Vol. 20, No. 3, 1967

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On the characterization of compact Hausdorff X for which C(X) is algebraically closed

Roger Countryman

Vol. 20 (1967), No. 3, 433–448

Although the problem considered here has its origins in Functional Analysis, the viewpoint and methods of this paper are purely topological. The problem is to give a completely topological characterization of those compact Hausdorff spaces X for which the algebra C(X) of all complex-valued continuous functions on X is algebraically closed, i.e. for which each polynomial over C(X), whose leading coefficient is constant, has a root in C(X).

A necessary condition in order that C(X) be algebraically closed is obtained here and it is proven that, in the presence of first countability, the condition is also sufficient. The necessary condition requires that X be hereditarily unicoherent and that each discrete sequence of continua in X have a degenerate or empty topological limit inferior. A rather general sufficient condition is also proved. It states essentially that each component of X have an algebraically closed function algebra and that each point of X be of finite order in the sense of Whyburn.

Mathematical Subject Classification
Primary: 46.25
Secondary: 54.60
Received: 20 June 1965
Revised: 9 February 1966
Published: 1 March 1967
Roger Countryman