Corollary 1

For every family (Y_λ)_{λ ∈ L} of closed subsets of X, J(∩_{λ ∈ L} Y_λ) is the radical of the sum of the ideals J(Y_λ).

It follows from Proposition 11 (iii) that J(∩_{λ ∈ L} Y_λ) is the smallest ideal which is equal to its radical and contains all the J(Y_λ); this ideal then contains ∑_{λ ∈ L} J(Y_λ) and therefore also the radical of ∑_{λ ∈ L} J(Y_λ) (§ 2, no. 6, Corollary 2 to Proposition 13), whence the corollary.