Sequential properties of function spaces with the compact-open topology

TL;DR

This paper investigates the sequential properties of function spaces with the compact-open topology, revealing conditions under which these spaces are sequential and characterizing their topological structure.

Contribution

It establishes new criteria for the sequentiality of C_k(X,2) spaces and describes their topological structure for various classes of spaces X.

Findings

C_k(M,2) is sequential and contains a closed Arens space S_2

C_k(X,2) is sequential iff X is locally compact or its derived set is compact

All C_k(X,2) spaces for certain non-locally compact Polish X are homeomorphic

Abstract

Let M be the countably infinite metric fan. We show that C_k(M,2) is sequential and contains a closed copy of Arens space S_2. It follows that if X is metrizable but not locally compact, then C_k(X) contains a closed copy of S_2, and hence does not have the property AP. We also show that, for any zero-dimensional Polish space X, C_k(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 C_k(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.