Quasi-metrizability

Quasi-metrizability of a topological space X is equivalent to the availability on X of a decreasing neighbourhood base 〈(x)n〉 at every x ϵ X, so constituted that, for every countable and relatively locally finiteA ⊂ X and n ϵ ω (writing, for each B ⊂ X and each m ϵ ω, (B)m for ∪{(x)m: x ϵ B}), we have ((A)v)v ⊂ (A)n for some v ϵ ω (dependent on A and n). By comparison, metrizability of T0-spaces, X is equivalent to the availability on X of a decreasing sequence 〈(x)n〉 of neighbourhoods at every x ϵ X, so constituted that, for every A ⊂ X, we have ∩{(A)n: n ϵ ω} = Cl A = ∩{Cl(A)n: n ϵ ω} (Hung, 1977).