A topological space X is called weakly first countable if there exists a function V:X→N→P(X) such that: (i) for every x∈X and n∈N, we have x∈V(x,n); (ii) for every x∈X and n∈N, V(x,n+1)⊆V(x,n); and (iii) a subset O⊆X is open if and only if for every x∈O there exists k∈N such that V(x,k)⊆O. A topological space is first countable if every point has a countable neighborhood basis. Does there exist in ZFC a weakly first countable compact Hausdorff space that is not first countable?