2010/05/30 by Gary Gruenhage, Gruenhage, Gary, Boaz Tsaban +3
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Mathematical and Theoretical Analysis #math.GN
paper · pdf · doi:10.48550/arxiv.1005.5542
Comments are welcome
arxiv created 2010/05/30 · openalex publication_date 2010/05/30 · arxiv updated 2010/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 n-th one nowhere dense in the (n+1)-st.