Sequential properties of function spaces with the compact-open topology

The main results of the paper are:(1) s 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; y 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 aces 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.