On The Root Closedness Of Continuous Function Algebras

摘要

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.