Notes on Fréchet spaces (Q1587830)

Summary: First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions on Fréchet spaces using these structures. Second, we give sufficient conditions, for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we give a condition, sufficient for a sequential space to be Fréchet, for a weakly first countable space to be first countable, and for a symmetrizable space to be semi-metrizable.