EN
Let 𝐾 be a family of compact sets in a Banach algebra A such that 𝐾 is stable with respect to finite unions and contains all finite sets. Then the sets $U(K) := {I ∈ Id(A): I ∩ K = ∅}$, K ∈ 𝐾 define a topology τ(𝐾) on the space Id(A) of closed two-sided ideals of A. 𝐾 is called normal if $I_i → I$ in (Id(A),τ(𝐾)) and x ∈ A╲I imply $lim inf_i∥x + I_i∥ > 0$. (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then Id(A) is second countable. (2) If the family of countable compact sets is normal and A is separable then there is a countable subset S ⊂ A such that for all closed two-sided ideals I we have $\overline{I ∩ S} = I$. Examples are separable C*-algebras, the convolution algebras $L^p(G)$ where 1 ≤ p < ∞ and G is a metrizable compact group, and others; but not all separable Banach algebras share this property.