  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.
