Extension of continuity of maps between axiomatic locally finite spaces

The paper studies the extension problem of continuous maps between axiomatic locally finite spaces (for short, ALF spaces). Indeed, an ALF space is a topological space satisfying a set of axioms suggested in . Further, an ALF space is defined by using a special kind of neighborhood different from the topological neighborhood in classical topology so that the continuity of maps between ALF spaces can be defined by preserving the neighborhood relation (see Definition 10). Therefore, it is necessary to develop the notions of continuity, homeomorphism and local homeomorphism for such spaces by using the neighborhood relation, which can be applicable in computer science. In the study of a deformation of an ALF space, we can develop a special kind of retract on ALF spaces. By using the retract, we can efficiently deal with the extension of continuous maps between ALF spaces