On jakovlev spaces

A topological space X is called weakly first countable, if for every point x there is a countable family {C_n^x | n ∈ w} such that x ∈ C_n+1^x ⊆ C_nⁿ and such that U ⊂ X is open iff for each x ∈ U some C_n^x is contained in U. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b = c, and also by forcing.