A countable Frechet-Urysohn space of uncountable character

摘要

We shall construct a countable Frechet-Urysohn α_4 not α_3 space X such that all finite powers of X are Frechet-Urysohn. It is well known that Frechet-Urysohn property is not productive. Arhangel'skii's α_i classification says that with increasing i the productivity of Frechet-Urysohn property gets worse, and especially, Frechet-Urysohn α_4-spaces behave very badly: They have a Frechet-Urysohn product with a convergent sequence (or with a 1st countable space), but the product two Frechet-Urysohn α_4-spaces may fail to be Frechet-Urysohn even in the presence of compactness. This perhaps explains why we tried to find the simplest possible example, a countable space with just one non-isolated point, which is Frechet-Urysohn in all finite powers, consequently α_4, and which still fails to be α_3.