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Abstract: We generalise some results of R. E. Stong concerning finite spaces to widersubclasses of Alexandroff spaces. These include theorems on function spaces,cores and homotopy type. In particular, we characterize pairs of spaces X,Ysuch that the compact-open topology on CX,Y is Alexandroff, introduce theclasses of finite-paths and bounded-paths spaces and show that everybounded-paths space and every countable finite-paths space has a core as itsstrong deformation retract. Moreover, two bounded-paths or countablefinite-paths spaces are homotopy equivalent if and only if their cores arehomeomorphic. Some results are proved concerning cores and homotopy type oflocally finite spaces and spaces of height 1. We also discuss a mistake foundin an article of F.G. Arenas on Alexandroff spaces.It is noted that some theorems of G. Minian and J. Barmak concerning the weakhomotopy type of finite spaces and the results of R. E. Stong on finiteH-spaces and maps from compact polyhedrons to finite spaces do hold for widerclasses of Alexandroff spaces.Since the category of T 0 Alexandroff spaces is equivalent to the category ofposets, our results may lead to a deeper understanding of the notion of a coreof an infinite poset.

Autor: Michał Kukieła

Fuente: https://arxiv.org/

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