Sequential properties of function spaces with the compact-open topology

The main results of the paper are: (1) If X is metrizable but not locally compact topological space, then C-k(X) contains a closed copy of S-2, and hence does not have the property AP; (2) For any zero-dimensional Polish X, the space C-k(X, 2) is sequential if and only if X is either locally compact or the derived set X' is compact; and (3) All spaces of the form C-k(X, 2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n + 1)st. (C) 2010 Elsevier B.V. All rights reserved.