The main results of the paper are:. (1)If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S2, and hence does not have the property AP;(2)For any zero-dimensional Polish X, the space Ck(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 Ck(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.
Bibliographical noteFunding Information:
✩ The third named author acknowledges the support of the FWF grant P19898-N18. We thank the referees for comments and suggestions.
- Arens space
- Compact-open topology
- Polish space
- Pytkeev property
- Sequential fan