On homotopy types of Alexandroff spaces

TL;DR

This paper extends known results about finite Alexandroff spaces to broader classes, characterizing homotopy types, cores, and function spaces, and clarifies the relationship between bounded-paths spaces and their cores.

Contribution

It introduces new classes of Alexandroff spaces, such as finite-paths and bounded-paths spaces, and establishes their homotopy equivalences and core properties, generalizing prior finite space results.

Findings

Bounded-paths and countable finite-paths spaces have cores as strong deformation retracts.

Homotopy equivalence of bounded-paths spaces is characterized by homeomorphic cores.

The compact-open topology on C(X,Y) is Alexandroff for certain space pairs.

Abstract

We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. These include theorems on function spaces, cores and homotopy type. In particular, we characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, introduce the classes of finite-paths and bounded-paths spaces and show that every bounded-paths space and every countable finite-paths space has a core as its strong deformation retract. Moreover, two bounded-paths or countable finite-paths spaces are homotopy equivalent if and only if their cores are homeomorphic. Some results are proved concerning cores and homotopy type of locally finite spaces and spaces of height 1. We also discuss a mistake found in an article of F.G. Arenas on Alexandroff spaces. It is noted that some theorems of G. Minian and J. Barmak concerning the weak homotopy type of finite spaces and the results of R. E. Stong on finite H-spaces and maps from compact polyhedrons to finite spaces do hold for wider classes of Alexandroff spaces. Since the category of T_0 Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.