Connectedness and covering properties via infra topologies with application to fixed point theorem

A new generalization of classical topology, namely infra topology was introduced. The importance of studying this structure comes from two matters, first preserving topological properties under a weaker condition than topology, and second, the possibility of applying infra-interior and infraclosure operators to study rough-set concepts. Herein, we familiarize new concepts in this structure and establish their master properties. First, we introduce the notions of infra-connected and locally infra-connected spaces. Among some of the results we obtained, the finite product of infra-connected spaces is infra-connected, and the property of being a locally infra-connected space is an infra-open hereditary property. We successfully describe an infra-connected space using infra-open sets, which helps to study concepts given in this section under certain functions. Then, we determine the condition under which the number of infra-components is finite or countable. Second, we define the concepts of infra-compact and infra-Lindelöf spaces and study some of their basic properties. With the help of a counterexample, we elucidate that the infra-compact subset of an infra-T₂ space is not infra-closed, in general. We end this work by one of the interesting topics in mathematics “fixed point theorem”, we show that when the infra-continuous function defined on an infra-compact space has a unique fixed point. To elucidate the topological properties that are invalid in the frame of infra topology, we provide some counterexamples.