A space is Frechet–Urysohn if each point in the closure of a set is a limit of a sequence from the set. A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. Applying methods of selection principles, we solve the following problem.

 

Problem (Gartside—Lo—Marsh [1]). Is there, consistently, a Tychonoff space X of cardinality at least p such the space Cp(X) is strongly sequentially separable but not Frechet–Urysohn?

 

This is a joint work with Alexander V. Osipov and Boaz Tsaban.

 

References

 

[1] P. Gartside, J. Lo, A. Marsh, Sequential density, Topology and its Applications 130 (2003), 75–86. 

[2] A. Osipov, P. Szewczak, B. Tsaban, Strongly sequentially separable function spaces, via selection principles, Topology and its Applications 270 (2020), 107048.