On the root closedness of continuous function algebras

For a compact Hausdorff space X, C ( X ) denotes the algebra of all complex-valued continuous functions on X. For a positive integer n, we say that C ( X ) is n-th root closed if, for each f ∈ C ( X ) , there exists g ∈ C ( X ) such that f = g n . It is shown that, for each integer m ⩾ 2 , there exists a compact Hausdorff space X m such that C ( X m ) is m-th root closed, but not n-th root closed for each integer n relatively prime to m. This answers a question posed by Countryman Jr. [R.S. Countryman Jr., On the characterization of compact Hausdorff X for which C ( X ) is algebraically closed, Pacific J. Math. 20 (1967) 433–438] et al.