Obtaining a weaker condition that preserves some inspired topological properties is always desirable. As a result, we introduce the concept of infra-fuzzy topology, which is a subset family that degrades the concept of fuzzy topology by omitting the condition of closedness under arbitrary unions. Fundamental properties of infra-fuzzy topological spaces are investigated, including infra-fuzzy open and infra-fuzzy closed sets, infra-fuzzy interior and infra-fuzzy closure operators, and the infra-fuzzy boundary of a fuzzy set. It is not possible to expect the latter concepts to have properties identical to those in ordinary fuzzy topological spaces. More precisely, the infra-fuzzy interior of a set need not be infra-fuzzy open, and the infra-fuzzy closure and boundary of a set may not be infra-fuzzy closed. Then, employing infra-fuzzy neighborhood systems, infra-fuzzy Q-neighborhood systems, the basis of infra-fuzzy topology, and infra-fuzzy relative topology, we propose several approaches for generating infra-fuzzy topologies. Finally, we define the notions of continuity, openness, closedness, and homeomorphism of mappings in the context of infra fuzziness and investigate some of their properties and characterizations. We show that the usual characterization of earlier notions in the infra-fuzzy structure is incorrect. We demonstrate that the family of all infra-fuzzy homeomorphisms on an infra-fuzzy topological space forms a group under mappings composition. We finish this work by proving that each infra-fuzzy homeomorphism between two infra-fuzzy topological spaces produces an isomorphism on groups of infra-fuzzy homeomorphisms of the corresponding spaces.

<abstract><p>This paper begins with an introduction to some soft topological operators that will be used to characterize several soft separation axioms followed by their main properties. Then, we define a new soft separation axiom called "soft $ T_D $-space" and analyze its main properties. We also show that this space precisely lies between soft $ T_0 $ and soft $ T_1 $-spaces. Finally, we characterize soft $ T_i $-spaces, for $ i = 0, 1, D $, in terms of the stated operators.</p></abstract>

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology.

In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft T 1 , while a maximal soft connected topology is soft T 0 . Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal.

The paper studies relations between types of generalized closed sets in topological spaces. It also answers an open question posed by Erdal.