The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm parallel to.parallel to(infinity). The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.
Also we consider continuous approximate roots of the equation z" - f = 0 with respect to z, where f is an element of C(X), and provide a topological characterization of compact Hausdorff space X with dim X <= 1 such that the above equation has an approximate root in C(X) for each f is an element of C(X), in terms of the first tech cohomology of X. (C) 2006 Elsevier B.V. All rights reserved.