The proper homotopy of Alexandroff spaces

Abstract

This paper introduces the study of the infinity in the class of Alexandroff spaces by establishing the proper category of locally finite Alexandroff spaces. We observe that the McCord and Clader theorems are available in this setting. The paper can be considered as the starting point of a seemingly new research: relating the combinatorics and topology of the proper category of Alexandroff spaces to the proper invariants used in the geometric topology of non-compact polyhedra and manifolds.