Sequential Properties of Function Spaces with the Compact-open Topology

Let M be the countably infinite metric fan. We show that Ck(M, 2) is sequential and contains a closed copy of Arens space S2. It follows that if X is metrizable but not locally compact, then Ck(X) contains a closed copy of S2, and hence does not have the property AP. We also show that, for any zero-dimensional Polish space X, Ck(X, 2) is sequential if and only if X is either locally compact or the derived set X′ is compact. In the case that X is a non-locally compact Polish space whose derived set is compact, we show that all spaces Ck(X, 2) are homeomorphic, having the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n + 1)st.