Continuity and separation axioms via infra-topological spaces

In order to investigate a particular topic in mathematics, more specifically, general topology, it is always desirable to find a weaker condition. This work is planned to study a weak (topological) structure named infra-topological space. An infra-topological space is the collection of subsets of a universe that includes the empty set and is closed under finite intersections. The continuity, openness, and homeomorphism of mappings between infra-topological spaces are explored. Through the use of some examples, analogous properties and characterizations of ordinary mappings cannot be hopped on infra-topological structures. Then, the concepts of product and coproduct of infra-topological spaces are analyzed. Furthermore, the notion of infra-quotient topologies, which are inspired by infra-continuity, is introduced. The essential properties indicate that infra-quotient topologies and ordinary quotient topologies act in parallel. The final part of this paper is devoted to the investigation of infra separation axioms (infra T_i-spaces, i = 0, 1, …, 4). The behaviour of ordinary separation axioms cannot be translated to an infra-topological structure. More precisely, infra-T₃ and infra-T₄-spaces are independent, and singletons need not be infra-closed in infra-T₁-spaces.