mathematics Article Connectedness and Local Connectedness on Infra Soft Topological Spaces Tareq M. Al-shami 1, * and El-Sayed A. Abo-Tabl 2,3   Citation: Al-shami, T.M.; Abo-Tabl, E.-S.A. Connectedness and Local Connectedness on Infra Soft Topological Spaces. Mathematics 2021, 9, 1759. https://doi.org/10.3390/ math9151759 Academic Editor: Francisco Gallego Lupia ˇ nez Received: 29 June 2021 Accepted: 14 July 2021 Published: 26 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Departmentof Mathematics, Sana’a University, Sana’a P.O. Box 1247, Yemen 2 Department of Mathematics, College of Arts and Science, Methnab, Qassim University, Buridah 51931, Saudi Arabia; a.adotabl@qu.edu.sa 3 Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt * Correspondence: t.alshami@su.edu.ye Abstract: This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paperto presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components. Keywords: separated soft sets; infra soft connected space; infra soft locally connected space; compo- nent; infra soft topology 1. Introduction Nowadays, researchers deal with the complexities of modeling vague/uncertain problems in different fields such as engineering, economics, medical science, computer science and sociology on a daily basis. Since classical methods are not always successful in addressing these types of problems, some novel approaches were proposed such as fuzzy set, rough set and soft set. Soft set, the core idea of this manuscript, was introduced by Molodtsov [1] in 1999. Since then, researchers have applied soft sets to different areas including decision-making problems [2], medical science [3–6] and computer science [7]. Then, Maji with their co-authors [8] put forward the basic concepts of soft set theory. They defined the operators of the intersection, union and difference between two soft sets and a complement of a soft set. To remove the shortcomings observed in Maji et al.’s work, some researchers and scholars reformulated some of their operators and presented some new types. For example, Ali et al. [9] provided new kinds of these operators in a way that enabled keeping some of the properties and results of crisp set theory in soft set theory. Other contributions have been made to define several types of soft equality relations such as those given in [10,11]. In 2011, Ça ˇ gman et al. [12] and Shabir and Naz [13] hybridized soft sets and A general topology to initiate the concept of soft topological spaces. They followed different methods to define a soft topology. Ça ˇ gman et al. defined a soft topology over an absolute soft set and different sets of parameters. However, Shabir and Naz formulated a soft topology over fixed sets of the universe and parameters. In this article, we followed the definition of Shabir and Naz. Al-shami and Koˇ cinac [14] discussed the interchangeable properties between general topology and extended soft topology. Alcantud [15] constructed soft Mathematics 2021, 9, 1759. https://doi.org/10.3390/math9151759 https://www.mdpi.com/journal/mathematics