL, Campus Gustavo Galindo, km. 30.5 v´ıa Perimetral, Guayaquil, Ecuador Jos´e Sanabria Department of Mathematics, Faculty of Education and Sciences, Universidad de Sucre, Sincelejo, Colombia Ennis Rosas Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla, Colombia 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 425 432 3553 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.250436 https://www.americaspg.com/journal/21/article/3553 https://www.americaspg.com/storage/71738596356.pdf