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.
