Extension of continuity of maps between axiomatic locally finite spaces

Abstract

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 Kovalevsky [Axiomatic digital topology, J. Math. Imag. Vis. 26 (2006), pp. 41–58; Geometry of Locally Finite Spaces, Monograph, Berlin, 2008]. Further, an ALF space is defined by using a special kind of neighbourhood different from the topological neighbourhood in classical topology so that the continuity of maps between ALF spaces can be defined by preserving the neighbourhood relation (see Definition 10). Therefore, it is necessary to develop the notions of continuity, homeomorphism and local homeomorphism for such spaces by using the neighbourhood 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.

Keywords:

• locally finite space
• axiomatic locally finite (ALF) space
• smallest open neighbourhood
• frontier
• neighbourhood relation
• homeomorphism
• local homeomorphism
• combinatorial homeomorphism
• abstract cell complex
• extension problem
• retract

2000 AMS Subject Classifications :

• 54C20
• 54C08
• 57Q05
• 26A15

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010002325). This paper was supported by the selection of research-oriented professor of Chonbuk National University in 2011.