Products of compact Fréchet spaces (Q1089952)

A space is a Fréchet space if each accumulation point of a subset is the limit of some sequence from the subset. Using a technique by Gruenhage, the author constructs (under MA), for each \(n\geq 2\), a compact Fréchet space X such that \(X^ n\) is a Fréchet space but \(X^{n+1}\) is not a Fréchet space. This answers a question posed by Nogura (a generalization of a question originally asked by Michael in 1972, and answered by several people under different set-theoretic assumptions). The construction uses a decomposition of \(\beta\) N.